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+{"text":"\\section{Introduction}\n\nThis paper is part of a study of certain  $C^*$-algebras which can be associated to a\nhyperbolic homeomorphism of a compact space, $(X,f)$.  They are  called  the the stable and unstable\nRuelle algebras, $\\mathcal{R}^s$ and $\\mathcal{R}^u$, and are higher dimensional generalizations of\nCuntz-Krieger algebras.  This means that if the dimension of $X$ is\nzero, then $\\mathcal{R}^u \\cong O_{A^T} \\otimes \\mathcal{K}$ and $\\mathcal{R}^s \\cong O_A \\otimes \\mathcal{K}$.  One of the basic\nresults of the theory is a duality relation between $\\mathcal{R}^u$ and $\\mathcal{R}^s$.\nIn the present paper we prove this explicitely in the zero dimensional\ncase.  Our reason for doing this is to bring out the use of Fock space to construct\nthe K-theory class implementing the duality in the zero dimensional case.      \n\n\n\n\n\nThe notion of Spanier-Whitehead duality in topology has a very natural\ngeneralization to K-theory of $C^*$-algebras.  Briefly, it says that\ntwo algebras, $A$ and $B$, are dual if there are duality classes\n$\\Delta \\in KK^i(A \\otimes B, \\mathbb{C})$ and $\\delta \\in KK^i(\\mathbb{C}, A \\otimes \\mathcal{B})$ which\ninduce an isomorphism between the K-theory of $A$ and the K-homology of\n$B$ via Kasparov product.  It is closely related to the notion of\nPoincar\\'e duality used by Connes in his study of the standard model\nof particle physics, ~\\cite{connes:book} .  We will describe it\nin more detail in Section 2.  A useful result proved in the paper\nis a criterion, presented in Section 3, for deciding when one has a duality\nbetween two algebras.  It is applicable when two duality classes such\nas $\\Delta$ and $\\delta$ are given and one wants to show that they induce \nduality isomorphisms.  Section 4 and Section 5 apply this criterion to\nthe case of two algebras associated to a hyperbolic dynamical system.\nIf $A$ is an $n \\times n$ aperiodic matrix then one can associate to it\nthe subshift of finite type, $(\\Sigma_A, \\sigma_A)$.  There are two\n$C^*$-algebras that can be constructed from this data--the\nCuntz-Krieger algebras $O_A$ and $O_{A^T}$.  We show that these algebras\nare dual.  In Section 6 we will discuss some further applications and\nmake some concluding remarks. \n\nIn a later paper we will establish duality for the stable and\nunstable Ruelle algebras associated to any hyperbolic homeomorphism of a\ncompact space, (a Smale space).  Ruelle algebras were introduced by\nthe second author in ~\\cite{ putnam}.  They can be thought of as\nhigher dimensional generalizations of Cuntz-Krieger algebras.  They\nare constructed by defining two equivalence relations on the Smale\nspace, stable and unstable equivalence.  One takes the $C^*$-algebras\nassociated to them and then takes the crossed products by the\nautomorphism induced by the homeomorphism.\n\n\nThe stable and unstable equivalence classes in a Smale space behave\nvery much like transverse foliations.  Because of that, and the fact\nthat the homeomorphism is contracting along the stable leaves, one\nobtains a duality in K-theory for the algebras.\n\nCuntz-Krieger algebras are special cases of Ruelle algebras, so the\nduality established here would follow from the more general theory.\nHowever, there is an intriguing aspect to this which as yet has no\nanalogue in the general case. Namely, the duality classes have\nrepresentatives constructed using  Fock space.  These classes are obtained\nin a natural manner following work of D. Evans, ~\\cite{ evans} and\nD. Voiculescu, ~\\cite{ voiculescu}.  This provides potential\nconnections with physics (c.f. ~\\cite{ jorgensen, dykema-n, faddeev}) and Voiculescu's work on free products which we\nhope to pursue in the future.  We would like to thank Dan Voiculescu\nand Marius Dadarlat for very helpful conversations.\n\nIt should be noted that the general duality theory for Smale spaces\nrequires a different approach which is based on the notion of\nasymptotic morphism.  The final version of the duality theorem uses\nthese methods, ~\\cite{ kaminker-p2}.\n\n\\section{K-theoretic duality for $C^*$-algebras}\n\nIn this paper we will be describing an example of some $C^*$-algebras\nthat are dual with respect to K-theory.  This notion of duality has\nappeared several times in the past, ~\\cite{kasparov:invent,kahn-k-s,parker} and\nrecently was used by Connes ~\\cite{connes:book}.  We present the\ndefinitions here and list some basic facts.  More details can be found\nin ~\\cite{ kaminker-p1}.  \n\nWe will use the following conventions.  Let $\\mathcal{S}$ denote $C_0(\\mathbb{R})$.\nThen $KK^1(A,B)$ will be, by definition, equal to $KK(\\mathcal{S} \\otimes A, B)$.\nFor $A$ and $B$ separable, and $A$ nuclear one has that $KK^1(A,B)\n\\cong Ext(A,B)$.\nWe establish some additional notation.  If $\\sigma$ is a permutation, and $A_1, \\ldots\nA_n$ are algebras, then we will also use $\\sigma$\n to denote the isomorphism\n\\begin{equation*} \nA_1 \\otimes \\cdots \\otimes A_n \\to A_{\\sigma (1)} \\otimes \\cdots \\otimes A_{\\sigma(n)}.\n\\end{equation*}  \nIf $\\sigma$ is a transposition interchanging $i$ and $j$, we will write\n$\\sigma_{ij}\\colon KK^*( \\cdots \\otimes A_i \\otimes \\cdots \\otimes A_j \\otimes\\cdots , B) \\to\nKK^*(\\cdots \\otimes A_j \\otimes \\cdots \\otimes A_i \\otimes \\cdots, B) $ for the\nhomomorphism  induced by $\\sigma$ on the first variable of the Kasparov groups, \nand $\\sigma^{ij}$ for the corresponding map induced on the second variable.  Let $\\tau_D \\colon KK^i(A,B) \\to KK^i(A \\otimes D, B \\otimes D)$ and\n$\\tau^D \\colon KK^i(A,B) \\to KK^i(D \\otimes A, D \\otimes B)$ denote the standard\nmaps, ~\\cite{ kasp}.  \n\nAlso, we will have need of the following version of Bott\nperiodicity.  Let \n\\begin{equation}\n\\label{toep} \n\\begin{CD}\n0 @>>> \\mathcal{K}(\\ell^2(\\mathbb{N})) @>>> \\mathcal{T} @>{\\sigma_{\\mathcal{T}}}>> C(S^1) @>>> 0\n\\end{CD}\n\\end{equation}\nbe the Toeplitz extension.  We will denote it by $\\mathcal{T} \\in KK^1(C(S^1),\n\\mathbb{C}) $ and its restriction to $\\mathcal{S}$ by $\\mathcal{T}_0 \\in KK(\\mathcal{S} \\otimes \\mathcal{S}, \\mathbb{C})$.\nLet $\\beta \\in KK(\\mathbb{C}, \\mathcal{S} \\otimes \\mathcal{S})$ be the Bott element.\nThen the following holds. (c.f. ~\\cite{ blackadar}) \n\\begin{theorem} One has\n$\\beta \\otimes_{\\mathcal{S} \\otimes \\mathcal{S}} \\mathcal{T}_0 = 1_{\\mathbb{C}}$ and $\\mathcal{T}_0\n\\otimes \\beta = 1_{\\mathcal{S} \\otimes \\mathcal{S}}$, .\n\\end{theorem}\n\nWe describe next the notion of duality we will be using.\n\\begin{definition}\nLet $A$ and $B$ be  $C^{*}$-algebras.  Suppose that, for $n\n= 0$ or $1$, two classes,\n$\\Delta \\in KK^n(A \\otimes B, \\mathbb{C})$ and $\\delta \\in KK^n(\\mathbb{C}, A \\otimes B)$ ,  are given.\nDefine homomorphisms $\\Delta_i \\colon K_i(A) \\to\nK^{i+n}(B)$ and $\\delta_i \\colon K^{i+n}(B) \\to K_i(A)$ in the  following way.\nIn $n = 1$ set\n\\begin{equation} \n   \\Delta_i(x) = \n\\begin{cases}\nx \\otimes_{A}\\Delta&  \\text{if $ i = 0$},\\\\\n\\beta \\otimes_{\\mathcal{S} \\otimes \\mathcal{S}} ( \\sigma_{12}(x \\otimes_{A} \\Delta))&  \\text{if $i = 1$} \n\\end{cases}\n\\end{equation}\nand let\n\\begin{equation} \n\\delta_i(y) =\n\\begin{cases}\n\\beta \\otimes_{\\mathcal{S} \\otimes \\mathcal{S}} (\\delta \\otimes_{B} y)&  \\text{if $i = 0$}, \\\\\n\\delta \\otimes_{B} y&  \\text{if $ i = 1$},\n\\end{cases}\n\\end{equation}\n\nIf $n = 0$ set\n\\begin{equation} \n   \\Delta_i(x) = \n\\begin{cases}\nx \\otimes_{A}\\Delta&  \\text{if $ i = 0$},\\\\\n \\sigma_{12}(x \\otimes_{A} \\Delta)&  \\text{if $i = 1$} \n\\end{cases}\n\\end{equation}\nand let\n\\begin{equation} \n\\delta_i(y) =\n\\begin{cases}\n\\delta \\otimes_{B} y&  \\text{if $i = 0$}, \\\\\n\\beta \\otimes_{\\mathcal{S} \\otimes \\mathcal{S}} (\\delta \\otimes_{B} y)&  \\text{if $ i = 1$},\n\\end{cases}\n\\end{equation}\n\nWe say that $A$ and $B$ are\ndual if  \n\\begin{equation*} \n\\Delta_i :K_i(A) \\to K^{i+n}(B)\n\\end{equation*}\nand\n\\begin{equation*} \n\\delta_i :K^{i+n}(B) \\to K_i(A)\n\\end{equation*}\nare inverse isomorphisms.  Given $A$, if such an algebra $B$ exists it is called a dual of $A$\nand it is denoted $\\EuScript{D} A$.    \n\\end{definition}\nIn this generality a dual is not unique, so care must taken with the\nnotation $\\EuScript{D} A$.  We will only use it if a specific dual is in hand.\nHowever, it is easy to see that a dual is unique up to\nKK-equivalence.  Indeed, $\\sigma^{12}(\\delta ') \\otimes_{A} \\Delta \\in\nKK(B',B)$ and $\\sigma^{12}(\\delta) \\otimes_{A} \\Delta ' \\in KK(B,B')$ yield\nthe required KK-equivalence.\n\nThe form of the definition of the homomorphisms $\\Delta_{*}$ and\n$\\delta_{*}$ is forced by our convention that $KK^1(A,B) = KK(A \\otimes\n\\mathcal{S}, B)$.  It is an interesting point that when dealing with an odd\ntype duality one must bring in some form of Bott periodicity\nexplicitely.  Either one can incorporate it into the definition of the\nhomomorphisms as we have done, or one can modify the definitions of\nthe K-theory groups.  As the reader will see, our choice is the most convenient one for\nthe proofs we are giving.  Note also that we are working only in the\nodd case, (i.e. $n=1$), in this paper.\n\n\n\nFor a specific algebra $A$ it is not clear\nwhether a dual, $\\EuScript{D} A$, exists.  In general, the existence of $\\EuScript{D} A$\nwith prescribed properties, such as separability, is a strong\ncondition.  If one can take $\\EuScript{D} A$ equal to $A$ then this\nagrees with what Connes has developed as Poincar\\'e duality in\n~\\cite{connes:book}.  \n\n\nIf one requires only the existence of $\\Delta$ and the fact that it\nyields an isomorphism in the definition above, then there is no\nguarantee that a class $\\delta$ exists to give the inverse\nisomorphism.  If $A$ was $C(X)$, with $X$ a finite complex, then the\nexistence of $\\delta$ would follow from that of $\\Delta$.  However, in\ngeneral this need not hold.\n\n\nThe origin of this notion is in Spanier-Whitehead duality in topology,\n~\\cite{ spanier}.  Recall that if $X$ is a finite complex then there is a dual\ncomplex, $DX$, along with class $\\Delta\\in H_m(X\\wedge\nDX)$ \nsatisfying that\n$\\backslash \\Delta :H^i(X) \\to H_{m-i}(DX)$\nis an isomorphism.  The space $DX$ is called the\nSpanier-Whitehead dual of $X$.  It is unique up to stable homotopy. If\n$M$ is a closed manifold of dimension $n$ embedded in $\\mathbb{R}^m$, \nthen $DM$ can be taken to be $(\\nu M)^{+}$, the Thom space of \nthe normal bundle of M. It is interesting to note that there is a relation\nbetween Spanier-Whitehead duality, the Thom isomorphism, $\\phi$,  \nand Poincar\\'e duality \n\\begin{equation*} \n\\begin{CD}\nH_{n-i}(M) @<<{\\backslash\\Delta}< H^{i+m-n}((\\nu M)^+)\\\\\n@A{\\cap [M]}AA@AA{\\phi}A\\\\\nH^{i}(M)@>>>H^{i}(M),\n\\end{CD}\n\\end{equation*}\nwhere $[M] = U \\backslash \\Delta$, $U$ the Thom class.  Of course,\n$\\EuScript{D} (C(X)) = C(D(X))$ for $X$ a finite complex.\n\nIf one works in the class $\\mathcal{N}$\nintroduced by Rosenberg and Schochet in their study of the Universal\nCoefficient Theorem, ~\\cite{ rosenberg-s}, the theory simplifies and\nthere is a strong analogy with the commutative case.  (However, in general, the\nrestriction that the algebras lie in $\\mathcal{N}$ is too strong.  In several\nimportant examples this does not hold.)  Recall that $ \\mathcal{N}$ is defined to be\nthe smallest class of separable, nuclear $C^*$-algebras  containing $\\mathbb{C}$ and\nclosed under forming extensions, direct limits, and KK-equivalence.\nThe Universal Coefficient Theorem for KK-theory holds for $KK(A,B)$ if\n$B$ is separable and $A \\in \\mathcal{N}$. Let $\\EuScript{D}\\mathcal{N}$ be the subclass of $\\mathcal{N}$\nconsisting of algebras $A$ in $\\mathcal{N}$ for which a dual $\\EuScript{D} A$ exists and\nis also in $\\mathcal{N}$.  For algebras in $\\EuScript{D}\\mathcal{N}$ the following facts are easy\nconsequences of the properties of the Kasparov product and the\nUniversal Coefficient Theorem.\n\n\\begin{enumerate}[i)]\n\\item If $A$ is dual to $B$, then $B$ is dual to $A$.\n\\item If $A \\in \\EuScript{D}\\mathcal{N}$, then  $\\EuScript{D}(\\EuScript{D} A)$ is KK-equivalent to $A$.\n\\item If $A \\in \\mathcal{N}$, then $A \\in \\EuScript{D}\\mathcal{N}$ if and only if $K_*(A)$ is\n  finitely generated.\n\\item Let $E, D \\in \\mathcal{N}$ and $A \\in \\EuScript{D}\\mathcal{N}$.  Then\n\\begin{equation*}\n\\Delta_* \\colon KK^*(E, D \\otimes A) \\to KK^{*+n}(E \\otimes \\EuScript{D} A, D)\n\\end{equation*}\nand \n\\begin{equation*} \n\\delta_* \\colon KK^*(E \\otimes \\EuScript{D} A, D) \\to KK^{*+n}(E, D \\otimes A)\n\\end{equation*}\nare inverse isomorphisms.\n\\item If $A$ has a dual, and $A'$ is KK-equivalent to A, then $A'$ has\n  a dual which is KK-equivalent to the dual of $A$.\n\\end{enumerate}\nFor details and further development, see ~\\cite{k-p2}. \nIt is not apparent if an algebra has a dual or not. Indeed, the main\ngoal of this paper is to exhibit an example of a class of algebras\nwith specific types of duals which have a geometric and dynamical\norigin.  However, one can start to build up a class of algebras which\nhave duals in an elementary way.  For example, if $X$ is a finite\ncomplex, then $\\EuScript{D} X$ exists.  If $A \\in \\mathcal{N}$ and $K_*(A)$ is finitely\ngenerated, then $A$ is KK-equivalent to $C(X)$ where $X$ is a finite\ncomplex, and hence $A$ has a dual.  Moreover, Connes has shown that\n$\\mathcal{A}_{\\theta}$ is self-dual for $\\theta$ irrational.  \n\nThe largest subclass of $\\mathcal{N}$ for which $\\EuScript{D}$ is involutive modulo\nKK-equivalence is $\\EuScript{D}\\mathcal{N}$.  This can be compared with a result of M.\nBoardman, ~\\cite{boardman}, which states that the largest category on\nwhich Spanier-Whitehead duality is involutive is the homotopy category\nof finite complexes.  Thus, $\\EuScript{D}\\mathcal{N}$ has a formal similarity with the\nhomotopy category of finite complexes.  This itself does not clarify\nthe issue of which $C^*$-algebras should play the role of\nnon-commutative finite complexes, but it is suggestive.  \nThis will be discussed further in ~\\cite{k-p2}.\n  \n \n\nOne may view duality as being of even or odd type depending on whether\n$\\Delta$ belongs to $KK^n(A \\otimes \\EuScript{D} A,\\mathbb{C})$ for $n$ even or odd.  We\nwill discuss the odd type of duality here. However, in connections to\nthe Novikov Conjecture, ~\\cite{ kaminker-p1}, and physics, ~\\cite{ connes:book}, the even\ntype naturally appears. \n\n\\section{Criterion for duality classes}\n\nIn this section we will present a technical result, Proposition ~\\ref{hyp}, which gives a criterion for when two classes $\\Delta$ and\n$\\delta$ yield duality isomorphisms. This is essentially the same as\nthe condition given by Connes, ~\\cite[p. 588]{connes:book}, except\nthat our duality is in the odd case and this requires adjusting the\narguments for Bott periodicity.  This technicality is actually what\nallows us to obtain the duality isomorphisms in the case of shifts of\nfinite type.  \n\nThus, we shall give useable conditions under which $\\Delta_* \\circ \\delta_* = 1$\nand $\\delta_* \\circ \\Delta_* = 1$.  This breaks into two parts.  The first is an\nuncoupling step and the second is a type of cancellation.  In the\nfollowing sections we apply this to the case of Cuntz-Krieger\nalgebras.\n\nWe will first prove in detail that $\\delta_0 \\Delta_0 = 1_{K_0(A)}$.\nThe statement, $\\delta_1 \\Delta_1 = 1_{K_1(A)}$, follows in a similar\nmanner.  We then sketch the proof that $ \\Delta_0 \\delta_0 =\n1_{K_0(B)}$.  To start with we will perform the uncoupling step.  Let $x\n\\in K_0(A) =  KK(\\mathbb{C},A)$.  Then we have\n\\begin{equation*} \n  \\delta_0 \\Delta_0 (x) = \\beta \\otimes _{\\mathcal{S} \\otimes \\mathcal{S}} (\\delta \\otimes_{B}\n  (x \\otimes_{A} \\Delta)).\n\\end{equation*}\nConsider, first, the factor $(\\delta \\otimes_{B} (x \\otimes_{A}\n\\Delta))$.  We have\n\\begin{align*}\n  (\\delta \\otimes_{B} (x \\otimes_{A} \\Delta)) &= \\tau_{\\mathcal{S}} (\\delta) \\otimes (\\tau^{A}\n  \\tau_{\\mathcal{S}} \\tau_{B} (x) \\otimes \\tau^{A} (\\Delta)) \\\\ &= (\\tau_{\\mathcal{S}}\n  (\\delta) \\otimes (\\tau^{A} \\tau_{\\mathcal{S}} \\tau_{B} (x)) \\otimes \\tau^{A} (\\Delta).\n\\end{align*}\nNow, a direct computation yields that\n\\begin{align*}\n  (\\tau_{\\mathcal{S}} (\\delta) \\otimes (\\tau^{A} \\tau_{\\mathcal{S}} \\tau_{B} (x)) \\otimes \\tau^{A}\n  (\\Delta) &= (\\tau_{\\mathcal{S}} \\tau^{\\mathcal{S}} (x) \\otimes {\\sigma_{12} \\sigma^{24}} \\tau^{A}\n  \\tau^{\\mathcal{S}} (\\delta)) \\otimes \\tau^{A} (\\Delta) \\\\ &= \\tau_{\\mathcal{S}} \\tau^{\\mathcal{S}}\n  (x) \\otimes ({\\sigma_{12} \\sigma^{24}} \\tau^{A} \\tau^{\\mathcal{S}} (\\delta)) \\otimes \\tau^{A}\n  (\\Delta)).\n\\end{align*}\nPutting $\\beta$ back into the product and simplifying, one obtains\n\\begin{equation*} \n  \\beta \\otimes _{\\mathcal{S} \\otimes \\mathcal{S}} (x \\otimes_{A} (\\delta \\otimes_B \\Delta)) = x \\otimes_A\n  (\\beta \\otimes _{\\mathcal{S} \\otimes \\mathcal{S}} (\\delta \\otimes_B \\Delta)).\n\\end{equation*}\nThis accomplishes the uncoupling.\n\n\\begin{prop}\nOne has $\\delta_0 \\Delta_0 (x) = x \\otimes_A (\\beta \\otimes _{\\mathcal{S} \\otimes \\mathcal{S}}\n(\\delta \\otimes_B \\Delta))$.\n\\end{prop}\nWhat one would hope is that\n\\begin{equation} \n\\label{hope}\n\\beta \\otimes _{\\mathcal{S} \\otimes \\mathcal{S}} (\\delta \\otimes_B \\Delta) = 1_A \\in KK(A,A),\n\\end{equation}\nthus yielding\n\\begin{equation*}\n\\delta_0 \\Delta_0 (x) = x.\n\\end{equation*}\nIndeed, if $\\delta \\otimes_B \\Delta = \\tau_A ( \\mathcal{T}_0)$, then $\\beta _{\\mathcal{S} \\otimes\n\\mathcal{S}} \\tau^A (\\mathcal{T}_0) = 1_A$ by Bott periodicity.  However, this need not be the case.  This is because $\\delta$ and\n$\\Delta$ behave like K-theory fundamental classes and may differ by a\nunit from ones which would yield ~\\eqref{hope}.  There is a way to\ncompensate for this which we address next.\n\n\\begin{prop}\n\\label{hyp}\nSuppose that there are automorphisms $\\Theta_A \\colon A \\otimes \\mathcal{S} \\to A \\otimes\n\\mathcal{S}$ and $\\Theta_B \\colon B \\otimes \\mathcal{S} \\to B \\otimes \\mathcal{S}$ such that \n\\begin{align*}\n  (\\Theta_A )_i \\colon K_i(A \\otimes \\mathcal{S}) \\to K_i(A \\otimes \\mathcal{S}) \\\\ (\\Theta_B )_i\n  \\colon K_i(B \\otimes \\mathcal{S}) \\to K_i(B \\otimes \\mathcal{S})\n\\end{align*}\nare the identity map, for $i= 0,1$, and, further,\n\\begin{align}\n\\label{thiii}\n\\sigma_{12}(\\delta \\otimes_B \\sigma_{12}(\\Delta)) &= \\Theta_A \\otimes_{A \\otimes \\mathcal{S}} \\tau^A\n(\\mathcal{T}_0) \\\\\n\\sigma_{12}(\\delta \\otimes_A \\sigma_{12}(\\Delta)) &= \\Theta_B \\otimes_{B \\otimes \\mathcal{S}} \\tau^B\n(\\mathcal{T}_0) \n\\end{align}\nThen,\n\\begin{equation*} \n\\delta_i \\Delta_i \\colon K_i(A) \\to K_i(A)\n\\end{equation*}\nis the identity for $i=0,1$\n\\end{prop}\n\\begin{proof}\nWe will give the proof for $\\delta_0 \\Delta_0$, the other case being\nsimilar.\nCondition ~\\eqref{thiii} states that \n\\begin{equation*} \n\\sigma_{12} \\tau_{\\mathcal{S}} \\tau_{A} (\\delta) \\otimes (\\tau_{A} (\\sigma_{12}(\\Delta))) =\n\\tau^{\\mathcal{S}} (\\Theta_A ) \\otimes \\tau^{A} (\\mathcal{T}_0).\n\\end{equation*}\nThus, one has\n\\begin{equation*} \n\\beta \\otimes _{\\mathcal{S} \\otimes \\mathcal{S}} (\\delta \\otimes_B (\\sigma_{12}(\\Delta))) = \\tau^A (\\beta) \\otimes\n\\tau_{\\mathcal{S}}(\\Theta_A) \\otimes \\tau^A (\\mathcal{T}_0). \n\\end{equation*}\nNow, \n\\begin{align*}\n\\tau^A(\\beta) \\otimes \\tau_{\\mathcal{S}} (\\Theta_A) &= (\\Theta_A)_* (\\tau^A(\\beta)) \\\\\n&= \\tau^A (\\beta),\n\\end{align*}\nso we obtain\n\\begin{align*}\n\\beta \\otimes _{\\mathcal{S} \\otimes \\mathcal{S}} (\\delta \\otimes_B \\sigma_{12}(\\Delta) &= \\tau^A (\\beta) \\otimes\n\\tau^A (\\mathcal{T}_0) \\\\\n&= 1_A,\n\\end{align*}\nwhich yields the desired result.\n\\end{proof}\n\nFor the composition $\\Delta_* \\delta_*$ we have a similar result.\n\\begin{prop}\nUnder the hypothesis of Proposition ~\\ref{hyp}, we have that \n\\begin{equation} \n\\Delta_i \\delta_i \\colon K^{i+1}(B) \\to K^{i+1}(B)\n\\end{equation}\nis the identity for $i=0,1$.\n\\end{prop}\n\\begin{proof}\nThe proof is obtained from the previous one by making obvious changes.\n\\end{proof}\n\nThe other cases follow in the same way.  Thus,  showing that one has a\nduality between algebras reduces to constructing the maps $\\Theta_A$\nand $\\Theta_B $ satisfying the conditions above.  In the next two sections\nwe will do this for the case of the stable and unstable Ruelle\nalgebras associated to a subshift of finite type.     \n\n\\section{Construction of duality classes for shifts of finite type}\n\nIn this section we will construct the classes in KK-theory needed to\nexhibit the duality between $O_A \\otimes \\mathcal{K}$ and $O_{A^T} \\otimes \\mathcal{K}$.  Let $A$\nbe an $n \\times n$ matrix with entries which are all zero or one.  We\nassume that $A$ has no row or column consisting entirely of\nzeros and that the associated shift space is a Cantor set.  \n\nThe Cuntz-Krieger algebra, $O_A$, is the universal $C^*$-algebra\ngenerated by partial isometries $s_1, \\dots , s_n$ satisfying \n\\begin{enumerate}[i)]\n\\item the projections $s_1 s_{1}^{*}, \\dots , s_n s_{n}^{*}$ are\npairwise orthogonal and add up to the identity of $O_A$,\n\\item for $k = 1, \\dots ,n$ one has\n\\begin{equation} \ns_{k}^{*} s_k = \\sum_i A_{ki} s_i s_{i}^{*}.\n\\end{equation}\n\\end{enumerate}\nThe condition above, that the shift space be a Cantor set, guarantees\nthat the algebra described does not depend on the choice of the\npartial isometries, ~\\cite{cuntz-k1}.\nIf $A_{ij} = 1 $ for all $i,j$, then the algebra $O_A$ is denoted\n$O_n$.\n\nIn a similar manner we consider $O_{A^T}$, with generators $t_1,\n\\dots, t_n$ satisfying\n\\begin{equation} \nt_{k}^{*} t_k = \\sum_i A_{ik} t_i t_{i}^{*}\n\\end{equation}\nfor $k = 1, \\dots ,n$.\n\nOur aim in this section is to explicitely construct the elements\n\\begin{equation*}\n\\delta \\in KK^1(\\mathbb{C}, O_A \\otimes O_{A^T})\n\\end{equation*}\nand \n\\begin{equation*}  \n\\Delta \\in KK^1(O_A \\otimes O_{A^T}, \\mathbb{C})\n\\end{equation*}\nwhich are needed to show that $O_A$ and $O_{A^T}$ are dual.\n\nThe construction of $\\delta$ is the easier of the two, (c.f. ~\\cite{cuntz-k2}).\nLet\n\\begin{equation} \nw = \\sum_{i=1}^{n} s_{i}^{*} \\otimes t_i \\in O_A \\otimes O_{A^T}.\n\\end{equation}\nThen one has\n\\begin{equation*} \nw^* w = w w^* = \\sum_{i,j} A_{ij} s_j s_{j}^{*} \\otimes t_i t_{i}^{*}.\n\\end{equation*}\nWe let $\\bar w \\colon C(S^1) \\to O_A \\otimes O_{A^T}$ denote both the\n(non-unital) map defined by\n\\begin{equation} \n\\bar w (z) = w\n\\end{equation}\nas well as its restriction to $C_0(\\mathbb{R}) \\subseteq C(S^1)$.  \n\\begin{definition}\nLet $\\delta \\in KK^1(\\mathbb{C}, O_A \\otimes O_{A^T})$ be the element determined by\nthe homomorphism $\\bar w$.\n\\end{definition}\nThe element $\\Delta$ is constructed using the full Fock space of a finite\ndimensional Hilbert space.  (For related constructions see the papers\nof D. Evans and D. Voiculescu, ~\\cite{voiculescu,evans}.)\n\nLet $\\mathcal{H}$ denote an n-dimensional Hilbert space with orthonormal basis\n$\\xi_1, \\dots, \\xi_n$.  Let $\\mathcal{H}^{\\otimes m} = \\mathcal{H} \\otimes \\cdots \\otimes \\mathcal{H}$ be the\nm-fold tensor product of $\\mathcal{H}$ and let $\\mathcal{H}_0$ be a one dimensional\nHilbert space with unit vector $\\Omega$.  Then the full Fock space of $\\mathcal{H}$,\n$\\mathcal{F}$, is defined to be \n\\begin{equation*} \n\\mathcal{F} = \\mathcal{H}_0 \\oplus (\\bigoplus_{n=1}^{\\infty} \\mathcal{H}^{\\otimes n})\n\\end{equation*}\nThere is a natural orthonormal basis for $\\mathcal{F}$,\n\\begin{equation*} \n\\{\\Omega, \\xi_{i_1}\\otimes \\cdots \\otimes \\xi_{i_m}|m=1,2,\\dots,\\qquad  1\\leq i_j\n\\leq n\\}. \n\\end{equation*}\nDefine the left and right creation operators, $L_1, \\dots , L_n$ and\n$R_1, \\dots ,R_n$, on $\\mathcal{F}$ by \n\\begin{equation*} \nL_k \\Omega = \\xi_k = R_k \\Omega\n\\end{equation*}\nand\n\\begin{align}\nL_k(\\xi_{i_1}\\otimes \\cdots \\otimes \\xi_{i_m} ) &= \\xi_k \\otimes \\xi_{i_1}\\otimes\n\\cdots \\otimes \\xi_{i_m} \\\\\nR_k(\\xi_{i_1}\\otimes \\cdots \\otimes \\xi_{i_m} ) &= \\xi_{i_1}\\otimes \\cdots \\otimes\n\\xi_{i_m} \\otimes \\xi_k\n\\end{align}\nNext, we bring in the matrix $A$.  Let $\\mathcal{F}_A \\subseteq \\mathcal{F}$ denote the\nclosed linear span of the vectors $\\Omega$ and those $\\xi_{i_1}\\otimes\n\\cdots \\otimes \\xi_{i_m} $ satisfying the condition that\n$A_{{i_j},{i_{j+1}}} = 1$ for all $j = 1, \\dots , m-1$.  Let $P_A$\ndenote the orthogonal projection of $\\mathcal{F}$ onto $\\mathcal{F}_A$.  Let \n\\begin{align*}\nL_k^A &= P_A L_k P_A \\in \\mathcal{B}(\\mathcal{F}_A) \\\\\nR_k^A &= P_A R_k P_A \\in \\mathcal{B}(\\mathcal{F}_A)\n\\end{align*}\nfor $k= 1, \\dots, n$.\n\nIt is easily checked that one has the following formulas.\n\\begin{align*}\nL_k^A \\xi_{i_1}\\otimes \\cdots \\otimes \\xi_{i_m} &= A_{k,i_1} \\xi_k \\otimes \\xi_{i_1}\\otimes\n\\cdots \\otimes \\xi_{i_m} \\\\\nR_k^A \\xi_{i_1}\\otimes \\cdots \\otimes \\xi_{i_m} &= A_{i_m,k}  \\xi_{i_1}\\otimes\n\\cdots \\otimes \\xi_{i_m} \\otimes \\xi_k \\\\\n(L_k^A)^* \\xi_{i_1}\\otimes \\cdots \\otimes \\xi_{i_m} &= A_{k,i_1}  \\xi_{i_2}\\otimes\n\\cdots \\otimes \\xi_{i_m} \\\\ \n(R_k^A)^* \\xi_{i_1}\\otimes \\cdots \\otimes \\xi_{i_m} &= A_{i_m,k}  \\xi_{i_1}\\otimes\n\\cdots \\otimes \\xi_{i_{m-1}}. \n\\end{align*}\nFrom this one easily obtains the following result.\n\\begin{prop}\n\\label{formulas}\nThe operators $R^A_k$ and $L^A_k$ are partial isometries and satisfy\n\\begin{enumerate}[i)]\n\\item $(L^A_k)^* L^A_k = \\sum_i A_{ki} L^A_i (L^A_i)^* + P_\\Omega$\n\\item $(R^A_k)^* R^A_k = \\sum_i A_{ik} R^A_i (R^A_i)^* + P_\\Omega$\n\\item $[L^A_k, R^A_l] = 0$\n\\item $[(L^A_k)^* , R^A_l] = \\delta_{kl} P_\\Omega$\n\\end{enumerate}\n\\end{prop}\nWe are now able to construct the element $\\Delta$.  Let $\\mathcal{E} \\subseteq\n\\mathcal{B}(\\mathcal{F}_A)$ be the $C^*$-algebra generated by $\\{R^A_1, \\dots, R^A_n,\nL^A_1, \\dots , L^A_n\\}$.  By Proposition~\\ref{formulas} the operator\n$P_\\Omega$, which is compact, is in $\\mathcal{E}$.  It is easy to check that\nthere is no non-trivial $\\mathcal{E}$-invariant subspace of $\\mathcal{F}_A$.  Thus, $\\mathcal{E}$\ncontains the compact operators, $\\mathcal{K}(\\mathcal{F}_A)$.\n\nModulo the ideal $\\mathcal{K}(\\mathcal{F}_A)$ the elements  $L^A_1, \\dots , L^A_n$ and\n$R^A_1, \\dots, R^A_n$ satisfy the relations for $O_A$ and $O_{A^T}$\nrespectively.  Moreover, the $L^A_i$'s and the $R^A_j$'s commute\nmodulo $\\mathcal{K}(\\mathcal{F}_A)$.  It follows that the $C^*$-algebra $\\mathcal{E} \/ \\mathcal{K}(\\mathcal{F}_A)$\nis a quotient of $O_A \\otimes O_{A^T}$.  In fact they are isomorphic.  This\nfollows since both $O_A$ and $O_{A^T}$ are nuclear and the ideal\nstructure of their tensor product may be completely described in terms\nof the ideals of $O_A$ and $O_{A^T}$.  These, in turn, have been\ncompletely described in ~\\cite{ cuntz-k2}.  It is then straightforward to\nverify that the generators of the ideals of $O_A \\otimes O_{A^T}$ give rise\nto non-compact operators (via the $L^A_k$ and $R^A_k$) and thus $\\mathcal{E}\n\/\\mathcal{K}(\\mathcal{F}_A) \\cong O_A \\otimes O_{A^T}$.\n\\begin{definition}\nLet $\\Delta \\in KK^1(O_A \\otimes O_{A^T}, \\mathbb{C})$ be the class determined by the\nexact sequence\n\\begin{equation} \n\\begin{CD}\n0 @>>> \\mathcal{K}(\\mathcal{F}_A) @>>> \\mathcal{E} @>{\\pi_{A}}>> \\to O_A \\otimes O_{A^T} \\to 0.\n\\end{CD}\n\\end{equation}\n\\end{definition}\nNote that one has\n\\begin{equation*} \n\\pi_A (R^A_k) = 1 \\otimes t_k\n\\end{equation*}\nand \n\\begin{equation*} \n\\pi_A (L^A_k) = s_k \\otimes 1.\n\\end{equation*}\n\n\n\n\n \\section{Duality for Cuntz-Krieger algebras}\nIn this section we will show that the duality classes constructed in\nthe previous section actually implement a duality isomorphism for the\nalgebras $O_A$ and $O_{A^T}$.  According to Proposition ~\\ref{hyp}, it\nwill be sufficient to construct homomorphisms\n\\begin{equation}\n\\begin{align}\n\\Theta_{O_A} \\colon O_A \\otimes \\mathcal{S} \\to O_A \\otimes \\mathcal{S} \\\\\n\\Theta_{O_{A^T}} \\colon O_{A^T} \\otimes \\mathcal{S} \\to O_{A^T} \\otimes \\mathcal{S}\n\\end{align}\n\\end{equation}\nwhich satisfy the conditions stated there.  That is, we\nmust show that $\\Theta_{O_A}$ and $\\Theta_{O_{A^T}}$ induce the identity\nhomomorphism on K-theory and satisfy the second condition in\nProposition ~\\ref{hyp} which states\n\\begin{align*}\n\\sigma_{12}(\\delta \\otimes_{O_{A^T}} \\sigma_{12}(\\Delta)) &= \\Theta_{O_A} \\otimes_{{O_A} \\otimes \\mathcal{S}} \\tau^{O_A}\n(\\mathcal{T}_0) \\\\\n\\sigma_{12}(\\delta \\otimes_{O_A} \\sigma_{12}(\\Delta)) &= \\Theta_{O_{A^T}} \\otimes_{{O_{A^T}} \\otimes \\mathcal{S}} \\tau^{O_{A^T}}\n(\\mathcal{T}_0)\n\\end{align*}\nWe will work out the details only for $\\Theta_{O_A}$, the other case\nbeing similar.\n\nTo define $\\Theta_{O_A}$ we first set\n\\begin{equation*}\n\\bar{\\Theta} \\colon O_A \\otimes C(S^1) \\to O_A \\otimes C(S^1)\n\\end{equation*}\nby\n\\begin{align*}\n\\bar{\\Theta} (1 \\otimes z) = 1 \\otimes z \\\\\n\\bar{\\Theta} (s_i \\otimes 1) = s_i \\otimes z.\n\\end{align*}\nThen $\\bar{\\Theta} $ extends to an automorphism of $O_A \\otimes\nC(S^1)$, as follows from the universal property of $O_A$.\nThe diagram\n\\begin{equation*}\n\\begin{CD}\nO_A \\otimes C(S^1) @>{\\bar{\\Theta}}>> O_A \\otimes C(S^1)\\\\\n@V{1_{O_A}} \\otimes \\pi VV    @V{1_{O_A}} \\otimes \\pi VV \\\\\nO_A @>id>> O_A\n\\end{CD}\n\\end{equation*}\ncommutes, where $\\pi \\colon C(S^1) \\to \\mathbb{C}$ is defined by $\\pi_1 (z) =\n1$.  It follows that we may define $\\Theta_{O_A} = \\bar{\\Theta} |\n\\ker(1_{O_A} \\otimes \\pi)$.  It is\nan automorphism of $O_A \\otimes \\mathcal{S}$.  We now must show that\n$\\Theta_{O_A}$ satisfies the necessary conditions.\n\\begin{theorem}\nThe maps\n\\begin{equation*}\n{\\Theta_{O_A}}_* \\colon K_i(O_A \\otimes \\mathcal{S}) \\to K_i(O_A \\otimes \\mathcal{S})\n\\end{equation*}\nare the identity for $i = 0 , 1$\n\\end{theorem}\n\\begin{proof}\nRecall that\n\\begin{equation*}\nO_A \\otimes \\mathcal{K} \\cong \\bar F_A \\rtimes_{\\sigma_A} \\mathbb{Z}\n\\end{equation*}\nwhere $\\bar F_A$ is a stable $AF$-algebra with automorphism\n$\\sigma_A$.  In this situation, $O_A$ is actually a full corner in\n$\\bar O_A$ and compressing $\\bar F_A$ to this corner yields $\\bar F_A \\subseteq O_A $ which is the\nclosure of the ``balanced words'' in the $s_i$'s as described in\n~\\cite{ cuntz-k1}.  Observe that the restriction of $\\bar{\\Theta} $ to\n$F_A \\otimes C(S^1)$ is the identity.  We will apply the\nPimsner-Voiculescu exact sequence to compute $K_*(O_A \\otimes \\mathcal{S})$,\nmaking necessary modifications since $\\bar F_A$ is not unital and then\nstudy ${\\Theta_{O_A}}_*$.\n\nLet $B$ denote the multiplier algebra of $\\bar F_A \\otimes \\mathcal{S} \\otimes \\mathcal{K}$\nwhere $\\mathcal{K} = \\mathcal{K}(l^2(\\mathbb{N}))$.  Let $e_{ij}$ denote the standard matrix units in\n$\\mathcal{K}$.  Define $\\rho \\colon \\bar F_A \\otimes \\mathcal{S} \\to B$ by\n\\begin{equation*}\n\\rho ( a \\otimes b) = \\sum_{i \\in \\mathbb{N}} \\sigma^{i}_A (a) \\otimes f \\otimes e_{ii}\n\\end{equation*}\nwhere the sum is taken in the strict topology.  Let $S$ denote the\nunilateral shift on $\\ell^2(\\mathbb{N})$.  Let $D$ denote the\n$C^*$-algebra generated by $\\bar F_A \\otimes \\mathcal{S} \\otimes \\mathcal{K}$, $1 \\otimes 1 \\otimes S$\nand $\\{ \\rho(a \\otimes f) | f \\in \\mathcal{S}, a \\in \\bar F_A\\}$.  Let $D_0$ be\nthe ideal in $D$ generated by $\\bar F_A \\otimes \\mathcal{S} \\otimes \\mathcal{K}$ and $\\{ \\rho(a\n\\otimes f) | f \\in \\mathcal{S}, a \\in \\bar F_A\\}$.  There is an exact sequence\n\\begin{equation*}\n0 \\to \\bar F_A \\otimes \\mathcal{S} \\otimes \\mathcal{K} \\to D_0 \\to \\mathcal{S} \\otimes (\\bar F_A \\rtimes \\mathbb{Z})\n\\to 0.\n\\end{equation*}\nMoreover, the two maps\n\\begin{equation*}\nj \\colon \\bar F_A \\otimes \\mathcal{S} \\to \\bar F_A \\otimes \\mathcal{S} \\otimes \\mathcal{K}\n\\end{equation*}\ndefined by $j(a \\otimes f) = a \\otimes f \\otimes e_{11}$\nand\n\\begin{equation*}\n\\rho \\colon \\bar F_A \\otimes \\mathcal{S} \\to D\n\\end{equation*}\nboth induce isomorphisms on K-theory.\n\nFinally, we have\n\\begin{equation}\nK_0(\\bar F_A \\otimes \\mathcal{S}) \\cong K_1(\\bar F_A)= 0\n\\end{equation}\nsince $\\bar F_A$ is an AF-algebra.  Putting this together, we obtain\nthe Pimsner-Voiculescu sequence for the $\\bar O_A$'s:\n\\begin{equation*}\n0 \\to K_1(\\bar O_A \\otimes \\mathcal{S}) \\to K_1(\\bar F_A \\otimes \\mathcal{S}) \\to K_1(\\bar F_A \\otimes\n\\mathcal{S}) \\to K_0(\\bar O_A \\otimes \\mathcal{S}) \\to 0\n\\end{equation*}\n\nWe define an automorphism $\\tilde \\Theta$ of $D$ by\n\\begin{equation}\n\\tilde \\Theta = ad (\\sum_{i \\in \\mathbb{N}} 1 \\otimes z^i \\otimes e_{ii})\n\\end{equation}\nwhere, again, the sum is in the strict topology.  Notice that $\\tilde\n\\Theta \\circ \\rho = \\rho$ and $\\tilde \\Theta | (\\bar F_A \\otimes \\mathcal{S} \\otimes \\mathcal{K})$\nis approximately inner and hence trivial on K-theory.  Also observe\nthat $\\tilde \\Theta | (\\bar F_A \\otimes \\mathcal{S} \\otimes \\mathcal{K}) = \\bar F_A \\otimes \\mathcal{S} \\otimes\n\\mathcal{K}$ and that the automorphism of the quotient of $D_0$ by $\\bar F_A\n\\otimes \\mathcal{S} \\otimes \\mathcal{K}$ induced by $\\tilde \\Theta $ is precisely $\\Theta_{O_A}\n$,\nafter identifying this quotient with $\\bar O_A \\otimes \\mathcal{S}$ and restricting to\n$O_A \\otimes \\mathcal{S} \\subseteq \\bar O_A \\otimes \\mathcal{S}$.  We have a commutative diagram\n\\begin{equation*}\n\\begin{CD}\n0 @>>> K_0(\\bar O_A \\otimes \\mathcal{S}) @>>> K_1(\\bar F_A \\otimes \\mathcal{S}) @>>> K_1(\\bar F_A\n\\otimes \\mathcal{S}) @>>> K_1(\\bar O_A \\otimes \\mathcal{S}) @>>>0 \\\\\n@. @VV{{\\Theta_{O_A}}_*}V   @VV{\\tilde \\Theta_*}V  @VV{\\tilde \\Theta_*}V\n@VV{{\\Theta_{O_A}}_*}V\\\\\n0 @>>> K_0(\\bar O_A \\otimes \\mathcal{S}) @>>> K_1(\\bar F_A \\otimes \\mathcal{S}) @>>> K_1(\\bar F_A\n\\otimes \\mathcal{S}) @>>> K_1(\\bar O_A \\otimes \\mathcal{S}) @>>>0\n\\end{CD}\n\\end{equation*}\nFrom the observations above, we have both maps $\\tilde \\Theta_* = id$,\nand it follows that ${\\Theta_{O_A}}_* $ is the identity.\n\\end{proof}\nIt remains for us to verify that  condition\n~\\eqref{thiii} is satisfied.  To that end we observe first that\n\\begin{equation*}\n\\sigma_{12}(\\delta \\otimes_{O_{A^T}} \\sigma_{12}(\\Delta)) = \\Theta_{O_A} \\otimes_{{O_A} \\otimes \\mathcal{S}} \\tau^{O_A}\n(\\mathcal{T}_0)\n\\end{equation*}\nis equivalent to\n\\begin{equation}\n\\label{81}\n\\tau_{\\mathcal{S}}((\\Theta_{O_A})^{-1}) \\otimes \\sigma_{12} \\tau_{\\mathcal{S}} \\tau_{O_A}\n(\\delta) \\otimes \\tau^{O_A} (\\sigma_{12}(\\Delta) ) =  \\tau^{O_A}\n(\\mathcal{T}_0).\n\\end{equation}\nThus, we will prove the latter statement.\n\n\nNow, $\\tau^{O_A}(\\sigma_{12}(\\Delta) ) \\in KK^1(O_A \\otimes O_{A^T} \\otimes O_A, O_A)$ was obtained from the\nextension\n\\begin{equation*}\n0 \\rar{}{} \\mathcal{K} \\otimes O_A \\rar{}{} \\mathcal{E} \\otimes O_A \\rar{\\pi_A \\otimes 1_{O_A}}{} O_A \\otimes O_{A^T} \\otimes O_A \\rar{}{} 0.\n\\end{equation*}\nMoreover, the remaining term\n\\begin{equation*}\n\\tau_{\\mathcal{S}}((\\Theta_{O_A})^{-1}) \\otimes \\sigma_{12} \\tau_{\\mathcal{S}} \\tau_{O_A} (\\delta)\n\\end{equation*}\nactually yields a $*$-homomorphism from $ O_A \\otimes \\mathcal{S} \\otimes \\mathcal{S}$ to $\nO_A \\otimes O_{A^T} \\otimes O_A \\otimes \\mathcal{S}$.  Thus, the left side of ~\\eqref{81} is\nrepresented by applying $\\tau_{\\mathcal{S}}$ to the element represented by the top row of the following diagram\n\\begin{equation*}\n\\begin{CD}\n0 @>>> \\mathcal{K} \\otimes O_A @>>> \\mathcal{E}' @>>> O_A \\otimes \\mathcal{S} @>>>0 \\\\\n@. @VVV @VVV @VV{1_{O_A} \\otimes i}V \\\\\n0 @>>> \\mathcal{K} \\otimes O_A @>>> \\mathcal{E}'' @>>> O_A \\otimes C(S^1) @>>>0 \\\\\n@. @VVV @VVV @VV{\\bar \\alpha}V \\\\\n0 @>>> \\mathcal{K} \\otimes O_A @>>> \\mathcal{E} \\otimes O_A @>>> O_A \\otimes O_{A^T} \\otimes O_A @>>> 0\n\\end{CD}\n\\end{equation*}\nwhere $\\alpha = (\\Theta_{O_A})^{-1} \\otimes  \\tau_{O_A} (\\delta) = \\bar\n\\alpha \\circ (1_{O_A} \\otimes i)$, and $\\alpha \\otimes 1_{\\mathcal{S}} = \\tau_{\\mathcal{S}}((\\Theta_{O_A})^{-1}) \\otimes \\sigma_{12} \\tau_{\\mathcal{S}} \\tau_{O_A} (\\delta)$.\n\n\nThe crucial step is to untwist the middle row by finding an isomorphism $\\mathcal{E}'' \\cong \\mathcal{T} \\otimes O_A$ so\nthat the following diagram commutes\n\\begin{equation*}\n\\begin{CD}\n0 @>>> \\mathcal{K} \\otimes O_A @>>> \\mathcal{E}'' @>>> O_A \\otimes C(S^1) @>>> 0 \\\\\n@. @V{=}VV @V{\\cong}VV @V{\\sigma_{12}}VV \\\\\n0 @>>> \\mathcal{K} \\otimes O_A @>>> \\mathcal{T} \\otimes O_A @>>{\\pi_{\\mathcal{T}}} \\otimes 1_{O_A}> C(S^1) \\otimes O_A @>>> 0,\n\\end{CD}\n\\end{equation*}\nwhere $\\mathcal{T}$ is the Toeplitz extension.\n\nAssuming this, the proof can be completed as follows.  We have\n\\begin{equation*}\n\\tau_{O_A}(\\mathcal{T}) = \\sigma_{12} \\bar \\alpha^* (\\tau^{O_A} (\\sigma_{12}(\\Delta) )).\n\\end{equation*}\n  Hence, one has\n\\begin{align*}\n\\tau_{O_A}(\\mathcal{T}_0) &=  (i \\otimes 1_{O_A})^* (\\tau^{O_A}(\\mathcal{T})) \\\\\n&= (i \\otimes 1_{O_A})^* \\sigma_{12} \\bar \\alpha^* (\\tau^{O_A}(\\sigma_{12}(\\Delta) )) \\\\\n&= \\sigma_{12} (1_{O_A} \\otimes i)^* \\bar \\alpha^* (\\tau^{O_A}(\\sigma_{12}(\\Delta) )) \\\\\n&= \\sigma_{12} \\alpha^* (\\tau^{O_A}(\\sigma_{12}(\\Delta) )).\n\\end{align*}\nThus, substituting in for $\\alpha$, we obtain\n\\begin{equation*}\n\\tau_{O_A}(\\mathcal{T}_0) = \\tau_{\\mathcal{S}} ((\\Theta_{O_A})^{-1}) \\otimes \\sigma_{12}\n\\tau_{\\mathcal{S}} \\tau^{O_A}(\\delta) \\otimes \\tau^{O_A}(\\sigma_{12}(\\Delta) ),\n\\end{equation*}\nwhich is the desired formula.\n\nWe now turn to the issue of obtaining the explicit isomorphism between\n$\\mathcal{E}''$ and $\\mathcal{T} \\otimes O_A$.  For convenience, we will suppress the $A$ in our\nnotation from the elements such as ${R_i}^A$, and ${L_i}^A$.  Define $W$ in $\\mathcal{E} \\otimes\nO_A$ by\n\\begin{equation*}\nW = \\sum_{i=1}^{n} R_i \\otimes {s_i}^*.\n\\end{equation*}\nWe will need two technical lemmas.\n\\begin{lem}\n\\label{43}\nOne has\n\\begin{enumerate}[i)]\n\\item $\\pi \\otimes 1_{O_A} (W) = \\bar \\alpha (1 \\otimes z)$.\n\\label{i}\n\\item $W^* W = \\sum_{i,j} A_{ji} R_j {R_j}^* \\otimes s_i {s_i}^* +\nP_{\\Omega} \\otimes 1$.\n\\item $[W^* , W] = P_{\\Omega} \\otimes 1$.\n\\item $(P_{\\Omega} \\otimes 1) W = 0$.\n\\item $[W, L_k \\otimes 1] = 0$ for $k = 1,\\dots,n$.\n\\item $[W^* , L_k \\otimes 1] = P_{\\Omega} \\otimes s_k$ for $k = 1,\\dots,n$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nFor (\\ref{i}), one proceeds as follows.  Note first that \n\\begin{align*}\n  \\bar \\alpha ( 1 \\otimes z) &= \\sigma^{23} (\\bar \\Theta_{O_A}^{-1})^* (1\n  \\otimes \\bar \\omega (z)) \\\\\n&= \\sigma^{23} (1 \\otimes \\bar \\omega (z)) \\\\\n&= \\sigma^{23} ( \\sum_i 1 \\otimes s_{i}^{*} \\otimes t_i) \\\\\n&= \\sum_i 1 \\otimes t_i \\otimes s_{i}^{*}.\n\\end{align*}\nMoreover, \n  \\begin{align*}\n    (\\pi \\otimes 1_{O_A})(W) &= (\\pi \\otimes 1_{O_A}) (\\sum_i R_i \\otimes\n    s_{i}^{*}) \\\\\n    &= \\sum_i \\pi(R_i) \\otimes s_{i}^{*} \\\\\n    &= \\sum_i 1 \\otimes t_i \\otimes s_{i}^{*}.\n  \\end{align*}\nThe remaining parts of lemma can be verified in a routine manner.\n\\end{proof}\n\nThe remaining facts we need are incorporated into the following.\n\\begin{lem}\n\\label{44}\nLet $V_k = W^* (L_k \\otimes 1)$, for $k = 1 \\dots n$.  Then we have, for\neach $k$,\n\\begin{description}\n\\item[i)] $\\pi \\otimes 1_{O_A} (V_k) = \\bar \\alpha (s_k \\otimes 1)$,\n\\label{ii}\n\\item[ii)] $\\sum_j V_j V^{*}_{j} = W^* W$,\n\\item[iii)] $V_{k}^{*} V_k = \\sum_j A_{kj} V_j V^{*}_{j} $,\n\\item[iv)] $[W, V_k] = 0$,\n\\item[v)] $[W^* , V_k] = 0$.\n\\end{description}\n\\end{lem}\n\\begin{proof}\n  As in the previous lemma, we will verify (\\ref{ii}) and leave the\n  remaining parts of the proof to the reader, since they are\n  essentially routine.  For ~\\ref{ii}, we check\n    \\begin{align*}\n  (\\pi \\otimes 1_{O_A}) (V_k) &= (\\pi \\otimes 1_{O_A})(W^*)(\\pi \\otimes\n  1_{O_A})(L_k \\otimes 1) \\\\\n  &= \\bar \\alpha (1 \\otimes z) (s_k \\otimes 1 \\otimes 1),\n  \\end{align*}\n  and  \n      \\begin{align*}\n      \\bar \\alpha (s_k \\otimes 1) &= \\sigma^{23} (1_{O_A} \\otimes \\bar w)\n      (\\bar \\Theta_{A}^{-1})^* (s_k \\otimes 1)  \\\\\n      &= \\sigma^{23} (1_{O_A} \\otimes \\bar w) (s_k \\otimes 1)  \\\\\n      &= \\sigma^{23} (1_{O_A} \\otimes w) (s_k \\otimes 1 \\otimes 1)  \\\\\n      &= \\bar \\alpha (1 \\otimes z) (s_k \\otimes 1 \\otimes 1).\n    \\end{align*}\n  \\end{proof}\n\nNow we may define the isomorphism from $\\mathcal{T} \\otimes O_A$ to $\\mathcal{E} ''$.  Let\n$S$ denote the unilateral shift. The required map is defined by\nsending $S \\otimes 1$ to $W$, and $1\n\\otimes S_k$ to $V_k$.  Note that the unit of $\\mathcal{T} \\otimes O_A$ is mapped to\n$W^* W$ in $\\mathcal{E}''$.  The fact that this assignment extends to a *-homomorphism\nfollows from the universal properties of $\\mathcal{T}$ and $O_A$.  The fact\nthat it is onto follows from observing that $\\mathcal{E} ''$ is generated by $\\{\nW, V_1, \\dots , V_k \\}$ which is straightforward.  Finally, the\nfact that the appropriate diagram commutes follows from ~\\ref{44} and\n~\\ref{43}.\n\n\n\n\\section{Final comments}\n\\begin{enumerate}\n\\item As mentioned earlier, the duality theorem holds for the stable\n  and unstable Ruelle algebras associated to a Smale\n  space,~\\cite{kaminker-p2}.  However, the duality classes, $\\Delta$\n  and $\\delta$, must be constructed in a different way.  This is done\n  using asymptotic morphisms and uses the fact that locally the Smale\n  space decomposes into a product of expanding and contracting sets.\n  It would be very interesting to have a Fock space construction of\n  the more general classes as well.\n\\item The duality result for Cuntz-Krieger algebras sheds some light\n  on the computations of the K-theory of $O_A$'s as in\n  ~\\cite{cuntz-k2}.  Recall that if $A$ is an $n \\times n$ aperiodic\n  matrix of $0$'s and $1$'s , then there are {\\em canonical}\n  isomorphisms\n\\begin{align*}\nK_0(O_A) \\cong \\mathbb{Z}^n \/ (1-A^T) \\mathbb{Z}^n \\\\\nK_1(O_A) \\cong \\ker(1-A^T) \\\\\nK^0(O_A) \\cong \\ker(1-A) \\\\\nK^1(O_A) \\cong \\mathbb{Z}^n \/ (1-A) \\mathbb{Z}^n.\n\\end{align*}\nNote that $\\mathbb{Z}^n \/ (1-A) \\mathbb{Z}^n \\cong \\mathbb{Z}^n \/ (1-A^T) \\mathbb{Z}^n$ by the\nstructure theorem for finitely generated abelian groups, but the\nisomorphism is not natural.\nThe explanation for why one has $A^T$ in the formulas now comes from\nduality, since one has the diagram\n\\begin{equation*}\n\\begin{CD}\nK_0(O_A) @>\\cong>> K^1(O_{A^T}) \\\\\n@VVV  @VVV \\\\\n\\mathbb{Z}^n \/ (1-A^T) \\mathbb{Z}^n @>{=}>> \\mathbb{Z}^n \/ (1-A^T) \\mathbb{Z}^n.\n\\end{CD}\n\\end{equation*}\n\\end{enumerate}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION }\n\nIn  cosmological models inflation is realized  by a slowly rolling scalar field, the so called  inflaton, whose energy density  dominates  the early history Universe \\cite{Guth:1980zm,Linde:1981mu,Mukhanov:1981xt,Albrecht:1982wi}. \nAmong  several suggestions regarding its origin, the economical scenario  that  this field can be identified with the \nStandard Model (SM) Higgs state $\\mathrm{h}$, has received considerable attention\\cite{Bezrukov:2007ep}.  In this approach, the\n Higgs field  drives inflation  through its strong coupling,  $\\upxi \\mathrm{h}^2 R$,  where $R$ is the Ricci scalar and $\\upxi$\n is a dimensionless parameter that acquires a large value,  $\\upxi\\gtrsim 10^4$. \n  \n\n  \n\nIn  modern particle physics theories,  cosmological  inflation is usually described within  the framework of supergravity or \nsuperstring grand unified theories (GUTs). In these theories the SM is embedded in a higher gauge symmetry and the field content including \nthe  Higgses  are incorporated in representations of the higher symmetry which includes the SM gauge group.  In this context, \nseveral new facts and constraints should be taken into account. For instance,  since new symmetry breaking stages are involved,  \nthe Higgs sector is usually  extented and  alternative possibilities for identifying the inflaton emerge. In addition, the effective \npotential has a specific structure  constrained from fundamental principles of the theory. In string theory effective models, for\n example, in a wide class of compactifications the scalar potential  appears with a no-scale structure  as in standard supergravity \ntheories \\cite{Cremmer:1983bf, Lahanas:1986uc}.   In general, the scalar potential is a function of the various fields which enter in a complicated \nmanner through the superpotential $W$ and the K\\\"ahler potential $K$.  Thus, a  rather detailed investigation is required to determine \nthe conditions for slow roll inflation and  ensure a stable  inflationary trajectory in such models. Modifications of the basic no-scale K\\\"ahler potential and various choices for the superpotential have been\nstudied leading to a number of different inflationary cases \\cite{Ellis:2013xoa}-\\cite{Romao:2017uwa}, while studies of inflation within supergravity in a model independent way can be found in \\cite{Covi:2008cn, Hardeman:2010fh}.\n\n\nIn the present work we implement the scenario of Higgs inflation in a model based on the Pati-Salam gauge symmetry $SU(4)_{C}\\times SU(2)_L\n\\times SU(2)_R$ \\cite{Pati:1974yy} (denoted for brevity with 4-2-2). This model has well known  attractive features (see for example the \nrecent review \\cite{Pati:2017ysg}) and has been successfully rederived  in superstring and D-brane \ntheories \\cite{Antoniadis:1988cm, Cvetic:2004ui, Anastasopoulos:2010ca, Cvetic:2015txa}. Early universe cosmology and inflationary predictions of the model (or its extensions) have been discussed previously in several works \\cite{Jeannerot:2000sv, Pallis:2011gr, Bryant:2016tzg}.  Here we consider a supersymmetric version of the 4-2-2 model where the breaking down to the SM gauge group takes place in two steps. First $SU(4)$ breaks  \nspontaneously at the usual supesymmetric GUT scale $M_{GUT}\\gtrsim 10^{16}$ GeV, down to the \\emph{left-right} group\\footnote{For a recent discussion on left-right models based on GUTs, see \\cite{Chakrabortty:2017mgi}. Inflation from an $SO(10)$ model with left-right intermediate symmetry is analysed in \\cite{Garg:2015mra}.} via the adjoint representation. Then,  depending on the specific structure of the Higgs\nsector, the $SU(2)_R$ scale can break either at the GUT scale, i.e., simultaneously  with $SU(4)$, or at some lower, intermediate energy scale. \nThe variety of possibilities are reflected back to the effective field theory model  implying  various interesting phenomenological \nconsequences. Regarding the Higgs inflation scenario, in particular, the inflaton field can be identified with the neutral components of the $SU(2)_{R}$ doublet fields\nassociated with the intermediate scale symmetry breaking.  In this work we will explore alternative  possibilities to realise inflation \nwhere the inflaton is identified with the $SU(2)_{R}$ doublets. We also examine the case of inflation in the presence of  the adjoint representation.\n\n\nThe layout of the paper is as follows. In section 2, we present a brief description of the 4-2-2 model, focusing in its particle content\nand the  symmetry breaking pattern. In sections 3 we present the  superpotential and the emergent no-scale supergavity K\\\"ahler potential\nof the effective model. We derive the effective potential and analyse the predictions on inflation when either the $SU(2)_{R}$ doublets or the adjoint play the r\\^ole of the inflaton. We present our conclusions in section 4. \n\n\n\n\n\n\\section{DESCRIPTION OF THE MODEL}\nIn this section we highlight the basic ingredients of  the model with gauge symmetry,\n\\be \\label{psgroup}\n SU(4)_{C}\\times{SU(2)_{L}}\\times{SU(2)_{R}}~\\cdot\n \\ee\n \\noindent This model unifies each family of quarks and leptons into two irreducible representations,\n  $F_{i}$ and $\\bar{F}_{i}$  transforming as \\cite{King:1997ia}\n \\[F_{i}=(4,2,1)_{i}\\quad{\\text{and}}\\quad \\bar{F}_{i}=(\\overline{4},1,2)_{i}~,\\]\n\n\\noindent under the corresponding factors of the gauge group~(\\ref{psgroup}). Here the subscript $i$ ($i=1,2,3$) denotes family index. \n Note that $F+\\bar{F}$ comprise the $16$ of $SO(10)$,   $16\\rightarrow{(4,2,1)+(\\overline{4},1,2)}$.\nThe explicit embedding  of the SM matter fields, including the right-handed neutrino is as follows:\t\n\\be\nF_{i}=\n\\begin{pmatrix} \nu_r & u_g & u_b & \\nu  \\\\\nd_r & d_g & d_b & e\n\\end{pmatrix}_{i}\\quad{,}\\quad{\\bar{F}_{i}=\n\\begin{pmatrix} \nu^{c}_r & u^{c}_g & u^{c}_b & \\nu^{c}  \\\\\nd^{c}_r & d^{c}_g & d^{c}_b & e^{c}\n\\end{pmatrix}_{i}}~,\n\\ee\n\n\n\\noindent where the subscript $(r,g,b)$ are color indices.\n\n The  symmetry breaking \n\\be\n SU(4)_{C}\\times{SU(2)_{R}}\\rightarrow{SU(3)_{C}\\times{U(1)_{Y}}}~,\n\\ee\n\n\n\\noindent is achieved by introducing two Higgs multiplets \n\n\\be\\label{HiggsofPS}\nH=(\\overline{4},1,2)=\n\\begin{pmatrix} \nu_{H}^{c} & u_{H}^{c} & u_{H}^{c} & \\nu_{H}^c  \\\\\nd_{H}^{c} & d_{H}^{c} & d_{H}^{c} & e_{H}^{c}\n\\end{pmatrix}\\quad{,}\\quad{\\bar{H}=(4,1,2)=\n\\begin{pmatrix} \n\\ov{u}_{H}^{c} & \\ov{u}_{H}^{c} & \\ov{u}_{H}^{c} & \\ov{\\nu}_{H}^c  \\\\\n\\ov{d}_{H}^{c} & \\ov{d}_{H}^{c} & \\ov{d}_{H}^{c} & \\ov{e}_{H}^{c}\n\\end{pmatrix}}\n\\ee\n which descend from the $16$ and $\\overline{16}$ of $SO(10)$ respectively. \n\n\nAn alternative way to break the gauge symmetry arises in the case where the adjoint \nscalar $\\Sigma=(15,1,1)$ is included in the spectrum.\n We parametrise $\\Sigma$  with a singlet scalar field  $S$ \n\\ba \n\\Sigma\\equiv (15,1,1)  &=&\n\\frac{S}{2\\sqrt{3}}\\left(\n\t\\begin{array}{cccc}\n\t\t1&0 & 0 &0\\\\\n\t\t0 &1& 0&0 \\\\\n\t\t0 & 0 &  1&0\\\\\n\t\t0 & 0 &0& -3\n\t\\end{array}\n\t\\right)~,\\label{Adj}\n\\ea\nwhich acquires a GUT scale vacuum expectation value (vev) $\\langle{S}\\rangle\\equiv\\upsilon\\simeq{3\\times{10^{16}}}$ GeV \n breaking  $SU(4)\\to SU(3)\\times U(1)$. The breaking leads to the left-right symmetric group, \n $SU(3)_{C}\\times{SU(2)_{L}}\\times{SU(2)_{R}}\\times{U(1)_{B-L}}$, and the decomposition of the Higgs fields  $H, \\bar{H}$ \n is as follows:\n\\begin{eqnarray}\\label{Hbreaking}\n\\begin{split}\nH(\\ov{4},1,2)&\\rightarrow{Q_{H}(\\ov{3},1,2)_{-1\/3}+L_{H}(1,1,2)_{1}}\\\\\n\\bar{H}(4,1,2)&\\rightarrow{\\ov{Q}_{H}(3,1,2)_{1\/3}+\\ov{L}_{H}(1,1,2)_{-1}}\n\\end{split}\n\\end{eqnarray}\n\n\n\n\n\\noindent where $Q_{H}=(u_{H}^{c}\\quad{d_{H}^{c}})^{T}$, $\\ov{Q}_{H}=(\\ov{u}_{H}^{c}\\quad{\\ov{d}_{H}^{c}})$  \nand $L_{H}=(\\nu_{H}^{c}\\quad{e_{H}^{c}})^{T}$, $\\ov{L}_{H}=(\\ov{\\nu}_{H}^{c}\\quad{\\ov{e}_{H}^{c}})$. \n\nThe right-handed doublets $L_{H},\\ov{L}_{H}$, acquiring vev's along their neutral components ${\\nu}_{H}^c ,\n \\ov{\\nu}_{H}^c $ and as a result they break the\n$SU(2)_R$ symmetry at some scale $M_R$. This way we obtain the symmetry breaking pattern~\\cite{Anastasopoulos:2010ca}:\n\\[\nSU(4)_{C}\\times{SU(2)_{R}}\\times{SU(2)_{L}}\\rightarrow{SU(3)_{C}\\times{U(1)_{B-L}}}\\times{SU(2)_{R}}\\times{SU(2)_{L}}\\to \n{SU(3)}\\times{SU(2)_{L}}\\times{U(1)_{Y}}.\n\\]\n The two scales $M_{GUT}$ and $M_R$ are not related to each other and it is in principle possible to \n take $M_R$ at some lower scale provided there is no conflict with observational data such as \n flavour changing neutral currents and lepton or baryon number violation.  \nRegarding the fast proton decay problem, in particular, in 4-2-2 models,  due to absence of the associated gauge bosons\nthere are no contributions from dimension six (d-6) operators, and related issues from d-5 operators can be remedied with \nappropriate symmetries in the superpotential. \n\n\nThe remaining spectrum and its $SO(10)$ origin is as follows:  The decomposition of the $10$ representation of $SO(10)$, \ngives  a bidoublet and a sextet field, transforming under  the 4-2-2 symmetry as follows\n\n\\be \n10\\rightarrow{h(1, 2, 2)+D_{6}(6,1,1)}~\\cdot \\label{10toHD}\n\\ee \n\n\\noindent\nThe two Higgs doublets of the minimal supersymmetric standard model (MSSM) descend from the bidoublet\n\n\\be\nh=(1,2,2)=\n\\begin{pmatrix}\nh_{2}^{+} & h_{1}^{0}\\\\\nh_{2}^{0} & h_{1}^{-}\n\\end{pmatrix}.\n\\ee\n\n\\noindent Also, the sextet of (\\ref{10toHD}) decomposes into a pair of coloured triplets: $D_{6}\\rightarrow{D_{3}(3,1,1)+\\overline{D}_{3}(\\ov{3},1,1)}$.\n\nCollectively we have the following SM assignments:\n\n\\begin{equation}\n\\begin{split}\nF&=(4,2,1)\\rightarrow Q(3,2,\\frac{1}{6})+L(1,2,-\\frac{1}{2})\\\\\n\\bar{F}&=(\\ov{4},1,2)\\rightarrow u^{c}(\\ov{3},1,-\\frac{2}{3})+d^{c}(\\ov{3},1,\\frac{1}{3})+e^{c}(1,1,1)+\\nu^{c}(1,1,0)\\\\\nh&=(1,2,2)\\rightarrow H_{u}(1,2,\\frac{1}{2})+H_{d}(1,2,-\\frac{1}{2})\\\\\nH&=(\\ov{4},1,2)\\rightarrow u^{c}_{H}(\\ov{3},1,-\\frac{2}{3})+d^{c}_{H}(\\ov{3},1,\\frac{1}{3})+e^{c}_{H}(1,1,1)+\\nu^{c}_{H}(1,1,0)\\\\\n\\bar{H}&=(4,1,2)\\rightarrow \\ov{u}^{c}_{H}(3,1,\\frac{2}{3})+\\ov{d}^{c}_{H}(3,1,-\\frac{1}{3})+\\ov{e}^{c}_{H}(1,1,-1)+\\ov{\\nu}^{c}_{H}(1,1,0)\\\\\nD_{6}&=(6,1,1)\\rightarrow{D_{3}(3,1,-\\frac{1}{3})+\\overline{D}_{3}(\\ov{3},1,\\frac{1}{3})}\n\\end{split}\n\\end{equation}\n\n\nFermions receive Dirac type masses from a common tree-level invariant term, $F\\bar{F}h$, whilst right-handed (RH) neutrinos receive heavy Majorana contributions from\nnon-renormalisable terms, to be discussed in the next sections. In addition, the colour triplets $d_{H}^{c}$ and $\\ov{d}_{H}^{c}$ are combined with the $D_{3}$ and $\\ov{D}_{3}$ states via the trilinear operators $HHD_{6}+\\bar{H} \\bar{H}D_{6}$ and  get masses near the GUT scale.\n\n\n\nAfter the short description of the basic features of the model, in the following sections we  investigate various inflationary scenarios in the context of no-scale supergravity, by applying the techniques presented in \\cite{Ellis:2014dxa, Ellis:2016spb}.\n\n\n\\section{INFLATION IN NO SCALE SUPERGRAVITY}\n\nIn this section we consider  the 4-2-2 model as an  effective string theory model  and study the implications of  Higgs inflation. \n The `light' spectrum in these constructions contains the MSSM states in representations transforming non-trivially under the \n gauge group and a number of moduli fields associated with the particular compactification. We will focus on the superpotential and the K\\\"ahler potential which are essential for the study of inflation. \n\nThe   superpotential is a holomorphic function of the fields. Ignoring Yukawa interaction terms,  the most general superpotential up to dimension four which is relevant to our discussion is \n\n \n \n \n \\begin{eqnarray}\n \\begin{split}\\label{wscalar}\n W&=M\\bar{H}H + \\mu\\bar{h}h + m \\tr(\\Sigma)^{2}+n \\bar{H}\\Sigma H+ c\\tr\\left(\\Sigma^{3}\\right)  \\\\\n &-\\alpha \\left(\\bar{H} H\\right)^{2}-\\beta\\left(\\bar{h}h\\right)^{2} -\\beta '\\left(\\bar{H}H\\right)\\left(\\bar{h}h\\right)- \\kappa \\tr\\left(\\Sigma^{4}\\right)-\\lambda \\bar{H} \\tr(\\Sigma^{2})H\n \\end{split}\n \\end{eqnarray}\n\n \n \n\\noindent where from now on  we set the reduced Planck mass to unity, $M_{Pl}=1$.  We focus on the dynamics of inflation during the first symmetry breaking stages at high energy scales.  For this reason we ignore all the terms involving the bi-doubled since this state mostly contribute in low energies by ginving mass to the MSSM particles and do not play an important r\\^ole during inflation. In addition we impose a $Z_{2}$ symmetry, under which $\\Sigma$ is odd and all the other fields are even. As a result the trilinear terms $\\bar{H}\\Sigma H$ and  $\\tr\\left(\\Sigma^{3}\\right)$ are eliminated from the superpotential in (\\ref{wscalar}). The elimination of these trilinear terms of the superpotential is important, since if we use $\\bar{H}\\Sigma H$ and  $\\tr\\left(\\Sigma^{3}\\right)$ instead of $\\bar{H} \\tr(\\Sigma^{2})H$ and $\\tr\\left(\\Sigma^{4}\\right)$, the shape of the  resulting potential is not appropriate and it leads to inconsistent results with respect to the cosmological bounds while at the same time returns a low scale value for the parameter $M$ in the superpotential, which usually expected to be close to the GUT scale.  Then, using (\\ref{Adj}) and (\\ref{Hbreaking}) the superpotential takes the following form: \n \n \\begin{eqnarray}\n \\begin{split}\\label{superpotential2}\n W &\\supset \\left(M-\\frac{\\tilde{\\lambda}}{9}S^{2}\\right)\\ov{Q}_{H}Q_{H}+\\left(M-\\tilde{\\lambda}S^{2}\\right)\\ov{L}_{H}L_{H}-\\alpha (\\ov{Q}_{H}Q_{H}+\\ov{L}_{H}L_{H})^{2}+mS^{2}-\\tilde{\\kappa}S^{4}\\\\\n & \\quad\\qquad\n \\end{split}\n \\end{eqnarray}\n \n \\noindent where $\\tilde{\\lambda}=\\frac{3\\lambda}{4}$ and $\\tilde{\\kappa}=\\frac{7\\kappa}{12}$. From the phenomenological point of view we expect $\\langle{S}\\rangle=v$ to be at the GUT scale. By assuming $v\\simeq{3\\times{10^{16}}}$GeV and using the minimization condition $\\partial{W}\/\\partial{S}=0$, we  estimate  that $m\\simeq{2\\tilde{\\kappa}v^{2}}$ which, for $\\tilde{\\kappa}=1\/2$, gives $m\\sim{10^{14}}$ GeV.\n \n In the two step breaking pattern that we consider here, $\\ov{L}_{H}$ and $L_H$ must remain massless at this scale in order to break the $SU(2)_R$ symmetry at a lower scale. The $SU(2)_{R}$ breaking scale should not be much lower than the  GUT scale in order to have a realistic heavy Majorana neutrino scenario. In addition we have to ensure that the coloured triplets $\\ov{Q}_{H}$ and $Q_{H}$ will be heavy. In order to keep the $\\ov{L}_{H}$, $L_H$ doublets at a lower scale, and at the same time the coloured fields $\\ov{Q}_{H}$ and $Q_{H}$ to be heavy, we assume that $M\\thickapprox{\\tilde{\\lambda}\\langle{S}\\rangle^{2}}=\\tilde{\\lambda}\\upsilon^{2}$. In this case $\\ov{Q}_{H}$, $Q_{H}$ acquire GUT scale masses  $M_{Q_{H}}\\thickapprox{\\frac{8\\tilde{\\lambda}}{9}\\langle{S}\\rangle^2}$.\n \n \n During inflation the colored triplets $\\ov{Q}_{H}$, $Q_{H}$ and the charged components of the RH doublets, $\\ov{L}_{H}$ and $L_H$, do not play an important r\\^ole. The $SU(2)_R$ symmetry breaks via the neutral components\\footnote{Here and for the rest of the paper, for shorthand we remove the subscript \"c\" on the fields, i.e: $\\ov{\\nu}^{c}_{H}$, $\\nu^{c}_{H}\\rightarrow{\\ov{\\nu}_{H}, \\nu_{H}}$.} $\\ov{\\nu}_{H}$ and $\\nu_{H}$. In terms of these states the superpotential reads:\n \n \\begin{eqnarray}\n \\begin{split}\n W= \\tilde{\\lambda}\\left(\\upsilon^{2} -S^{2}\\right)\\ov{\\nu}_{H}\\nu_{H}-\\alpha (\\ov{\\nu}_{H}\\nu_{H})^{2}+mS^{2}-\\tilde{\\kappa}S^{4}\n \\end{split}\n \\end{eqnarray}\n \n\\noindent where we have made use of the relation $M\\simeq\\tilde{\\lambda}\\upsilon^{2}$.\n\n\n\\noindent  The K\\\"{a}hler potential  has a no-scale structure and is a hermitian function of the  fields and their conjugates. For the present \n  analysis, we will consider the dependence of the Higgs fields of the 4-2-2 gauge group and  the `volume' modulus $T$. \nTherefore, assuming the fields $\\phi_i=(S, T, H, h)$ and their complex conjugates,  we write\n\\begin{equation}\\label{kahler1}\n\\begin{split}\nK = -3 \\log \\left[T + T^{\\ast}- \\frac{1}{3}\\left(H H^{\\ast} + \\bar{H} \\bar{H}^{\\ast} +\\tr\\Sigma^{\\dagger}\\Sigma\\right) +\\frac{\\xi}{3}\\left(H \\bar{H} + H^{\\ast} \\bar{H}^{\\ast}\\right)+ \\frac{\\zeta}{3}\\left(h h^{\\ast}+\\bar{h} \\bar{h}^{\\ast}\\right)\\right]\n\\end{split}\n\\end{equation}\n\n\\noindent where $\\xi$ is a dimensionless parameter. In the expression (\\ref{kahler1}), we can ignore the last term which involves the bidoublet and in terms of $\\nu_{H}$, $\\ov{\\nu}_{H}$ and $S$, the K\\\"{a}hler potential reads: \n\\begin{equation}\\label{kahler2}\n\\begin{split}\nK = -3 \\log \\left[T + T^{\\ast}- \\frac{1}{3}\\left(|\\nu_{H}|^{2} + |\\ov{\\nu}_{H}|^{2} +S^{2}\\right) +\\frac{\\xi}{3}\\left(\\ov{\\nu}_{H}\\nu_{H} +(\\ov{\\nu}_{H})^{\\ast}(\\nu_{H})^{\\ast} \\right)\\right].\n\\end{split}\n\\end{equation}\nIn order to  determine  the effective potential we define the function\n\\[ G= K+\\log|W|^2\\equiv  K+\\log W+\\log W^*.\n\\]\nThen the effective potential  is given by \n\\ba \nV=e^G\\left(G_iG_{i j^*}^{-1}G_{j^*}-3\\right)+V_D\\label{VGK}\n\\ea \nwhere  $G_i (G_{j^*})$ is the derivative with respect to the field $\\phi_i\n(\\phi^*_j)$ \nand the indices $i,j$  run over the various fields. $V_D$ stands for the D-term contribution. \n\n\\noindent   Computing the derivatives and substituting\tin  (\\ref{VGK}) the  potential takes the form \n\t\t\n\\begin{eqnarray}\\label{fullpotential}\n\\begin{split}\nV[\\ov{\\nu}_{H},\\nu_{H},S]&=\\frac{9}{(-3+\\nu_{H}^{2}+\\ov{\\nu}_{H}^{2}+S^{2}-2\\xi\\ov{\\nu}_{H}\\nu_{H})^{2}}\\left[(\\tilde{\\lambda}\\upsilon^{2}-2\\alpha \\nu_{H}\\ov{\\nu}_{H})^{2}(\\nu_{H}^{2}+\\ov{\\nu}_{H}^{2})-8\\tilde{\\lambda}mS^{2}\\ov{\\nu}_{H}\\nu_{H}\\right.\\\\\n&-2\\tilde{\\lambda} S^{2}(\\tilde{\\lambda}\\upsilon^{2}-2\\alpha \\nu_{H}\\ov{\\nu}_{H})(\\nu_{H}^{2}+\\ov{\\nu}_{H}^{2})+4\\tilde{\\lambda}^{2}S^{2}(\\ov{\\nu}_{H}\\nu_{H})^{2}\\\\\n&\\left. +4m^{2}S^{2}-16\\tilde{\\kappa}S^{4}(m-\\tilde{\\lambda}\\ov{\\nu}_{H}\\nu_{H})+\\tilde{\\lambda}^{2}S^{4}(\\nu_{H}^{2}+\\ov{\\nu}_{H}^{2})+16\\tilde{\\kappa}^{2}S^{6}\\right]\n\\end{split}\n\\end{eqnarray}\n\n\\noindent where we have ignored the D-term contribution and we have assumed that the value of the $T$ modulus field is stabilized at $\\langle{T}\\rangle=\\langle{T^{*}}\\rangle=1\/2$, see  \\cite{Cicoli:2013rwa, Ellis:2013nxa}. Notice that in the absence of the Higgs contributions in the K\\\"ahler \npotential, the effective potential is exactly zero, $ V=0$ due to the well known property of the no-scale structure. \n\n\n We are going now to investigate two different inflationary cases: firstly, along H-direction and secondly along S-direction.\n\n\\subsection{INFLATION ALONG $H$-DIRECTION}\n\n We proceed by parametrizing the neutral components of the $L_{H}$ and $\\ov{L}_{H}$ fields as $\\nu_H=\\dfrac{1}{2}\\left(X+Y\\right)e^{i\\theta}$ and $\\ov{\\nu}_H=\\dfrac{1}{2}\\left(X-Y \\right) e^{i\\varphi}$,  respectively.  These  yield\n\n\\begin{equation}\nX  = \\mid \\nu_H \\mid + \\mid \\bar\\nu_{H} \\mid,\n \\qquad Y  =  \\mid \\nu_H \\mid - \\mid \\bar\\nu_{H} \\mid~\\cdot\n\\end{equation}\n\n\n\\noindent \nAssuming  $\\theta=0$ and $\\varphi=0$, along the D-flat direction,  $Y=0$, and the combination $X$ is identified with the inflaton. The shape of the potential, as a function of the fields $S$ and $X$, is presented in Figure \\ref{3Dplots}. In order to avoid singularities from the denominator we have assume a condition which is described in the following. \n\n\n\\begin{figure}[t!]\n \t\\begin{subfigure}{.5\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=.95\\linewidth]{3Dplot_1.pdf}\n \t\n \t\t\\label{3d1}\n \t\\end{subfigure}%\n \t\\begin{subfigure}{.5\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=.95\\linewidth]{3Dplot_2.pdf}\n \t\n \t\t\\label{3d2}\n \t\\end{subfigure}\t\n \t\\caption{\\small{Plots of the potential as a function of $S$ and $X$ and for appropriate values of the other parameters. The plot on the right displays a close-up view  of the  region with small values for $X$ and $S$. }}\n \t\\label{3Dplots}\n \\end{figure}\n\n\n\n\n The potential along the $S=0$ direction is:\n\\begin{equation}\\label{potential_3_6}\nV\\left(X\\right) =\\frac{\\tilde{\\lambda}^{2}\\upsilon^{4}X^{2}\\left(1-\\frac{\\alpha X^{2}}{2 \\tilde{\\lambda}\\upsilon^{2}}\\right)^{2}}{2\\left(1-\\left(\\frac{1-\\xi}{6}\\right)X^{2}\\right)^{2}} .\n\\end{equation}\n\nThe shape of the $V(X,S)$ scalar potential presented in Figure \\ref{3Dplots} along with the inflaton trajectory description and the simplified form in (\\ref{potential_3_6}) is similar with the one presented in \\cite{Ellis:2014dxa,Ellis:2016spb}. As it is usually the case in no-scale supergravity,  the effective potential displays a singularity when the denominator vanishes. The presence of these singularities lead to an exponentially steep potential which can cause violation of the basic slow-roll conditions (i.e. $\\varepsilon\\ll{1}$, $|\\eta|\\ll{1}$). Consequently, these singularities must be removed. In our specific model described by the potential \\eqref{potential_3_6},\n we first notice that for the special value $\\xi=1$  the potential is free from singularities. For generic values of $\\xi$ however, i.e. $\\xi\\ne 1$, the potential displays a singularity for $X=\\sqrt{\\frac{6}{1-\\xi}}$.  In order to  remove the zeros of the denominator in \\eqref{potential_3_6}, we assume the following condition \\cite{Ellis:2014dxa},\n\\ba\n\\alpha=\\frac{\\left(1-\\xi\\right)\\tilde{\\lambda}\\upsilon^{2}}{3}~\\cdot\\label{singularitycondition}\n\\ea \n\n\n\\noindent This is a strong assumption which relates parameters with different origins. Indeed, $\\alpha$ is a superpotential parameter while $\\xi$ descents from the Kahler potential. Since in our specific model the condition \\eqref{singularitycondition} lacks an explanation from first principles, it will be reasonable in the subsequent analysis to study the effects of a slightly relaxed version of \\eqref{singularitycondition}. This can be achieved by introducing a small parameter $\\delta$ (with $\\delta\\ll{1}$) and modifying the condition  as follows,\n\n\\ba\n\\label{singularitycondition2}\n\\alpha=\\frac{\\left(1-\\xi+\\delta\\right)\\tilde{\\lambda}\\upsilon^{2}}{3}~\\cdot \\label{dsingularitycondition}\n\\ea\n\n\n\n\\noindent In the remaining of this section, we are going to study the potential for special $\\xi$ values using the conditions~(\\ref{singularitycondition}) and (\\ref{dsingularitycondition}). \n\n\n\nWe will start by analysing some special cases first. By imposing (\\ref{singularitycondition}), which means $\\delta=0$ \nthe scalar potential simplifies to a quadratic monomial,\n\n\\begin{equation}\\label{quadraticform}\nV\\left(X\\right) = \\frac{\\tilde{\\lambda}^{2}\\upsilon^{4}}{2}X^{2}\n\\end{equation}\n\\noindent something that can be also seen from the  plots  in Figure \\ref{3Dplots}, where for small values of S (along the $S=0$ direction) the potential receives a quadratic shape form. The equation (\\ref{quadraticform}) shows the potential of a chaotic inflation scenario. However, at this stage,\n the inflaton field $X$ is not canonically normalized since its kinetic energy terms take the following form\n\\begin{equation}\n\\begin{split}\n\\mathcal{L}\\left(X\\right)= \\frac{ 1-\\frac{\\xi}{6}\\left(1-\\xi\\right) X^{2}}{2\\left(1-\\frac{1}{6}\\left(1-\\xi\\right) X^{2}\\right)^{2}} \\left(\\partial X \\right)^{2} -\\frac{\\tilde{\\lambda}^{2}\\upsilon^{4}}{2}X^{2} .\n\\end{split}\n\\end{equation}\nWe introduce a  canonically normalized field $\\chi$ satisfying \n\\begin{equation}\n\\begin{split}\n\\left(\\frac{d\\chi}{dX}\\right)^{2} = \\frac{ 1-\\frac{\\xi}{6}\\left(1-\\xi\\right) X^{2}}{\\left(1-\\frac{1}{6}\\left(1-\\xi\\right) X^{2}\\right)^{2}}.\n\\end{split}\n\\end{equation}\nAfter integrating, we obtain the canonically normalized field $\\chi$ as a function of $X$\n\n\\begin{equation}\\label{hfield}\n\\chi  =\\sqrt{6}\\tanh^{-1}\\left(\\frac{\\left(1 - \\xi\\right)X}{\\sqrt{6\\left(1-\\frac{\\xi\\left(1-\\xi\\right)X^{2}}{6} \\right)}}\\right)\n-\\sqrt{\\frac{6 \\xi}{1-\\xi}}\\sin^{-1}\\left(\\sqrt{\\xi \\left(\\frac{1-\\xi}{6}\\right)}X\\right).\n\\end{equation}\n\n \\noindent Next, we  investigate the implications of equation (\\ref{hfield}) by considering two different cases, for $\\xi=0$ and $\\xi\\neq{0}$. \n\n$\\bullet$ For $\\xi=0$ we have $X=\\sqrt{6}\\tanh\\left(\\frac{\\chi}{\\sqrt{6}}\\right)$ and the potential becomes,\n\n\\begin{equation}\\label{Tpotential}\nV= 3 \\tilde{\\lambda}^{2}\\upsilon^{4}  \\tanh^{2}\\left(\\frac{\\chi}{\\sqrt{6}}\\right),\n\\end{equation}\n\\noindent which is analogous to the conformal chaotic inflation model (or T-Model) \\cite{Kallosh:2013xya}.\n In these particular type of models the potential has the general form:\n\n\n\n\n\\be\\label{Tmodels}\n V(\\chi)=\\uplambda^{n}\\tanh^{2n}\\left(\\frac{\\chi}{\\sqrt{6}}\\right) \\quad\\text{where}\\quad n=1,2,3,...\n \\ee\n As we can see, for  $n=1$ we receive our result in (\\ref{Tpotential}) with  $\\uplambda=3\\tilde{\\lambda}^{2}\\upsilon^{4}$. This potential  can be further reduced to subcases depending upon the value of $\\chi$. For $\\chi\\geqslant1$ the potential in equation (\\ref{Tpotential}) reduces to Starobinsky model \\cite{Starobinsky:1980te}. In this case the inflationary observables have values $\\left(n_{s},r\\right)\\approx \\left(0.967,0.003\\right)$ and the tree level prediction for $\\xi=0$ is consistent with the latest {Planck} bounds \\cite{Ade:2015lrj}. This type of models will be further analysed  in the next section where inflation along the $S$-direction is discussed. \\\\\n \n\n$\\bullet$ The particular case of $\\xi=1$ implies a quadratic chaotic inflation and the tree-level inflationary prediction  $\\left(n_{s},r\\right)\\approx \\left(0.967,0.130\\right)$ is ruled out according to the latest \\emph{Planck} $2015$ results. For $0<\\xi<1$ , the prediction for $\\left(n_{s},r\\right)$, can be\n worked out numerically.  \n\nAfter this analysis we turn our attention to a numerical calculation. In our numerical analysis we imply the modified condition (\\ref{singularitycondition2}) were as mentioned previously a small varying parameter $\\delta$ has been introduced in order to soften the strict assumption \\eqref{singularitycondition}. By substitute the relaxed condition \\eqref{singularitycondition2} in \\eqref{potential_3_6} and neglecting $\\mathcal{O}(\\delta^{2})$, the potential receives the following form:\n\n\n\n\\begin{equation}\\label{potentiladelta}\nV(X)\\simeq{\\frac{\\tilde{\\lambda}^{2}\\upsilon^{4}}{2}X^{2}}\\left(1-\\frac{2\\delta X^{2}}{6+(\\xi-1)X^{2}}\\right).\n\\end{equation}\n\n\\noindent As we observe the first term in the above relation is the quadratic potential \\eqref{quadraticform}, while the second term encodes the effects of the small parameter $\\delta$. In addition, we note that the order of the singularity enhancement have been improved in comparison with the initial potential \\eqref{potential_3_6}. Next we present our numerical results where the r\\^ole of the parameter $\\delta$ is also discussed.\n\n\n\n\n\n\\subsection{NUMERICAL ANALYSIS }\n\nBefore presenting numerical predictions of the model it is useful to briefly review here the basic results of the slow roll assumption. The inflationary slow roll parameters are given by \\cite{DeSimone:2008ei, Okada:2010jf}:\n\\begin{equation}\n\\epsilon=\\dfrac{1}{2}\\left(\\frac{V^{\\prime}\\left(X\\right)}{V(X)\\chi^{\\prime}\\left(X\\right)}\\right)^{2} \\quad{,}\\quad \\eta=\\left(\\frac{V^{\\prime\\prime}\\left(X\\right)}{V(X)\\left(\\chi^{\\prime}\\left(X\\right)\\right)^{2}}-\\frac{V^{\\prime}\\left(X\\right)\\chi^{\\prime\\prime}\\left(X\\right)}{V(X)\\left(\\chi^{\\prime}\\left(X\\right)\\right)^{3}}\\right).\n\\end{equation}\nThe third slow-roll parameter is,\n\\begin{equation}\n \\varsigma^{2}=\\left(\\frac{V^{\\prime}\\left(X\\right)}{V(X)\\chi^{\\prime}\\left(X\\right)}\\right)\\left(\\frac{V^{\\prime\\prime\\prime}\\left(X\\right)}{V(X)\\left(\\chi^{\\prime}\\left(X\\right)\\right)^{3}}-3\\frac{V^{\\prime\\prime}\\left(X\\right)\\chi^{\\prime\\prime}\\left(X\\right)}{V(X)\\left(\\chi^{\\prime}\\left(X\\right)\\right)^{4}}+3\\frac{V^{\\prime}\\left(X\\right)\\left(\\chi^{\\prime\\prime}\\left(X\\right)\\right)^{2}}{V(X)\\left(\\chi^{\\prime}\\left(X\\right)\\right)^{5}}-\\frac{V^{\\prime}\\left(X\\right)\\chi^{\\prime\\prime\\prime}\\left(X\\right)}{V(X)\\left(\\chi^{\\prime}\\left(X\\right)\\right)^{4}}\\right)\n\\end{equation}\n\\noindent where a prime denotes a derivative with respect to $X$. The slow-roll approximation is valid as long as the conditions $\\epsilon\\ll1$,$\\mid \\eta\\mid\\ll1$ and $\\varsigma^{2}\\ll1$\n hold true. In this scenario the tensor-to-scalar ratio $r$, the scalar spectral index $n_{s}$ and the running of the spectral index $\\frac{dn_{s}}{d\\ln k}$ are given by\n \\begin{equation}\nr\\simeq16 \\epsilon \\quad{,}\\quad n_{s}\\simeq 1+2\\eta-6\\epsilon \\quad{,}\\quad \\frac{dn_{s}}{d\\ln k}\\simeq 16\\epsilon\\eta-24\\epsilon^{2}+2\\varsigma^{2}.\n \\end{equation}\n The number of e-folds is given by,\n \\begin{equation}\n N_{l}=\\int_{X_{e}}^{X_{l}}\\left(\\frac{V\\left(X\\right)\\chi^{\\prime}\\left(X\\right)}{V^{\\prime}(X)}\\right) dX,\n \\end{equation}\n\\noindent  where $l$ is the comoving scale after crossing the horizon, $X_{l}$ is the field value at the comoving scale and $X_{e}$ is the field when inflation ends, i.e $max\\left(\\epsilon\\left(X_{e}\\right),\\eta\\left(X_{e}\\right),\\varsigma\\left(X_{e}\\right)\\right)=1.$\\\\\nFinally, the amplitude of the curvature perturbation $\\Delta_{R}$\n is given by:\n  \\begin{equation}\n\\Delta_{R}^{2}=\\frac{V\\left(X\\right)}{24 \\pi^{2} \\epsilon\\left(X\\right)}.\n  \\end{equation}\n  \n  \n   Focusing now on the  numerical analysis, we see that we have to deal with three parameters: $\\xi, \\delta$ and $\\tilde{\\lambda}$. We took the number of e-folds ($N$) to be 60, and  in Figure \\ref{ns_vs_r_plots} we present two different cases in the $n_{s}-r$ plane, along with the Planck measurements (\\emph{Planck} TT,TE,EE+lowP)  \\cite{Ade:2015lrj}. Specifically, in Figure $1(a)$, we fixed  $\\xi$ and vary $\\tilde{\\lambda}$ and $\\delta$. The various colored (dashed) lines corresponds to different fixed $\\xi$-values. The green line corresponds to the limiting case with $\\xi=1$ and as we observe the results are more consistent with the Plank bounds (black solid contours) as the value of $\\xi$ decreases. Similar, in Figure $1(b)$ we treat $\\delta$ as a fixed parameter while we vary $\\xi$ and $\\tilde{\\lambda}$. Also, in this case,  we observe that for a significant region of the parameter space the solutions are in good agreement with the observed cosmological bounds. The green curve here corresponds to $\\delta=10^{-6}$. The special case with $\\delta=10^{-6}\\sim 0$ and $\\xi=1$ is represented by the black dot and as we discussed earlier is ruled out from the recent cosmological bounds. We observe from the plot that, as $\\xi$ approaches to unity the splitting between the curves due to different values of $\\delta$ is small and the solution converges to $\\delta\\sim{0}$ case. However, as we decrease the values of $\\xi$  we have splitting of the curves and better agreement with the cosmological bounds.  Finally in plots 1(c) and 1(d) we present values of the running of the spectral index with respect to $n_{s}$. We observe that the running of the spectral index, approximately receives values in the range  $-5\\times{10^{-4}}<\\frac{dn_{S}}{d\\ln{k}} <5\\times{10^{-4}}$.\n\n   \n   \n   \n \\begin{figure}[t!]\n \t\\begin{subfigure}{.5\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=.9\\linewidth]{ns_r_fixed_xi.pdf}\n \t\t\\caption{\\small{r vs $n_{s}$ for fixed values of $\\xi$}}\n \t\t\\label{ns_r_fixed_xi}\n \t\\end{subfigure}%\n \t\\begin{subfigure}{.5\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=.9\\linewidth]{ns_r_fixed_delta.pdf}\n \t\t\\caption{\\small{$n_{s}$ vs r for fixed values of $\\delta$}}\n \t\t\\label{ns_r_fixed_delta}\n \t\\end{subfigure}\t\n \t\\begin{subfigure}{.5\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=.9\\linewidth]{ns_dns_fixed_xi.pdf}\n \t\t\\caption{\\small{$\\frac{dn_{S}}{d\\ln{k}}$ vs $n_{s}$ for fixed values of $\\xi$}}\n \t\t\\label{ns_dns_fixed_xi}\n \t\\end{subfigure}%\n \t\\begin{subfigure}{.5\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=.9\\linewidth]{ns_dns_fixed_delta.pdf}\n \t\t\\caption{\\small{$\\frac{dn_{S}}{d\\ln{k}}$ vs $n_{s}$ for fixed values of $\\delta$}}\n \t\t\\label{ns_dns_fixed_delta}\n \t\\end{subfigure}%\n \t\\caption{\\small{The inflationary predictions ($r$-$n_{s}$) and ($\\frac{dn_{s}}{d\\ln{k}}-n_{s}$) of the model by varying the various parameters involved in to the analysis. In all cases we took the  number of e-folds, $N=60$. In plots (a) and (b) black solid contours represents the Planck constraints  (\\emph{Planck} TT,TE,EE+lowP)  at $68\\%$ (inner) and $95\\%$ (outer) confidence level \\cite{Ade:2015lrj}. In plots (a) and (c) we keep $\\xi$ constant for each curve and vary $\\tilde{\\lambda}$ and $\\delta$. While in plots (b) and (d) for each curve we fixed $\\delta$ and vary $\\tilde{\\lambda}$ and $\\xi$. The black dot solution corresponds to $\\xi=1$. }}\n \t\\label{ns_vs_r_plots}\n \\end{figure}\n       \n \n \n Next we present additional plots to better clarify the r\\^ole of the various parameters involved in the analysis.\n \n Firstly, we study the spectral index $n_{s}$ as a function of the various parameters. The results are presented in Figure \\ref{ns_plots}. In plots (a) and (b) we consider the cases with fixed values for $\\xi$ and $\\delta$ respectively, and we take variations for  $\\tilde{\\lambda}$.  We vary the parameter $\\xi$ \n  in the range $\\xi\\sim{[0.92,1]}$ with the most preferable solutions for $\\xi\\simeq[0.96, 1]$. In addition the two plots suggest that acceptable solutions \n  are found in the range $\\tilde{\\lambda}\\sim[10^{-2},10^{-1}]$.  In plots (c) and (d)  $n_s$ is depicted  in terms of $\\delta$ and $\\xi$ respectively. As we expected the dependence on $\\delta$ is negligible when it receives very small values, since we observe from plot 3(c) that the various curves are almost constant for very small $\\delta$ values. The results are become more sensitive on $\\delta$ as we decrease the value of $\\xi$. This behaviour can also be confirmed  from the potential \\eqref{potentiladelta}. As we can see for $\\xi\\sim{1}$ the second term is simplified and the potential receives a chaotic like form. In this case the effects of small $\\delta$ in the observables are almost negligible (green line). However as we decrease the value of $\\xi$ and we increase the values of $\\delta$ the second term becomes important and contributes to the results.\n\n\\begin{figure}[t!]\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\\includegraphics[width=.9\\linewidth]{ns_loglam_fixed_xi.pdf}\n\t\t\\caption{\\small{$n_{S}$ vs $\\log{\\tilde{\\lambda}}$}}\n\t\t\\label{ns_lam_fixed_xi}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\\includegraphics[width=.9\\linewidth]{ns_loglam_fixed_delta.pdf}\n\t\t\\caption{\\small{$n_{S}$ vs $\\log{\\tilde{\\lambda}}$}}\n\t\t\\label{ns_lam_fixed_delta}\n\t\\end{subfigure}\n\t\t\\medspace\\\\\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\\includegraphics[width=.9\\linewidth]{ns_logdelta.pdf}\n\t\t\\caption{\\small{$n_{S}$ vs $\\log\\delta$}}\n\t\t\\label{ns_delta}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.9\\linewidth]{ns_xi.pdf}\n\t\t\\caption{\\small{$n_{S}$ vs $\\xi$}}\n\t\t\\label{ns_xi}\n\t\\end{subfigure}%\n\t\\caption{\\small{Plots (a) and (c) shows how $n_{S} $ depends on $\\log{\\tilde{\\lambda}}$ and  $\\log{\\delta}$ respectively. For each curve in Plots (a),(c) we fixed the value of $\\xi$ and vary $\\tilde{\\lambda}$ and $\\delta$. Similarly, Plots (b) and (d), shows $n_{s} $ vs $\\log{(\\tilde{\\lambda})}$ and $n_{S} $ vs $\\xi$ respectively. In Plots (b) and (d) the value of $\\delta$ is fixed while we vary the other parameters.}}\n\t\\label{ns_plots}\n\\end{figure}\n\n\n\\noindent\n\n\n\n\nNext, in Figure \\ref{r_plots} we consider various cases for the tensor to scalar ratio, r. The description of the plots follows the spirit of those presented in Figure \\ref{ns_plots} for  the spectral index $n_{S}$. In particular, by comparing the plots 4(c) and 3(c) we notice that the dependence of $r$ on $\\delta$ is weaker in comparison with $n_{S}$. Thus the relaxation parameter $\\delta$ strongly affects the spectral index $n_{S}$ while for $\\delta<10^{-4}$ and fixed $\\xi$ the tensor-scalar ratio $r$ remains almost constant. In summary from the various figures presented so far we observe that  consistent solutions can be found in a wide range of the parameter space. We also note that the model predicts solutions with $r\\leq{0.02}$, which is a prediction that can be tested with the discovery of primordial gravity waves and with bounds of future experiments. \n\n \\begin{figure}[t!]\n \t\\begin{subfigure}{.5\\textwidth}\n \t\t\\centering\n \t\\includegraphics[width=.95\\linewidth]{r_loglam_fixed_xi.pdf}\n \t\\caption{ $r$ vs $\\log{\\tilde{\\lambda}}$}\n \t\\label{r_lam_fixed_xi}\n \t\\end{subfigure}%\n \t\\begin{subfigure}{.5\\textwidth}\n \t\t\\centering\n \t\\includegraphics[width=.95\\linewidth]{r_loglam_fixed_delta.pdf}\n \t\\caption{ $r$ vs $\\log{\\tilde{\\lambda}}$}\n \t\\label{r_lam_fixed_delta}\n \t\\end{subfigure}%\n \t\\medspace\\\\\t\n \t \t\\begin{subfigure}{.5\\textwidth}\n \t \t\t\\centering\n \t \t\t\\includegraphics[width=.95\\linewidth]{r_logdelta.pdf}\n \t \t\t\\caption{ $r$ vs $\\log\\delta$}\n \t \t\t\\label{r_delta}\n \t \t\\end{subfigure}%\n \t \t\\begin{subfigure}{.5\\textwidth}\n \t \t\t\\centering\n \t \t\t\\includegraphics[width=.95\\linewidth]{r_xi.pdf}\n \t \t\t\\caption{ $r$ vs $\\xi$}\n \t \t\t\\label{r_xi}\n \t \t\\end{subfigure}\n \t\\caption{\\small{Plots (a) and (c) shows $r$ vs $\\log{\\tilde{\\lambda}}$ and $r$ vs $\\log{\\delta}$ respectively. For each curve in Plots (a) and (c) we fixed $\\xi$  and vary $\\tilde{\\lambda}$ and $\\delta$. Similar, in Plots (b) and (d) we present  $r$ vs $\\log{\\tilde{\\lambda}}$ and $r$ vs $\\xi$. For each curve in these plots we fixed the value of $\\delta$ and vary $\\tilde{\\lambda}$ and $\\xi$.}}\n \t\\label{r_plots}\n \\end{figure}\n\n\nRegarding the superpotential parameter $\\tilde{\\lambda}$, we can see from the various plots that its value must be  within the range $\\tilde{\\lambda}\\sim{[10^{-2}, 10^{-1}}]$. Using this range of values for $\\tilde{\\lambda}$ and the fact that, $M_{Q_{H}}\\approx{\\frac{8\\tilde{\\lambda}}{9}\\upsilon^{2}}$, with $\\upsilon\\simeq{10^{-2}}$ in $M_{Pl}=1$ units we conclude that : $M_{Q_{H}}\\sim{[0.217, 2.17]\\times{10^{13}}}$ GeV. The fact that the mass value is small compare to the $\\mathcal{O}(M_{GUT})$ scale, can create tension with other phenomenological predictions of the model, like unification of gauge couplings. On the other hand, as already mentioned  , $Q_{H},\\ov{Q}_{H}$ triplet fields can be mixed with the triplets   $D_{3},\\ov{D}_{3}$ contained in the sextet $D_{6}$, something that is possible to lead in a significant lift to the mass value of the extra triplet fields. \n\n\nIt is also interesting to investigate the values of the Hubble parameter during inflation $H_{inf}$ in the model. In the slow-roll limit the Hubble parameter it depends on the value of $X$:\n\n\n\\begin{equation}\nH_{inf}^{2}=\\frac{V(X)}{3M_{Pl}^{2}}\n\\end{equation}\n\n\\noindent and we evaluate it at the pivot scale. In Figure \\ref{Hinf_ns_plots} we show the values of the Hubble parameter in the ($H_{inf}-n_{s}$) plane. We observe that the values of the Hubble parameter with respect to $n_{s}$ bounds are of order $10^{13}$ GeV.\n\n\n\\begin{figure}[t!]\n \t\\begin{subfigure}{.5\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=.9\\linewidth]{Hinf_ns_fixed_xi.pdf}\n \t\t\\caption{\\small{$H_{inf}$ vs $n_{s}$ for fixed values of $\\xi$}}\n \t\t\\label{Hinf_ns_fixed_xi}\n \t\\end{subfigure}%\n \t\\begin{subfigure}{.5\\textwidth}\n \t\t\\centering\n \t\t\\includegraphics[width=.9\\linewidth]{Hinf_ns_fixed_delta.pdf}\n \t\t\\caption{\\small{$H_{inf}$ vs $n_{s}$ for fixed values of $\\delta$}}\n \t\t\\label{ns_r_fixed_delta}\n \t\\end{subfigure}\t\n \t\\caption{\\small{Plots showing the values (in GeV) of the Hubble parameter with respect to the scalar spectral index $n_{s}$. For acceptable $n_{s}$ values we see that the Hubble parameter receives values of order~$10^{13}-10^{14}$ GeV.}}\n \t\\label{Hinf_ns_plots}\n \\end{figure}\n\n\n\n\n\n\\subsection{REHEATING}\n\nAs already have been discussed in Section 2, the quarks and leptons in the 4-2-2 model are unified under the representations $F_{i}=(4,2,1)$ and $\\bar{F}_{i}=(\\bar{4},1,2)$, where $i=1,2,3$ denote the families and the RH-neutrinos are contained in the $\\bar{F}$ representation. A heavy Majorana mass for the RH-neutrinos can be realized from the following non-renormalisable term \n\n\\be \\label{majorana}\nM_{\\nu^c} \\nu^c\\nu^c\\approx \n  \\gamma\\frac{\\bar{F}\\bar{F}\\bar{H}\\bar{H}}{M_{*}}\n  \\ee\n \n  \n  \n\\noindent where we have suppressed generation indices for simplicity, $\\gamma$ is a coupling constant and $M_{*}$ represents a high cut-off scale (for example the compactification scale in a string model or the Planck scale $M_{Pl}$). In terms of $SO(10)$  GUTs this operator descent from the following invariant operator \n\n\n\\[ 16_{F}16_{F}\\bar{16}_{H}\\bar{16}_{H}\\] \n\n  \\noindent and as described in \\cite{Leontaris:2016jty} can be used to explain the reheating process of the universe after the end of inflation. In our case the 4-2-2 symmetry breaking occur in two steps: first $G_{PS}\\xrightarrow{\\langle{S}\\rangle}G_{L-R}$ and then  $G_{L-R}\\xrightarrow{\\langle{\\nu_{H}}\\rangle,\\langle{\\bar{\\nu}_{H}}\\rangle}G_{SM}$. The first breaking is achieved via the adjoint of the PS group at the GUT scale while the second breaking occurs in an intermediate scale $M_{R}$. After the breaking of the L-R symmetry, the high order term in (\\ref{majorana}) gives the following Majorana mass term for the RH neutrinos\n  \n\\be  \n  \\gamma\\frac{\\langle{\\nu_{H}}\\rangle^{2}}{M_Pl}\\nu^{c}\\nu^{c}.\n  \\ee\n  \n  \n  \n\\begin{figure}[t!]\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.9\\linewidth]{ns_TRH_fixed_xi.pdf}\n\t\t\\caption{}\n\t\t\\label{ns_Trh_fixed_xi01}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.9\\linewidth]{ns_TRH_fixed_delta.pdf}\n\t\t\\caption{ }\n\t\t\\label{ns_Trh_fixed_xi05}\n\t\\end{subfigure}%\n\t\\medspace\\\\\t\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.9\\linewidth]{r_TRH_fixed_xi.pdf}\n\t\t\\caption{}\n\t\t\\label{r_Trh_fixed_xi}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.9\\linewidth]{r_TRH_fixed_delta.pdf}\n\t\t\\caption{ }\n\t\t\\label{r_Trh_fixed_delta}\n\t\\end{subfigure}\n\t\\caption{\\small{Plots (a) and (b) shows solutions in the $n_{s}-T_{RH}$ plane by varying the various parameters of the model, while plots (c) and (d) present solutions in the $r-T_{RH}$ plane. In all the cases for the coupling constant $\\gamma$ we choose the values $\\gamma=0.1$ (solid), $\\gamma=0.5$ (dashed) and $\\gamma=1$ (dotted).}}\n\t\\label{ns_Trh_plots}\n\\end{figure}  \n  \n  \n  \n  \n  \n\\noindent We can see that a heavy Majorana scale scenario implies that the $SU(2)_{R}$ breaking scale should not be much lower than the $SU(4)$ scale and also $\\gamma$ should not be too small. Another important role of the higher dimensional operators is that after inflation the\ninflaton $X$ decays into RH neutrinos through them to reheat the Universe. In addition the subsequent decay of these neutrinos can explain the baryon asymmetry via leptogenesis \\cite{Fukugita:1986hr, Lazarides:1991wu}\n. For the reheating temperature, we  estimate \\cite{Leontaris:2016jty} (see also \\cite{Lazarides:2001zd}) :\n\n\\begin{equation}\nT_{RH}\\sim \\sqrt{\\Gamma_{X} M_{Pl}}\n\\end{equation}\n\n \\noindent where the total decay width of the inflaton is given by\n \n \n \\be\n\\Gamma_{X}\\simeq{\\frac{1}{16\\pi}\\left(\\frac{M_{\\nu^{c}}}{M}\\right)^{2}M_{X}}  \n\\ee  \n\n\n\\noindent with $M_{\\nu^{c}}= \\gamma\\frac{\\langle{\\nu_{H}}\\rangle^{2}}{M_{Pl}}$ the mass of the RH neutrinos and $M_{X}$ the mass of the inflaton. The later is calculated from the effective mass matrix at the local minimum and approximately is   $M_{X}=2M\\simeq{2\\tilde{\\lambda }\\upsilon^{2}}$. Since $M\\simeq{10^{13}}$GeV, the decay condition $M_{X}>M_{\\nu^{c}}$ it is always satisfied for appropriate choices of the parameters $\\langle\\nu_{H}\\rangle$ and $\\gamma$. In Figures \\ref{ns_Trh_plots} we present solutions in $n_{s}-T_{RH}$ and $r-T_{RH}$ plane with respect to the various parameters of the model. For the computation of $T_{RH}$ we assume that $\\langle\\nu_{H}\\rangle =M\\simeq{\\tilde{\\lambda}v^{2}}$ and we present the results for  $\\gamma=0.1$ (solid), $\\gamma=0.5$ (dashed) and $\\gamma=1$ (dotted). In this range of $\\gamma$ values we have a Majorana mass, $M_{\\nu^{c}}\\sim{10^{6}-10^{7}}$ GeV, which decreases as we decrease the value of $\\gamma$. In addition, gravitino constraints implies a bound for the reheating temperature with $T_{RH}<10^{6}-10^{9}$ GeV and as we observe from the plots there are acceptable solutions in this range of values. More precisely, from plots (a) and (c) we see that for $\\xi>0.97$  and $\\gamma>0.5$ most of the results predict $T_{RH}>10^{9}$ GeV. However, it is clear that the consistency with the gravitino constraints strongly improves as we decrease $\\gamma$, since all the curves with $\\gamma=0.1$ (solid lines) predicts $T_{RH}\\lesssim{10^{9}}$ GeV. Similar conclusions can be derived from plots (b) and (d). In addition, from the $r-T_{RH}$ plots (c) and (d) we observe that for  $T_{RH}<10^{6}-10^{9}$ there are regions in the parameter space with $r\\sim{10^{-2}-10^{-3}}$. Furthermore, we observe from plot 6(c) that the tensor-scalar ratio and the reheating temperature are decreased as we decrease the value of $\\xi$ since the curves are shift to the left and down regions of the plot.\n\n\nA sample of the results have been discussed so far is presented in Table \\ref{mastertable}. The table is organized in horizontal blocks and each block contains three sets of values. For each set in a block we change only the coupling constant $\\gamma$ ($\\gamma=1,0.5,0.1$) while we keep $\\tilde\\lambda$, $\\xi$ and $\\delta$ constant. We observe that as we decrease the values of $\\tilde{\\lambda}$ and $\\xi$ the values of the tensor to scalar ratio ($r$) and the reheating temperature ($T_{RH}$) also decreased. \n\n\n\n\n\n\n\n\\begin{table}[t]\n\t\\resizebox{\\textwidth}{!}{\n\t\\begin{tabular}{|lc|cccc|cc|ccc|c|}\n\t\t\\hline\n\t\n\t\t\n\t\t$\\frac{X_{0}}{M_{Pl}}$&$\\frac{X_{e}}{M_{Pl}}$&$\\gamma$&$\\tilde{\\lambda}$ & $\\xi$ &$ \\delta$  & $\\frac{M_{Inf}}{M_{Pl}}$ & $\\frac{M_{\\nu^{c}}}{M_{Pl}}$ & $n_{s}$&$r$&$\\frac{dn_{s}}{dln\\kappa}$&$\\log{(T_{RH}\/ GeV)}$\\\\\n\t\t\\hline\n\t\t\t15.04 &1.41& 1&0.0384&0.9936& $10^{-6}$& $1.16\\times10^{-5}$& $3.4\\times10^{-11}$& 0.968& 0.1070& $-4.7\\times{10^{-4}}$&9.83\\\\\n\t\t\t15.04 &1.41& 0.5&0.0384&0.9936& $10^{-6}$& $1.16\\times10^{-5}$& $1.7\\times10^{-11}$& 0.968& 0.1070& $-4.7\\times{10^{-4}}$&9.53\\\\\n\t\t\t15.04 &1.41& 0.1&0.0384&0.9936&$10^{-6}$& $1.16\\times10^{-5}$& $3.4\\times10^{-12}$& 0.968& 0.1070& $-4.7\\times{10^{-4}}$&8.84\\\\\n\t\t\\hline\n13.848 &1.41& 1& 0.0304&0.98&  $10^{-4.61}$& $9.25\\times10^{-6}$& $2.139\\times10^{-11}$& 0.971& 0.057& $-2.87\\times{10^{-4}}$&9.683\\\\\n13.848 &1.41& 0.5& 0.0304&0.98&  $10^{-4.61}$& $9.25\\times10^{-6}$& $1.07\\times10^{-11}$& 0.971& 0.057& $-2.87\\times{10^{-4}}$&9.382\\\\\n13.848&1.41& 0.1& 0.0304&0.98&  $10^{-4.61}$& $9.25\\times10^{-6}$& $2.139\\times10^{-12}$& 0.971& 0.057& $-2.87\\times{10^{-4}}$&8.683\\\\\n\t\\hline\n\n\t\t\t\t\n\t\t\n\t\t\n\t\t\n\t\n\n\t\n\t\n\t\t\n\t\t\n\n\t\n\t\t12.83& 1.40&1& 0.02141& 0.97&  $10^{-4.22}$& $6.5\\times10^{-6}$&\n\t\t\t$1.05\\times10^{-11}$& 0.967& 0.0238&  $1.5\\times{10^{-6}}$& 9.45\\\\\n\t\t\t\n\t\t\t12.83& 1.40&0.5& 0.02141& 0.97&  $10^{-4.22}$& $6.5\\times10^{-6}$&\n\t\t\t$5.29\\times10^{-12}$& 0.967& 0.0238&  $1.5\\times{10^{-6}}$& 9.15\\\\\n\t\t\t\n\t\t\t\t12.83& 1.40&0.1& 0.02141& 0.97&  $10^{-4.22}$& $6.5\\times10^{-6}$&\n\t\t\t\t$1.05\\times10^{-12}$& 0.967& 0.0238&  $1.5\\times{10^{-6}}$& 8.45\\\\\n\t\t\t\t\\hline\n\t\t\t12.69& 1.40&1& 0.019& 0.97&  $10^{-3.72}$& $5.8\\times10^{-6}$&\n\t\t\t$8.4\\times10^{-12}$& 0.958& 0.018&  $2.3\\times{10^{-4}}$& 9.38\\\\\n\t\t\t\n\t\t\t12.69& 1.40&0.5& 0.019& 0.97&  $10^{-3.72}$& $5.8\\times10^{-6}$&\n\t\t\t$4.2\\times10^{-12}$& 0.958& 0.018&  $2.3\\times{10^{-4}}$& 9.08\\\\\n\t\t\t\n\t\t\t12.69& 1.40&0.1& 0.019& 0.97&  $10^{-3.72}$& $5.8\\times10^{-6}$&\n\t\t\t$8.4\\times10^{-13}$& 0.958& 0.018&  $2.3\\times{10^{-4}}$& 8.3\\\\\n\t\t\t\\hline\n\t\t\t11.85& 1.40&1& 0.0118&0.96&  $10^{-4.82}$& $3.57\\times10^{-6}$& \n\t\t\t$3.2\\times10^{-12}$& 0.966& 0.0061&  $5.1\\times{10^{-5}}$& 9.065\\\\\n\t\t\t\n\t\t\t\t11.85& 1.40&0.5& 0.0118&0.96&  $10^{-4.82}$& $3.57\\times10^{-6}$& \n\t\t\t\t$1.6\\times10^{-12}$& 0.966& 0.0061&  $5.1\\times{10^{-5}}$& 8.76\\\\\n\t\t\t\t\n\t\t\t\t\t11.85& 1.40&0.1& 0.0118&0.96&  $10^{-4.82}$& $3.57\\times10^{-6}$& \n\t\t\t\t\t$3.2\\times10^{-13}$& 0.966& 0.0061&  $5.1\\times{10^{-5}}$& 8.065\\\\\n\t\t\t\t\\hline\n\t\t\t11.79& 1.40& 1&0.010& 0.96 & $10^{-4.397}$& $3.13\\times10^{-6}$ &\n\t\t\t$2.5\\times10^{-12}$& 0.957& 0.0050&  $2.1\\times{10^{-4}}$& 8.98\\\\\n\t\t\t\n\t\t\t\t11.79& 1.40&0.5& 0.010& 0.96 & $10^{-4.397}$& $3.13\\times10^{-6}$ &\n\t\t\t\t$1.2\\times10^{-12}$& 0.957& 0.0050&  $2.1\\times{10^{-4}}$& 8.67\\\\\n\t\t\t\n\t\t\t\t11.79& 1.40&0.1& 0.010& 0.96 & $10^{-4.397}$& $3.13\\times10^{-6}$ &\n\t\t\t\t$2.5\\times10^{-13}$& 0.957& 0.0050&  $2.1\\times{10^{-4}}$& 7.97\\\\\n\t\t\t\n\t\t\\hline\n\t\t\t11.64& 1.404&1&0.00891&0.958& ${10^{-4.5}}$& $2.71\\times10^{-6}$& \n\t\t\t$1.85\\times10^{-12}$& 0.957& 0.0034&  $1.8\\times{10^{-4}}$& 8.89\\\\\n\t\t\t\n\t\t\t11.64& 1.404&0.5&0.00891&0.958& ${10^{-4.5}}$& $2.71\\times10^{-6}$& \n\t\t\t$9.24\\times10^{-13}$& 0.957& 0.0034&  $1.8\\times{10^{-4}}$& 8.59\\\\\n\t\t\t\t11.64& 1.404&0.1&0.00891&0.958& ${10^{-4.5}}$& $2.71\\times10^{-6}$& \n\t\t\t\t$1.84\\times10^{-13}$& 0.957& 0.0034&  $1.8\\times{10^{-4}}$& 7.89\\\\\n\t\t\t\\hline\n\t\t\t11.59& 1.40&1&0.0084&0.958& ${10^{-4.5}}$& $2.6\\times10^{-6}$& \n\t\t\t$1.64\\times10^{-12}$& 0.956& 0.00299&  $1.9\\times{10^{-4}}$& 8.84\\\\\n\t\t\t\n\t\t\t11.59& 1.40&0.5&0.0084&0.958& ${10^{-4.5}}$& $2.6\\times10^{-6}$& \n\t\t\t$8.2\\times10^{-13}$& 0.956& 0.00299&  $1.9\\times{10^{-4}}$& 8.54\\\\\n\t\t\t\t11.59& 1.40&0.1&0.0084&0.958& ${10^{-4.5}}$& $2.6\\times10^{-6}$& \n\t\t\t\t$1.64\\times10^{-13}$& 0.956& 0.00299&  $1.9\\times{10^{-4}}$& 7.84\\\\\n\t\t\t\\hline\n\t\\end{tabular}}\n\t\n\t\\caption{ \\small{Inflationary predictions of the model for various values of $\\tilde{\\lambda}$, $\\xi$, $\\delta$ and $\\gamma$.  The number of e-folds is taken to be $N=60$.}}\n\t\\label{mastertable}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\\subsection{INFLATION ALONG S DIRECTION}\n\nHere we briefly discussed  the case where the $S$ field has the r\\^ole of the inflaton. In the potential (\\ref{fullpotential}) we put $\\langle\\nu_H\\rangle=0$ and \n$\\langle \\ov{\\nu}_{H}\\rangle=0$ so we have:\n\\begin{equation}\n\\begin{split}\nV= \\frac{144 \\tilde{\\kappa}^{2} S^{2}\\left( \\frac{m}{2 \\tilde{\\kappa}} - S^{2}\\right)^{2}}{\\left(3 - S^{2}\\right)^{2}}.\n\\end{split}\n\\end{equation}\nIn order to remove the singularity of the denominator, we take $m=6 \\tilde{\\kappa}$. In this case we get the following simple form\n\n\\begin{equation}\\label{S_chaotic}\nV= 144 \\tilde{\\kappa} ^{2}S^{2}\n\\end{equation}\n\n\\noindent which is of the form of  a chaotic-potential.\n\nNow the kinetic energy is defined as,\n\\begin{equation}\n\\begin{split}\n\\mathcal{L}=\\frac{1}{2} K^{j}_{i} \\left(\\partial S\\right)^{2}  -144 \\tilde{\\kappa} S^{2} \\quad \\text{where}\n \\quad K^{j}_{i}=\\frac{\\partial^{2} K}{\\partial S \\partial S^{*}}=\\frac{9}{\\left(3-SS^{*}\\right)^{2}}~\\cdot \n\\end{split}\n\\end{equation}\nLet $S=\\dfrac{X}{\\sqrt{2}}$ then the potential in (\\ref{S_chaotic}) becomes, $V= 72 \\tilde{\\kappa} ^{2}X^{2}$, and from the coefficient of the \nkinetic energy term we can find  $X$ in terms of a canonical normalized field $\\chi$:\n\\begin{equation}\n\\begin{split}\n X =\\sqrt{6} \\tanh\\left(\\frac{\\chi}{\\sqrt{6}}\\right).\n\\end{split}\n\\end{equation}\nThe potential in terms of the canonical normalized field reads as\n\\begin{equation}\\label{ValongS}\nV= 432 \\tilde{\\kappa}^{2}  \\tanh^{2}\\left(\\frac{\\chi}{\\sqrt{6}}\\right),\n\\end{equation}\n\n\\noindent which is analogous to the conformal chaotic inflation model or T-Model inflation already mentioned before. Potentials for the T-Model inflation are given in Equation (\\ref{Tmodels}).  For $n=1$ the potential  become, $V(\\chi)=\\uplambda \\tanh^{2}\\left(\\frac{\\chi}{\\sqrt{6}}\\right)$, which is similar to our potential in (\\ref{ValongS}) for $\\uplambda=432 \\tilde{\\kappa}^{2}$. We can understand the inflationary behaviour in these type of models, by considering two cases. \n\n First for $\\chi\\geqslant1$, by \nwriting the potential in  exponential form we have\n\n\\begin{equation}\nV= \\uplambda \\left(\\frac{1-e^{-\\sqrt{\\frac{2}{3}} \\chi}}{1+e^{-\\sqrt{\\frac{2}{3}} \\chi}}\\right)^{2}=\\uplambda \\left(1-\\frac{2e^{-\\sqrt{\\frac{2}{3}} \\chi}}{1+2e^{-\\sqrt{\\frac{2}{3}} \\chi}}\\right)^{2}=\\uplambda\\left(1-2e^{-\\sqrt{\\frac{2}{3}} \\chi}\\right)^{2}\n\\end{equation}\n\n\\noindent and for large values of $\\chi$ we can write\n\n\\begin{equation}\nV\\simeq\\uplambda\\left(1-4e^{-\\sqrt{\\frac{2}{3}} \\chi}\\right)~,\n\\end{equation}\n\n\\noindent where $\\uplambda=432 \\tilde{\\kappa}^{2}$. The slow roll parameters in terms of the field $\\chi$ and for  large number of e-folds ($N$)  are\n\n\\begin{equation} \\label{hN}\n\\frac{d\\chi}{dN}=\\frac{V^{\\prime}}{V} =4\\sqrt{\\frac{2}{3}}e^{-\\sqrt{\\frac{2}{3}} \\chi}.\n\\end{equation} \n\n\n\t\n\\noindent Integrating~(\\ref{hN}) we have $\\int{e^{\\chi\\sqrt{2\/3}}d\\chi}=\\int{4\\sqrt{\\frac{2}{3}}dN}$, which gives the relation\n\\begin{equation}\\label{nF}\n\\begin{split}\ne^{-\\sqrt{\\frac{2}{3}}\\chi} =\\frac{3}{8 N}.\n\\end{split}\n\\end{equation}\n\n\n\\noindent Using the relation above  we have for the  slow-roll parameter $\\epsilon$ that,\n\\begin{equation}\n\\begin{split}\n\\epsilon=\\dfrac{1}{2}\\left(\\frac{V^{\\prime}}{V}\\right)^{2}=\\dfrac{1}{2}\\left(4\\sqrt{\\frac{2}{3}}e^{-\\sqrt{\\frac{2}{3}} \\chi}\\right)^{2}=\\frac{3}{4 N^{2}}.\n\\end{split}\n\\end{equation}\nSimilarly the second slow-roll parameter $\\eta$ is found to be,\n\\begin{equation}\n\\eta=\\left(\\frac{V^{\\prime\\prime}}{V}\\right)=-\\frac{1}{N}.\n\\end{equation}\nFinally, the predictions for the tensor-to-scalar ratio $r$ and the natural-spectral index $n_{s}$ are,\n\\begin{equation}\n\\begin{split}\nr=\\frac{12}{N^{2}}\\quad,\\quad n_{s}=1+2\\eta-6\\epsilon=1-\\frac{2}{N}-\\frac{9}{4 N^{2}}\n\\end{split}\n\\end{equation}\n\\\\\n\\noindent and for  $N=60$ e-foldings we get $n_{s} \\simeq 0.9673$ and $r \\simeq 0.0032$.\n\nRegarding the case with $\\chi \\eqslantless 1$, we can see from the expression (\\ref{ValongS}) that the potential reduces to a quadratic chaotic form. The tree-level inflationary predictions in this case are $\\left(n_{s},r\\right)\\approx \\left(0.967,0.130\\right)$, which are ruled out with the latest \\emph{Planck} $2015$ results. \n\nThe discussion above strongly depends on the assumption $m=6\\tilde{\\kappa}$ that we imposed on the potential in order to simplify it. If we consider small variations of this assumption similar to \\eqref{singularitycondition2} and modify the condition as, $m=6\\tilde{\\kappa}+\\delta$, we will see that the parameter $\\delta$ contributes only to $n_{S}$ while the tensor-to-scalar ratio $r$ remains constant. \n\n\n\n\\section{CONCLUSIONS}\n\n\nIn the present work we have studied ways to realise the inflationary scenario in a no-scale supersymmetric model \n based on the Pati-Salam  gauge group $SU(4)\\times SU(2)_L\\times SU(2)_R$, supplemented with a $Z_2$ discrete  symmetry. The spontaneous  \n breaking of the group factor $SU(4)\\to SU(3)\\times U(1)_{B-L}$  is realised via the $SU(4)$ adjoint $\\Sigma=(15,1,1)$  and the \n breaking of the $SU(2)_{R}$ symmetry is achieved by non-zero vevs of the neutral components $\\nu_{H}, \\ov{\\nu}_{H}$ of the  Higgs fields\n  $(4,1,2)_H$ and $(\\bar 4,1,2)_{\\bar H}$. \n\n We have considered a no-scale structure K\\\"ahler potential and  assumed that the  Inflaton field  is a combination of  \n $\\nu_{H}, \\ov{\\nu}_{H}$  and find that the resulting potential is similar with the one presented in \\cite{Ellis:2014dxa, Ellis:2016spb} \n but our parameter space  differs substantially.  Consequently,  there are qualitatively  different solutions which are presented \n and analysed in the present work. The results strongly depend on the parameter $\\xi$ and for various characteristic values of the latter\n we obtain  different types of inflation models. In particular, for $\\xi=0$ and canonical normalized field $\\chi\\geq{1}$, the potential \n reduces to Starobinsky model and for $\\xi=1$ the model receives a chaotic inflation profile. The results for $0<\\xi<1$ have been analysed in detail while reheating via the decay of the inflaton in right-handed neutrinos is discussed.\n\n We also briefly discussed the alternative possibility where the $S$ field has the r\\^ole of the inflaton. In this case, the potential is exponentially \n flat for $\\chi\\geq{1}$.  Similar conclusions can be drawn for the Starobinsky model. On the other hand for small $\\chi$ it reduces to  a quadratic potential.\n\nIn conclusion, the $SU(4)\\times SU(2)_L\\times SU(2)_R$ model described in this paper can provide inflationary predictions consistent with the observations. Performing a detailed analysis we have shown  that consistent solutions with the Planck data are found for a wide range of the parameter space of the model. In addition the inflaton can provide masses to the right-handed neutrinos and depending   on the value of reheating temperature and the right-handed\nneutrino mass spectrum thermal\nor non-thermal leptogenesis is a natural outcome. Finally we mention that, in several cases the tensor-to-scalar ratio $r$, a canonical measure of primordial gravity waves,  is close to$\\sim{10^{-2}}-10^{-3}$ and can be tested in future experiments.\n\n\n\n\\vspace{1cm}\n{\\bf \\large Acknowledgements}\\quad\\quad\n\n\\noindent The authors are thankful to George K. Leontaris, Qaisar Shafi, Tianjun Li and  Mansoor Ur Rehman for helpful discussions and useful comments. WA would like to thank the Physics Department at University of Ioannina for hospitality and for providing conducive atmosphere for research where part of this work has been carried out. WA  was  supported  by  the  CAS-TWAS  Presidents  Fellowship  Programme.\n\n\n\n\n\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\\par Let $g_1$ and $g_2$ be eigencuspforms of the same weight $k\\geq 2$ and level $M_1$ and \n$M_2$ respectively. Throughout, we fix a prime $p\\geq 5$, and $\\mathfrak{p}|p$ a prime in the ring of integers $\\mathcal{O}_L$ of a suitably large number field $L$ containing the field of Fourier coefficients generated by $g_1$ and $g_2$. Assume that the $g_i$ have trivial nebentype character\nand that all but finitely many of the Hecke eigenvalues of the $g_i$ are\ncongruent modulo $\\mathfrak{p}$. Then, we say that the newforms\n$g_i$ are $\\mathfrak{p}$-congruent. In the situation of $\\mathfrak{p}$-congruent newforms, there is a general \nphilosophy that the critical values of any L-function functorially associated\nto $g_1$ and $g_2$ will also be $\\mathfrak{p}$-congruent. More generally, one \nexpects the $\\mathfrak{p}$-adic L-functions of $g_1$ and $g_2$, if they exist, to also be $\\mathfrak{p}$-congruent. Furthermore, the corresponding $p$-primary Selmer groups defined over the cyclotomic $\\mathbb{Z}_p$-extension of $\\mathbb{Q}$ should also be related.\n\n\\par Before we state our results, we require more precise notation. Let $g_1=\\sum_{n\\geq 1} a(n, g_1) q^n$ and \n$g_2 =\\sum_{n\\geq 1} a(n, g_2) q^n$ be the Fourier expansions\nof the $g_i$. We make the following assumptions for $i=1,2$:\n\\begin{itemize}\n    \\item $g_i$ is not of CM-type, and has trivial central character;\n    \\item $g_i$ is $\\mathfrak{p}$-ordinary, i.e., that $a(p, g_i)$ is a $\\mathfrak{p}$-adic unit;\n    \\item $g_i$ are $p$-stabilized newforms, meaning that the level\n$M_i$ of $g_i$ is divisible by precisely the first power of $p$; that each $g_i$ is an eigenvector for the $p$-th Hecke\noperator $U_p$ (with unit eigenvalue), and that \nthe level of the newform\nassociated to $g_i$ has level either $M_i$ or $M_i\/p$; and\n\\item the Galois representation into $\\op{GL}_2(\\bar{\\mathbb{F}}_p)$ associated to\n    $g_i$ is  irreducible and $\\mathfrak{p}$-distingushed.\n\\end{itemize}\n\n\\par Let $\\mathcal{O}$ denote the completion of $\\mathcal{O}_L$ at $\\mathfrak{p}$ and write $K$ for\nthe fraction field of $\\mathcal{O}$. The newforms $g_1$ and \n$g_2$ are said to be \\emph{$\\mathfrak{p}$-congruent} if $a(q, g_1)\\equiv a(q, g_2)\\mod{\\mathfrak{p}}$ for all primes \n$q\\nmid M_1 M_2 p$. We simply say that $g_1$ and $g_2$ are $p$-congruent if they are $\\mathfrak{p}$-congruent for some prime \n$\\mathfrak{p}|p$. \n\n\n\nIn \\cite{GV00}, Greenberg and the third named author of the present work studied the main conjecture of Iwasawa theory for \nstandard 2-dimensional representations\n$\\rho_i$ \nattached to $p$-congruent modular forms $g_i$ as above. In this case, the $p$-adic L-functions of the $g_i$ \nare the well-known $p$-adic L-functions arising from modular symbols, and \nthe Selmer groups are Greenberg's $p$-ordinary Selmer groups, arising \nfrom Galois cohomology. In this situation, the main results of \\cite{GV00}\nshow that the Selmer groups and $p$-adic L-functions inherit congruence properties\nfrom the congruent modular forms, just as predicted by the general philosophy.\nMore specifically, the authors of \\cite{GV00} \nstudied the relationship between the Iwasawa invariants associated to Galois representations arising from Hecke eigencuspforms \nthat are residually isomorphic,\nand proved certain explicit formulae relating the values of these invariants.\nThese results were used to deduce certain cases of the Main Conjecture of Iwasawa\ntheory, by combining the formulae for the invariants with\ndeep results of Kato.\n\nThe primary goal of the present paper is to generalize the explict relationship\nbetween the Iwasawa invariants of the congruent forms $g_1, g_2$ from \nthe case of the standard representation of dimension 2 to the case of\nthe symmetric square representation, which has dimension 3. A secondary \naccomplishment in this paper is the complete proof of the integrality  \nfor the $p$-adic functions of degree 3, since this result (although seemingly known to experts)\nseems not to be found in the literature. Implict in this discussion\nof integrality is a careful normalization of the periods appearing in the definition\nof the $p$-adic L-function. This is a subtle point: it turns out to be\nquite difficult to show that the normalization which gives rise to congruences\ncoincides with the canonical normalization given by Hida. We show that Hida's period\ngives the correct congruences if a certain variant of Ihara's lemma holds. This lemma \nis unknown in weight $k >2$ in the generality which we require, although it holds \nunconditionally in weight 2. We remark also that all the results in this paper are much\neasier to prove in the case of weight $2$ and the main novelty lies in the results for higher \nweight. \n\n\n\nTo continue, we require more notation. Let the $g_i$ be as above.\nLet $\\mathbb{Q}_{\\op{cyc}}$ denote the cyclotomic $\\mathbb{Z}_p$-extension of $\\mathbb{Q}$, i.e., the unique $\\mathbb{Z}_p$-extension of $\\mathbb{Q}$ contained in $\\mathbb{Q}(\\mu_{p^\\infty})$. Let \n$\\Lambda=\\mathcal{O}[[\\text{Gal}(\\mathbb{Q}_{\\op{cyc}}\/\\mathbb{Q})]]$ denote the usual\nIwasawa algebra.  For $i=1,2$, let\n\\[\\rho_{g_i}:\\op{Gal}(\\bar{\\mathbb{Q}}\/\\mathbb{Q})\\rightarrow \\op{GL}_2(\\mathcal{O})\\]be the associated Galois representation.\nIf we fix a rank-$2$ Galois-stable $\\mathcal{O}$-lattice $T_{g_i}$ in the $p$-adic representation associated to $g_i$, normalized as\nin \\cite{lz16},\nwe may view the representations $\\rho_{g_i}$ as taking values in $\\mathcal{O}$.  For $i=1,2$, the residual representation is \ndenoted $\\bar{\\rho}_{g_i}:=\\rho_{g_i}\\mod{\\mathfrak{p}}$. \nSince the modular forms $g_1$ and $g_2$ are $\\mathfrak{p}$-congruent,\nand since we will assume throughout this paper that $\\bar{\\rho}_{g_1}$ and $\\bar{\\rho}_{g_2}$ are \\emph{absolutely irreducible}, \nwe find that the the semisimplifications $\\bar{\\rho}_{g_1}^{\\op{ss}}$ \nand $\\bar{\\rho}_{g_2}^{\\op{ss}}$ are isomorphic.\n\n\nLet $\\psi$ denote a Dirichlet character of conductor $c_\\psi$, where $(c_\\psi, 2pM_1M_2)=1$. In this paper we will assume that\n\\begin{itemize}\n    \\item $\\psi$ is even, and non-quadratic, and\n    \\item the coefficient field $K$ contains the values of $\\psi$\n\\end{itemize}\nLet\n$r_{g_i}=\\op{Sym}^2(\\rho_i)$ denote the symmetric square representation\nfor $g_i$, with $i=1,2$, viewed as taking values in the symmetric square\nof the lattice $T_{g_i}$. \nIn this setting the representations\n$r_{g_i}\\otimes\\psi:\\op{Gal}(\\bar{\\mathbb{Q}}\/\\mathbb{Q})\\rightarrow \\op{GL}_3(\\mathcal{O})$ are residually isomorphic. Let $\\mathbf{A}_{i, \\psi}$ be the \n$p$-primary \nrepresentation associated to the underlying Galois stable $\\mathcal{O}$-lattice for $r_{g_i}\\otimes\\psi$. Note that since $g_i$ is \n$p$-ordinary, \nso is $r_{g_i}\\otimes\\psi$. We work with the primitive Selmer group $\\op{Sel}_{p^\\infty}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})$ as defined by Greenberg in \n\\cite{Gre89}. It is shown by Loeffler and Zerbes in \\cite{lz16} that $\\op{Sel}_{p^\\infty}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})$ \nis $\\Lambda$-cotorsion and this allows us to define a nonzero algebraic $p$-adic L-function \n$L^\\text{alg}(r_{g_i}\\otimes\\psi)\\in \\Lambda$. We also point out that the existence of $L^\\text{alg}(r_{g_i}\\otimes\\psi)\\in \\Lambda$ and part of the main conjecture is \nproven, in some cases, in unpublished\nwork of Urban \\cite{urban06}. The arguments in this paper do not assume the main conjecture. By the Weierstrass preparation theorem, \n\\[L^\\text{alg}(r_{g_i}\\otimes\\psi)=p^{\\mu} a(T) u(T),\\] where $a(T)$ is a distinguished polynomial and $u(T)$ is a unit in $\\Lambda$. The $\\mu$-invariant $\\mu^{\\op{alg}}(r_{g_i}\\otimes\\psi)$ is the number $\\mu$ in the above factorization, and the $\\lambda$-invariant $\\lambda^{\\op{alg}}(r_{g_i}\\otimes\\psi)$ is the degree of $a(T)$.\n\n\\par Next, we define a primitive $p$-adic L-function $L^\\text{an}(r_{g_i}\\otimes\\psi)\\in \\Lambda$, for $i=1,2$.\nThis is essentially done in old work of Schmidt and others; see \\cite{schmidt88} for the basic source,\nand the discussion in \\cite{lz16} for an account of the various refinements. For the most part, we shall adopt the notation of \\cite{lz16}, which is different from \nthat of \\cite{schmidt88}. The reason for this is\nthat the authors of \\cite{lz16} discuss and define the \n Selmer groups corresponding to their L-functions,\n while there is no convenient reference for the\nSelmer groups corresponding correctly to the L-functions as normalized in \\cite{schmidt88}. Under the present hypotheses, Schmidt proves the existence of an element $L^\\text{an}(r_{g_i}\\otimes\\psi)\\in \\Lambda\\otimes\\mathbb{Q}$ satisfying\na certain interpolation property with respect to special values of the complex symmetric square L-function. The authors of \\cite{lz16}\nprovide a convenient summary of the rather complicated history of this result. The interpolation property defining\nthe $p$-adic L-function\nis given at the end of Section 2 below. \n\nIt is important to remark that the definition of  $L^\\text{an}(r_{g_i}\\otimes\\psi)$\npresupposes the choice of certain transcendental period, and that the convention that we use differs in one important respect\nfrom that of  \\cite[section 2]{lz16}. As a result of our convention, the $p$-adic L-function we work with is actually contained in $\\Lambda$. We discuss this normalization in further detail in Section 2 of this paper, and summarize the key points later in this introduction.\n According to our normalization, the main conjecture then predicts the equality\n$$L^\\text{alg}(r_{g_i}\\otimes\\psi)= u(T)\\cdot L^\\text{an}(r_{g_i}\\otimes\\psi)\\in \\Lambda$$\nwhere $u(T)$ is a unit in $\\Lambda$. We remark that the congruence ideal which appears in the statement of the Main\nConjecture in \\cite{lz16} does not play a role here, since our definition of the period incorporates this factor. We remark also that in the convention of \\cite{lz16}, $\\Lambda$ refers to the completed\ngroup algebra of ${\\operatorname{Gal}}(\\mathbb{Q}(\\mu_{p^\\infty})\/\\mathbb{Q})\\cong\\mathbb{Z}_p^\\times$, whereas we have taken $\\Lambda$ to be the group algebra\nof ${\\operatorname{Gal}}(\\mathbb{Q}_{\\op{cyc}}\/\\mathbb{Q})\\cong \\mathbb{Z}_p\\cong 1+p\\mathbb{Z}_p$. In other words, we do not cover the case of nontrivial tame cyclotomic twists. The methods\nof this paper apply equally well in this excluded case, but we have chosen to avoid it, mostly for simplicity. \n\nAssuming the integrality of the $p$-adic L-function, we have \nwell-defined Iwasawa invariants \n$\\lambda^\\text{an}(r_{g_i}\\otimes\\psi), \\mu^\\text{an}(r_{g_i}\\otimes\\psi)\n\\in \\mathbb{Z}_{\\geq 0}$ on the \nanalytic side as well. The main conjecture implies that \n$$\\lambda^\\text{an}(r_{g_i}\\otimes\\psi)=\\lambda^{\\op{alg}}(r_{g_i}\\otimes\\psi)$$ and\n$$\\mu^\\text{an}(r_{g_i}\\otimes\\psi)\n=\\mu^{\\op{alg}}(\\rho_{g_i}\\otimes\\psi)$$ for $i=1,2$. \n\nOur goal is to relate the invariants for the congruent forms $g_1$ and $g_2$. However, the primitive invariants\nas defined above are not related at all; it is quite possible for the primitive invariants to be trivial in one case \nyet highly nontrivial in the other. The correct relationship, as discovered in \\cite{GV00}, is between\n\\emph{imprimitive} Iwasawa invariants, which we now proceed to define. A key to our construction \nis the specification of certain sets $\\Sigma$ of primes, which we now proceed to elaborate:\nwe let  $\\Sigma$ denote a finite set of prime\nnumbers $q\\neq p$ such that\n\\begin{itemize}\n\\item $2\\in\\Sigma$;\n\\item if $q\\in \\Sigma$ is odd, then $\\op{min}(\\op{ord}_q(M_1),\\op{ord}_q(M_2))< 2$, and \n\\item if $q\\vert M_1M_2$ is such that \n $\\op{min}(\\op{ord}_q(M_1),\\op{ord}_q(M_2))< 2 $, then $q\\in\\Sigma$.\n \\end{itemize}\nIn other words, if we write $a_i=\\op{ord}_q(M_i)$, then $\\Sigma$ includes $2$, and all odd $q$ for which $\\{a_1, a_2\\} = \\{0, n\\}$ or $\\{1, n\\}$ as \nunordered sets, for any $n>0$, as well\nas some other odd primes for which $a_1=a_2=0$.  Then one has the following basic lemma, which seems to be \ndue to Livn\u00e9 \\cite{livne}; it is stated in the form we need\nby Carayol \\cite{carayol}, page 789. \n\\begin{Lemma}\n\\label{dtlemma}\nFor an odd prime $q\\in \\Sigma$, we have that $\\op{max}(\\op{ord}_q(M_1),\\op{ord}_q(M_2))\\leq 2$.\nFurthermore, if $q\\notin\\Sigma$, then $\\op{ord}_q(M_1)= \\op{ord}_q(M_2)$. For $q=2$, we either\nhave  $\\op{max}(\\op{ord}_2(M_1),\\op{ord}_2(M_2))\\leq 2$, or else $\\op{ord}_2(M_1)=\\op{ord}_2(M_2)$.\n\\end{Lemma}\nThe essential point is that the integers $M_i$ are not too far from the Artin conductor\nof their common residual representation, and hence from each other. Otherwise stated, the \\emph{only} way the exponents\n$a_1, a_1$ can be different at \\emph{any} prime $q$, including $q=2$, is if $\\{a_1, a_2\\} $ is one of $\\{0, 1\\}$ or $\\{1, 2\\}$ or $\\{0, 2\\}$, \nas unordered sets.\nAn examination of Livne's proof shows that this follows from the fact\nthat the Swan conductor of the restriction of $\\rho_i$ to $\\op{Gal}(\\bar{\\mathbb{Q}}_q\/\\mathbb{Q}_q)$  coincides\nwith the Swan conductor of its reduction, since \nthe wild ramification is a pro-$q$ group. The importance of this lemma in our work cannot\nbe overstated -- our arguments for $p$-adic L-finctions\nwould fail if there were\nsome prime $q$ at which the levels of $g_1$ and $g_2$ differed by a high\npower. \n\nLet $N$ denote the least common multiple of $M_1, M_2$, and $\\prod_{q\\in\\Sigma} q^2$. In view of the lemma above, the \nprime factorization of $N\/M_i$ is of the form $\\prod_{q\\in\\Sigma}q^{e_i},$ with $e_i\\leq 2$. Let $S$\nbe the set of primes $q| N$ and the primes dividing the conductor of $\\psi$ (which is away from from $2N$). \nSet $S_0:=S\\backslash \\{p\\}$. For each $i$,\nwe have an imprimitive Selmer group, obtained by relaxing the local\nconditions at all the primes $q\\in S_0$. It is shown\nin \\cite{GV00} that the imprimitive\nSelmer group is cotorsion if and only if the primitive Selmer group\nis so. Thus, once again we have imprimitive invariants\n$\\lambda^{\\op{alg}}_{S_0}(r_{g_i}\\otimes\\psi)$ and $\\mu^{\\op{alg}}_{S_0}(r_{g_i}\\otimes\\psi)$. The basic result is the\nfollowing:\n\n\\begin{Proposition}\n\\label{algebraic-invariants-intro}\nThe following statements hold:\n\\begin{itemize}\n    \\item $\\mu_{S_0}^{\\op{alg}}(r_{g_i}\\otimes\\psi)=\n   \\mu^{\\op{alg}}(r_{g_i}\\otimes\\psi)$, and\n  \\item  $\\lambda_{S_0}^{\\op{alg}}(r_{g_i}\\otimes\\psi)=\n   \\lambda^{\\op{alg}}(r_{g_i}\\otimes\\psi) + \\sum_{q\\in S_0} \\sigma_i^{(q)}(\\psi).$\n\\end{itemize}\n\\end{Proposition}\n\nHere the integers $\\sigma_i^{(q)}(\\psi)$ are the degrees of certain polynomials\ncoming from applying the Weierstrass preparation theorem to the \nannihilators of certain local cohomology groups, which are known\nunconditionally to be torsion, and whose annihilators can be \ndescribed explicitly in terms of Euler factors. \nMost of this is carried through from \\cite{GV00}, where an analysis of the local conditions\nis made under very mild hypotheses. \n\nSimilar considerations apply to the $p$-adic L-function. Following\na construction due originally to Coates, Hida, and Schmidt, there\nexists an imprimitive $p$-adic L-function $L^{\\op{an}}_\n\\Sigma(r_{g_i}\\otimes\\psi)\\in \\Lambda$, with corresponding\ninvariants $\\lambda_{\\Sigma}^{\\op{an}}(r_{g_i}\\otimes\\psi)$ and $\\mu^{\\op{an}}_{\\Sigma}(r_{g_i}\\otimes\\psi)$. This L-function\nis produced by interpolation of special values of a degree three Euler product given by Shimura (see (\\ref{shimura-euler-product}) below)\n and deleting Euler factors\nat the primes in $\\Sigma$. An important observation is that, at each prime $q\\notin S_0\\backslash \\Sigma$, the levels\n$M_1$ and $M_2$ are divisible by $q^2$ . This is by our choice of $\\Sigma$. It follows from this, and  \nour assumption that the forms have trivial central character, that the Euler factor at  $q\\in S_0\\backslash \\Sigma$ in \nShimura's Euler product  \n  is trivial.  We will discuss this point further in the sketch of proof.\n  Therefore, we have  $L^{\\op{an}}_\\Sigma(r_{g_i}\\otimes\\psi)= L^{\\op{an}}_{S_0}(r_{g_i}\\otimes\\psi)\\in \\Lambda$ \nand \\[\\lambda_{\\Sigma}^{\\op{an}}(r_{g_i}\\otimes\\psi)=\\lambda_{S_0}^{\\op{an}}(r_{g_i}\\otimes\\psi)\\text{ and } \\mu^{\\op{an}}_{\\Sigma}(r_{g_i}\\otimes\\psi)=\\mu^{\\op{an}}_{S_0}(r_{g_i}\\otimes\\psi).\\] \n\nWe remark that\nthe logical structure of the argument to construct $p$-adic L-functions\nis the opposite of what one might imagine -- one starts with \nthe imprimitive L-function and then passes to the primitive object\nby dividing out by explicit Euler factors. The fact that the \nresulting primitive L-function, \\emph{a priori} meromorphic,\nis actually analytic, is proven in \\cite{schmidt88}, under our\nhypotheses that the $g_i$ are not CM forms.\n\nOur main result  in Section 2 is the following. As stated here, the result is dependent\non the validity of a certain variant of Ihara's lemma (see Hypothesis \\ref{hypothesis ihara}). The theorem\nholds unconditionally for $k=2$, and even for $k>2$, one can make an unconditional statement at the cost\nof introducing a certain ambiguity in the choice of periods. The only role played by Ihara's lemma\nis to remove the dependence of the period of the imprimitive L-function on the choice of the auxiliary\nprimes in the level, in the sense that the periods appearing in the imprimitive L-function may be different\nfrom those appearing in the primitive one. \n\\begin{Proposition} Assume that Hypothesis \\ref{hypothesis ihara} holds. The following statements hold.\n\\label{analytic-invariants-intro}\n\\begin{itemize}\n    \\item $\\mu_{S_0}^\\text{an}(r_{g_i}\\otimes\\psi)=\n   \\mu^{\\op{an}}(r_{g_i}\\otimes\\psi)$, and\n  \\item  $\\lambda_{S_0}^{\\op{an}}(r_{g_i}\\otimes\\psi)=\n   \\lambda^{\\op{an}}(r_{g_i}\\otimes\\psi) + \\displaystyle\\sum_{q\\in S_0} \\sigma_i^{(q)}(\\psi).$\n\\end{itemize}\n\\end{Proposition}\n\n\\noindent where the integers $\\sigma_i^{(q)}(\\psi)$ are the \\emph{same} as the ones occurring\nin the algebraic case, since, the Euler factors in the algebraic\nand analytic sides are exactly the same. This is a deep fact: \nthe equality between the Galois-theoretic Euler factors \nin the algebraic case and the complex Euler factors in the analytic\nL-function is the local Langlands correspondence for the \n3-dimensional representations $r_{g_i}\\otimes\\psi$; see\n\\cite{GJ78}, or \\cite{schmidt88}. \n\nWith this preparation, we may now state our remaining results. Recall\nthat the character $\\psi$ is assumed to be \neven and non-quadratic, and to have conductor $c_\\psi$ coprime to $M_1M_2$,\nas well as to the primes in $\\Sigma$. \n\n\\par Our first task is to deal with the integrality of the $p$-adic L-functions, since\nthis integrality property  is a folklore result for which a complete proof seems \nnot to have ever been written down. It is stated as Proposition 2.3.5\nin \\cite{lz16}, where it is attributed to Hida, but no reference is given. \nDespite searching many papers of Hida, we were only able to find a discussion for the case of weight $2$, in some notes from an instructional\nconference in India. Therefore, we provide a through discussion of integrality, and give an integral construction valid for all weights. The proof turns out to be somewhat delicate, once the weight is large compared to\n$p$. For the convenience of the reader, we will give the definition of the canonical period (associated to a choice of level) as part of the sketch of the proof later on in this introduction. \n\n\\begin{Th}\n\\label{integrality-thm-intro} \nLet the notation be as above.\nThen the primitive L-functions $L^\\text{an}(r_{g_i}\\otimes\\psi)\\in \\Lambda$ are integral, normalized with Hida's canonical period, as in \\cite{lz16}. \nThe imprimitive $p$-adic L-functions \n$L^\\text{an}_\\Sigma(r_{g_i}\\otimes\\psi)= L^\\text{an}_{S_0}(r_{g_i}\\otimes\\psi) \\in \\Lambda$ are integral as well, for the same choice\nof periods as in the primitive case. \n\\end{Th}\n\n\nWith regard to the Iwasawa invariants, our result is as follows. Once again, we include the Ihara Lemma\nas a hypothesis, since formulating a general result without it would give a clumsy statement. Consider the pair\n$g_1, g_2$ of $p$-congruent forms, of level $M_1, M_2$ respectively. Let $S$ denote any set of primes $q$ containing\n$q=2$ and all primes dividing $M_1M_2$. Let $S_0=S\\backslash \\{p\\}$. \n\\begin{Th}\n\\label{intro-thm}\nLet the notation be as above, and assume Hypothesis \\ref{hypothesis ihara}.\nThen the following statements hold.\n\\begin{enumerate}\n\\item If $\\mu_{S_0}^\\text{an}(r_{g_1}\\otimes\\psi) = 0$, we have\n$\\mu_{S_0}^\\text{an}(r_{g_2}\\otimes\\psi) = 0$, and \n$\\lambda_{S_0}^\\text{an}(r_{g_1}\\otimes\\psi) =\n\\lambda_{S_0}^\\text{an}(r_{g_2}\\otimes\\psi)$\n\\item If $\\mu_{S_0}^{\\op{alg}}(r_{g_1}\\otimes\\psi) = 0$, we have\n$\\mu_{S_0}^{\\op{alg}}(r_{g_2}\\otimes\\psi) = 0$, and \n$\\lambda_{S_0}^{\\op{alg}}(r_{g_1}\\otimes\\psi) =\n\\lambda_{S_0}^{\\op{alg}}(r_{g_2}\\otimes\\psi)$\n\\item If $\\mu_{S_0}^\\text{an}(r_{g_1}\\otimes\\psi) =  \\mu_{S_0}^{\\op{alg}}(r_{g_1}\\otimes\\psi) = 0$, and\n$\\lambda_{S_0}^\\text{an}(r_{g_1}\\otimes\\psi)=\n\\lambda_{S_0}^{\\op{alg}}(r_{g_1}\\otimes\\psi)$, then\n$\\mu_{S_0}^\\text{an}(r_{g_2}\\otimes\\psi) =  \\mu_{S_0}^{\\op{alg}}(r_{g_2}\\otimes\\psi) = 0$, and\n$\\lambda_{S_0}^\\text{an}(r_{g_2}\\otimes\\psi)=\n\\lambda_{S_0}^{\\op{alg}}(r_{g_2}\\otimes\\psi)$\n\\end{enumerate}\n\\end{Th}\n\n\nThe theorem holds unconditionally for $k=2$. As we have remarked, it is possible to give an unconditional statement for all weights, \nif one is willing to let the period depend on the level, or if $p >k$.  The essential point is that there exists\n\\emph{some} natural choice of periods \nthat gives ride to a congruence\nof $p$-adic L-functions, but it is not clear, without\nthe additional Hypothesis, \nthat the periods coincide with those specified by\nHida as in the previous theorem.\n\nIt is clear from the relationships given in Propositions \\ref{algebraic-invariants-intro} and \\ref{analytic-invariants-intro}, that the third statement follows from\nthe first two. Furthermore, it follows from the third statement\nthat if one knows the main conjecture and vanishing of the $\\mu$-invariants for $g_1$, that the same conclusions follow for\n$g_2$. Examples where the main conjecture is known for a particular\nform may be found in \\cite{lz16}. \n\nTo end this introduction, we give a sketch of the arguments and indicate the various difficulties\nand novelties needed to overcome them.\n\n\nThe main difficulties occur on the analytic side, starting with the integrality property described above.\nFurthermore, the proof of the required congruences of $p$-adic L-functions turns out to be \nsomewhat more delicate than in the case of \\cite{GV00}, and relies crucially on Lemma \\ref{dtlemma}\nand the very specific set $\\Sigma$ we have chosen. \nTo get the required results, one has to redo the Coates-Hida-Schmidt construction of the $p$-adic \nL-function, which goes back almost 25 years, \nand apply various subtle refinements that were not available at that time. \n\nWe briefly recall the steps in the construction. For notational simplicity, assume that $g=g_i$ is a $p$-stabilized\nnewform for some \nfixed value $i\\in \\{1,2\\}$ and set $M:=M_i$. Then $M$ is divisible by precisely the first power of $p$. \nIf  $g=\\sum a(n ,g)q^n$, then the Dirichlet series $\\sum_n a(n, g)n^{-s}$ has an Euler product of the form\n$$\\prod_q(1-\\alpha_qq^{-s})^{-1}(1-\\beta_qq^{-s})^{-1}$$\nwith certain parameters $\\alpha_q, \\beta_q$ at each prime $q$ (including $p$). If $q$ divides $M$, then\n one or both of these parameters may be zero. \n \nNow let $\\chi=\\psi \\eta$ be an even Dirichlet character of conductor $c_\\psi p^r$. \nHere, $\\eta$ is a finite order character of conductor $c_\\eta=p^r$. We assume that\n$\\chi$ is not quadratic. Let $T$ denote any set of prime numbers such that $p\\notin T$. \nThen the $T$-imprimitive naive symmetric square $L$-function of the  is defined as follows:\n\n\\begin{equation}\n\\label{shimura-euler-product}\n\\mathscr{L}_T(r_g\\otimes\\psi, s) = \n\\prod_{q\\notin T} \\left( (1-\\chi(q)\\alpha_q\\beta_q q^{-s})(1-\\chi(q)\\beta_q^2q^{-s})(1-\\chi(q)\\alpha_q^2q^{-s})\\right)^{-1},\n\\end{equation}\nwhere $\\alpha_q$ and $\\beta_q$ are determined from the degree 2 Euler product as above. \nObserve that since $g$ is assumed to have trivial central character, the Euler factor above is trivial as soon as $q^2$ divides the level $M$. \nThus we can enlarge the set $T$ by including all primes $q$ with $q^2\\vert M$ without changing the L-function. When $T$ is the empty\nset, we denote the resulting function as $D_g(\\chi, s)$. This is nothing but the function denoted by $D(s, g, \\chi)$ by Shimura in \\cite{shimura-holo}.  We remark  that Shimura never uses the fact that $g$ is a $p$-stabilized newform; rather he uses only that the standard degree 2 L-function\nassociated to $g$ admits an Euler product of the above shape, to get the parameters $\\alpha_q, \\beta_q$. This will be important below in dealing\nwith the imprimitive situatiuon. \n\nNow fix a set $\\Sigma$ satisfying the conditions above. In practice $\\Sigma$ will depend on $g_{i+1}$, where we read the indices modulo\n$2$, but we do not need that here.  If $g=\\sum a(n ,g)q^n$, let $f=\\sum_n a(n, f)q^n$, where $a(n, f)=0$ if $n$ is divisible by\nany prime in $\\Sigma$, and $a(n, f) = a(n, g)$ if not.  Let $N$ denote the level of $f$. Under our conditions on $\\Sigma$, \nwe have that if $q\\vert N\/M$, then $\\text{ord}_q(N)= 2$.  The form $f$  is an eigenvector for all the Hecke operators of level $N$, and \nthe eigenvalue of $U_q$ on $f$ is zero, for every prime $q\\neq p$ dividing $N$. \nThe standard $L$-function of $f$ admits an Euler product, and we get integers $\\alpha_q', \\beta_q'$ associated to $f$ just as before,\nso that $\\alpha_q=\\alpha'_q, \\beta_q=\\beta'_q$ if $q\\notin\\Sigma$, and $\\alpha_q=\\beta_q=0$ if not. \n\n\nThen we can follow Shimura and \ndefine a degree three Euler product for $f$ just as above, with $\\alpha_q',\\beta_q'$ instead of $\\alpha_q, \\beta_q$. This time \nwe take $T$ to be the empty set, so we get Shimura's $D_f(\\chi, s)$. \nIt is easy to see that  $D_f(\\chi, s)$ is nothing but $L_\\Sigma(r_g\\otimes\\psi, s) = L_{S_0}(r_g\\otimes\\psi, s)$. \nWith these notations, the imprimitive $p$-adic L-function $L^\\text{an}_\\Sigma(r_{g}\\otimes\\psi)=L^\\text{an}_{S_0}(r_{g}\\otimes\\psi)$ \nis defined via interpolation\nof  $L^\\text{an}_\\Sigma(r_{g}\\otimes\\psi, s) = L^\\text{an}_{S_0}(r_{g}\\otimes\\psi, s) = D(f,\\chi, s)$ \nat the critical values of $s$, namely, for $s=n\\in\\mathbb{Z}$ with\n$n$ odd and $0<  n < k$, and for $\\eta$ varying over cyclotomic characters of $p$-power\norder.\n\n\\begin{Remark}   \nSince it is a somewhat confusing point, we remark that the $L$-function $D_g(\\chi, s)$\nis \\emph{not} the primitive $L$-function  $\\mathscr{L}(r_g\\otimes\\psi, s)$ of degree three attached to the symmetric square lift of $g$ to $\\op{GL}_3$.\n This is because $\\mathscr{L}(r_g\\otimes\\psi, s)$ may have nontrivial Euler factors at the primes dividing the level $M_0$, even at primes $q$ with\n$\\alpha_q=\\beta_q=0$, so that Shimura's Euler\nproduct has the factor $1$. \nThus\nthe term ``imprimitive'' is used in \\cite{lz16} to refer \nto $D_g(\\chi, s)=\\mathscr{L}_T(r_g\\otimes\\psi, s)$ when $T$ is the empty set.  Our $D_f(\\chi, s)$ is even less primitive than  the already defective\n $D_g(\\chi, s)$. The function $D_f(\\chi, s)$ does not appear in \\cite{lz16}. \n\\end{Remark}\n\nFor the present, we focus on $D_f(\\chi, s)$. The starting point is Shimura's formula expressing \n $D_f(\\chi, s)$ in terms of the Petersson inner product of a theta series and an Eisenstein series:\n    \\[(4\\pi)^{-s\/2}\\Gamma(s\/2) D_f(\\chi, s) = \\langle f, \\theta_{\\overline\\chi}(z)\\Phi(z,\\overline\\chi, s)\\rangle_{N_\\chi},\\]see \\eqref{sturm relation} and the discussion preceding it. \n    \n\\par For an odd integer $n$ in the range\n$1\\leq n\\leq k-1$, set $H_{\\overline\\chi}(n) := \\theta_{\\overline\\chi}(z)\\Phi(z,\\overline\\chi, n)$. Shimura in \\cite{shimura-holo} has shown that $H_{\\overline\\chi}(n)$ is a nearly holomorphic modular form of level\n$N_\\chi$, weight $k$, and trivial character. Our first job is to prove that the values in question\nare algebraic and integral, when divided by Hida's canonical period. The algebraicity is well-known, and was basically \nproven by Shimura himself:\nit follows from the fact that the Fourier coefficients of $H_{\\overline\\chi}(n)$\ncan be calculated explicitly, and turn out to be algebraic (and even integral). Then one \nhas to replace the nearly holomorphic form with its holomorphic projection. This projection will\nhave algebraic Fourier coefficients, so Shimura's method shows\nthat the transcendental part of the inner product $(f, H_{\\overline\\chi}(n))$\nis just the Petersson inner product of $f$ with itself. This procedure is carried out in \\cite{schmidt86},\nand various other works. However, a certain amount of work beyond that of Shimura is required to deal with possible vanishing of the \nspecial values owing to zeroes of the Euler factors. \n\n\\begin{Remark} We note that  Schmidt works with $D_g(\\chi, s)$ and not $D_f(\\chi, s)$. \nHowever, it is clear that once one has a $p$-adic L-function interpolating values of $D_g(\\chi, s)$,\nthat one can get an interpolation of $D_f(\\chi, s)$: one has only to multiply by the finitely many Euler factors by which these objects\ndiffer. Alternatively, one can simply verify that Schmidt's construction works for $D_f(\\chi, s)$; this is essentially\ncarried out in Section 2 below. The hard part of Schmidt's work is to go from $D_g(\\chi, s)$ to the primitive $\\mathscr{L}(r_g\\otimes\\psi, s)$,\nwhich requires \\emph{division} by certain Euler factors, and it is highly nontrivial to show that the quotient is analytic. \nIn this paper, we shall take for granted the existence of the imprimitive\n$p$-adic L-function $L^{\\op{an}}_{S_0}(r_g\\otimes\\psi)$ interpolating the special values of $D_f(\\chi, s)$. \n\\end{Remark}\n\nWe now continue with the sketch, and give some idea of what is involved in normalizing the periods and verifying integrality.\nAn unfortunate feature of Schmidt's construction is that his holomorphic projection destroys integrality, since the holomorphic projection \nin general introduces\ndenominators dividing $k!$. This is the reason why Schmidt and subsequent authors are unable to\nprove integrality. To get around this, we have to change tactics, and use methods from $p$-adic\nmodular forms -- we replace the holomorphic projection with the ordinary projection, which is denominator-free. This is enough for our purposes, since we are dealing with $f$ ordinary. \n\nIn consideration of integrality, one has to of course be careful about the exact periods which\nappear. As is well-known (see the statement of Proposition 2.3.5 of \\cite{lz16}) one cannot\nsimply use the Petersson norm of $f$ or $g$ -- one has to scale by a certain congruence number. We now explain the definition of the periods, and show how the congruence number manifests itself. \n\nThe key idea (due to Hida) is that the Petersson inner product is related to a certain algebraic inner\nproduct, up to scalar multiple. Let $S_k(N,\\mathcal{O})$ be the space of cusp forms of weight $k$ and level $N$ with coefficients in $\\mathcal{O}$, \nand $\\textbf{T}$ the ring generated by Hecke operators acting on $S_k(N,\\mathcal{O})$. Let $\\mathcal{P}_f$ be the kernel of the map $\\textbf{T}\\rightarrow \\mathcal{O}$ associated to $f$ and $\\mathfrak{m}$ the unique maximal ideal of $\\textbf{T}$ generated by $\\mathcal{P}_f$ and $\\mathfrak{p}$. We have assumed that the residual representation associated to $f$ is absolutely irreducible, ordinary, and $p$-distinguished, so\nit follows that $\\textbf{T}_{\\mathfrak{m}}$ is Gorenstein. This induces an algebraic duality pairing \n\\[\n(\\;\\cdot, \\cdot)_N: S_k(N, \\mathcal{O})_\\mathfrak{m}\\times S_k(N,\\mathcal{O})_\\mathfrak{m} \\rightarrow \\mathcal{O},\\]\nsee \\eqref{integral-pairing} and the discussion preceding it for more details. We need to compare the algebraic pairing defined above to the usual Petersson inner product. We define a modified Petersson product on $S(N, \\mathbf{C})$ by setting\n\\begin{equation}\n\\label{modified-petersson-2}\n\\{v, w\\}_N = \\langle v, w^c\\vert W_N\\rangle_N\n\\end{equation}\nwhere the pairing on the right is the Petersson product. The superscript $c$ denotes complex conjugation\non the Fourier coefficients, and $W_N$ is the Atkin-Lehner involution. It is then shown that the two pairings are essentially scalar \nmultiples of each other, thus $\\{f , f\\}_N=\\Omega_N (f, f)_N$, where $\\Omega_N$ is the canonical period and \nwell-defined up to $\\mathbb{Z}_p$. Equivalently, $\\Omega_N = \\frac{(f, f)_N}{\\{f , f\\}_N}$. The numerator of this \nfraction is the so-called congruence number for $f$.\n\nAt this point, the reader will note that the entire analysis of special values as described above \nis carried out in the context of the imprimitive form $f.$ This raises red flags, since Hida's construction\nrelies crucially on some kind of multiplicity one result, for instance, \nto show that the pairings are scalar multiples of each other. Furthermore,\nit is not clear that the pairings $\\{f, f\\}_N$ or $(f, f)_N$ are  non-zero, without some kind \nof semisimplicity in the Hecke algebra. We are able to circumvent\nthis problem because we are restricting attention to maximal ideals $\\mathfrak{m}$ where\n$U_q\\in \\mathfrak{m}$ at all bad primes $q\\neq p$, and because  by Lemma \\ref{dtlemma} the auxiliary level $N\/M$ is cube-free,\nand divisible only by primes for which $N$ itself is cube-free. The point is that if $q\\nmid M$, where $M$ is the level\nof the $p$-stabilized newform $f$, then $U_q$  at level $N$ can have at most 3 different\neigenvalues: $\\alpha_q, \\beta_q, 0$, with $\\alpha_q\\beta_q =q^{k-1}$,  on any form obtained from $f$ by degeneracy maps.\nThus the nonzero eigenvalues of $U_q$ \nare $p$-adic units. The cube-free condition allows us to rule out any failure of semisimplicity in the \ngeneralized eigenspace for $U_q=0$, and the imprimitive form $f$ is chosen to lie in exactly this eigenspace. \nWe remark that this argument would fail in the presence \nof auxiliary level which were divisible by a cube.\n\nA further -- and more stubborn -- point arises from comparion of the periods of $f$ at level $N$ (which may vary depending\non $\\Sigma$, which in turn depends on $g_2$) with the canonical periods of the $p$-stabilized newform $g$ at level $M$. \nRelating these periods\nrequires us to assume that a certain version of Ihara's lemma is satisfied, see Hypothesis \n\\ref{hypothesis ihara}. The result is known unconditionally in the case when $k=2$, but is not fully resolved in all\nthe higher weight cases. We are therefore required to carry around  Hypothesis \\ref{hypothesis ihara}.\nIf the reader is willing to allow the periods to depend on the level, then our results become unconditional.\n\n Finally, for the purposes of relating the Iwasawa invariants of $g_1$ and $g_2$, we must\n show that if $g_1$ and \n $g_2$ are $\\mathfrak{p}$-congruent modular forms, then we may simultaneously add primes to the level to obtain \n imprimitive modular forms $f_1$ and $f_2$ of the same level for which \\emph{all} Fourier coefficients are \n $\\mathfrak{p}$-congruent, and for which all $U_q$ eigenvalues are zero. The fact that the level so obtained \n satisfies our cube-free condition is an application of \nLemma \\ref{dtlemma}, whose significance cannot be overstated. Once a level is determined, we use  properties \nof the algebraic pairing, to show\n that normalized special values of $p$-adic $L$-functions associated to  $r_{g_i}\\otimes\\psi$ and $r_{g_2}\\otimes\\psi$ \n are $\\mathfrak{p}$-congruent, see Theorem \\ref{special values congruence}. \nAs a result, we obtain a relationship for the analytic Iwasawa invariants associated with \n$r_{g_1}\\otimes\\psi$ and $r_{g_2}\\otimes\\psi$. \n\n\n\\par \nOn the algebraic side, there is little difficulty. The results on Galois cohomology\nin \\cite{GV00} are quite general,\nand apply to the situation treated here, so require little more than translation. In section \\ref{s 4}, we introduce the \\emph{fine Selmer group}. The residual Selmer group is seen to be finite precisely when $\\mu$-invariant vanishes. We relate the finiteness of the residual Selmer group to the vanishing of the $\\mu$-invariant of the fine Selmer group and establish a natural criterion for the finiteness of the residual Selmer group, see Theorem \\ref{muzeroconditions}. This foreshadows a residual Iwasawa theory purely associated with the residual representation.\n\n\\section*{Acknowledgments}\nThe authors would like to thank Haruzo Hida, Antonio Lei, Giovanni Rosso and Eric Urban for helpful comments.\n \n\n\n\\section{Congruences for symmetric square L-functions}\n\n\\subsection{Definitions and normalizations}\n\\label{assumptions-and-definitions} Let $p\\geq 5$ denote a prime, and fix a prime $\\mathfrak{p}$ of $\\bar{\\mathbb{Q}}$ with residue characteristic $p$. Let $M\\geq 1$ be an integer such that $M=M_0p$, where $p\\nmid M_0$.\nLet $g$ denote a $\\mathfrak{p}$-stabilized newform of even weight $k$ for the group $\\Gamma_0(M)$. Denote the newform associated to $g$ by $g_0$. Note that $g_0$ has level $M_0$ or $M_0p$. Assume throughout that $g_0$ is not of CM type, that is, $g_0\\otimes\\chi\\neq g_0$ for any Dirichlet character $\\chi$. Furthermore, assume that the nebentype character of $g$\nis trivial.\n\\par If $z$ denotes a variable in the upper\nhalf plane, and $q=e^{2\\pi i z}$, write the Fourier expansion of $g$ as $g(z)=\\sum a(n,g)q^n$. Then the $L$-function $L(s, g) = \\sum a(n,g) n^{-s}$ of $g$ has the formal Euler product expansion\n\\[L(s, g) = \\prod_q(1-\\alpha_q q^{-s})^{-1}(1-\\beta_q q^{-s})^{-1},\\]where the product is taken over all prime numbers $q$, and $\\alpha_q, \\beta_q$ are certain complex numbers. For $q \\nmid M$, we have $\\alpha_q\\beta_q=q^{k-1}$, but if $q\\mid M$, then one or both \n$\\alpha_q, \\beta_q$ is zero. In fact, since the character of $g$ is trivial, the formulae of Miyake \\cite[Theorem 4.6.17]{miy89} for $q\\neq p$ show\nthat if $M$ is divisible by precisely the first power of $q$, then $\\alpha_q^2= q^{k-2}$ and $\\beta_q=0$, while if $q^2\\vert M$, \nthen both are zero. In the special case that $q=p$, we have $\\alpha_p\\neq 0$, $\\beta_p=0$, \nand $\\alpha_p$ is a $\\mathfrak{p}$-adic unit. \n\n\\par We follow the classical\nconventions when speaking of Hecke operators acting on modular forms. Let $m$ be any positive integer. For $q\\nmid m$ (resp. $q\\mid m$) the Hecke operator $T_q$ (resp. $U_q$) of level $m$ corresponds\nto the \\emph{right} action of the double coset \n$\\Gamma_0(m)\\begin{pmatrix} q & 0 \\\\ 0 & 1\\end{pmatrix}\\Gamma_0(m)$, acting via the usual slash operator. With these normalizations, the eigenvalue\nof $T_q$ on $f$ is $a_q = \\alpha_q + \\beta_q$, for $q\\nmid M$, and we have $\\alpha_q\\beta_q= q^{k-1}$. If $m$ is divisible by precisely the first power of $q$,\nthen the eigenvalue of $U_q$ is $\\alpha_q=\\pm \\sqrt{q^{k-2}}$, and if $q^2\\vert M$, then the eigenvalue of $U_q$ is zero.\n\\par We set the Petersson product of modular forms $v, w$ of weight $k\\geq 2$\non $\\Gamma_0(m)$ (at least one of which is cuspidal) to be as follows:\n$$\\langle v, w\\rangle_m = \\int_{B(m)}v(z)\\overline{w(z)} y^{k-2} dx dy,$$ and the integral is taken over a fundamental domain $B(m)$ for $\\Gamma_0(m)$. The Hecke operators $T_q$ are self-adjoint with respect to the Petersson inner-product on $\\Gamma_0(m)$. Write $W_m$ for\nthe operator on modular forms of level $m$ induced\nby the action of the matrix $\\begin{pmatrix} 0 & 1 \\\\  -m & 0\\end{pmatrix}$. Recall that\nthe adjoint of $U_q$ acting on cuspforms of level $m$ is the operator $U_q^*$ given by $W_m U_q W_m$. Note that the matrix $W_m$\nnormalizes $\\Gamma_0(m)$, and that the adjoint of $U_q$ depends on the level, although $U_q$ itself does not.\n\n\\par Let $\\Sigma$ be a finite set of primes, and define the modular form \n \\[f(z) :=\\sum_{(n,\\Sigma)=1}a(n,g) q^n,\\] where the sum is restricted to indices $n$ that \n are indivisible by each prime in $\\Sigma$. Then one has the following formula for the L-function\n \\begin{equation}\n \\label{deg-two-product}\n L(f,s) =\\sum_{(n,\\Sigma)=1}a(n,g) n^{-s} = \\left(\\prod_{(q,\\Sigma)=1}(1-\\alpha_q q^{-s})(1-\\beta_q q^{-s})\\right)^{-1},\n \\end{equation}\nwhere the product is taken over primes away from $\\Sigma$. The modular form $f$ has level\nat most $M\\prod_{q\\in \\Sigma} q^2$.\n\\begin{assumption}\\label{assumptions on Sigma}\nWe make the following assumptions on $\\Sigma$:\n\\begin{itemize}\n\\item $2\\in \\Sigma$;\n\\item $p\\notin \\Sigma$;\n\\item if $q\\in\\Sigma$, then $q^2\\nmid M$; and \n\\item if $q$ exactly divides $M$, then $q\\in \\Sigma$.\n\\end{itemize}\n\\end{assumption}\n\nAs in the introduction, $\\Sigma$ contains the prime $2$, together with all primes $q$ that divide $M$ to precisely the first power, together with\nother primes that do not divide $M$. If $q^2$ divides $M$, then $q\\notin\\Sigma$, unless $q=2$.\n In our applications, we will have to\nchoose $\\Sigma=\\Sigma_i$ for $g_i$ in a way that depends on the congruent form $g_{i+1}$, where we read the indices modulo 2.\nIn particular, $\\Sigma_1$ and $\\Sigma_2$ may be different. For now, we simply fix one form $g=g_i$. \nThroughout, $\\psi$ is an even and non-quadratic character of conductor $c_\\psi$ coprime to $N$. It follows from the definition of $\\Sigma$\nand the description of $\\alpha_q, \\beta_q$ given above \nthat we have\n\\begin{Lemma} Under the assumptions above, the modular form $f$ has level $N$ given by $N = M \\cdot \\prod_{q\\in\\Sigma} q^{e(q)}$, where\nfor $q$ odd we have\n$e(q)=2$ if $q\\nmid M$, and $e(q) =1$ if $M$ is divisible by precisely the first power of $q$. For $q=2$, we have $e_2=2$ if $M$ is odd,\n$e_2=1$ if $2$ exactly divides $M$, and $e_2=0$ if $4 \\vert M$. \n\\end{Lemma}\n\nAssumption \\ref{assumptions on Sigma} leads to the following elementary consequences:\n\\begin{enumerate}\n\\item $N=pN_0$, where $(N_0, p)=1$;\n\\item $4\\vert N$;\n\\item $f$ is an eigenvector\nof the Hecke operators $T_q$ for $q\\nmid N$ and for $U_q$ when $q\\vert N$;\n\\item the eigenvalue $\\alpha_p$ of the $U_p$ operator  on $f$ is a $\\mathfrak{p}$-adic unit;\n\\item  for $q\\vert N_0, q\\neq p$, the eigenvalue of $U_q$ on $f$ is zero;\n and\n\\item the product in (\\ref{deg-two-product}) may be taken over $q$ away from $N_0$.\n\\end{enumerate}\nThe final statement follows from the fact that if $q\\neq p$ divides $N_0$ and $q\\notin\\Sigma$, then $q^2\\vert M$, so that $\\alpha_q=\\beta_q=0$.\nIf necessary, we enlarge $K$ to \ncontain the Fourier coefficients of $g$ as well as the number $\\alpha_p$.  As before, we write\n ${\\mathcal O}$ to denote the completion of the ring of integers of $K$ at the prime ${\\mathfrak p}$. \n\n\\subsection{Dirichlet series, the naive symmetric square, and the Petersson product formula}\n\nLet $\\chi$ be a Dirichlet character of the form $\\psi\\eta$, where $\\psi$ has conductor away from $N$, and $\\eta$ has $p$-power conductor.\nLet  $f$ be as defined previously. Then the naive  $\\chi$-twisted symmetric square $L$-function of $f$  is as follows:\n\\begin{equation}\n\\label{naive-product}\nD_f(\\chi, s) = \\prod_{q\\nmid N_0} \\left( (1-\\chi(q)\\alpha_q\\beta_q q^{-s})(1-\\chi(q)\\beta_q^2q^{-s})(1-\\chi(q)\\alpha_q^2q^{-s})\\right)^{-1}.\n\\end{equation}\n\nIn the language of the introduction,\n$D_f(\\chi, s) $ is Shimura's $D(s, f, \\chi) =  \\mathscr{L}_\\Sigma(r_g\\otimes\\chi, s) =  \\mathscr{L}_{S_0}(r_g\\otimes\\chi, s)$, where\n$S_0=S\\backslash\\{p\\}$, and $S$ is the set of primes dividing $N$. Here we are repeatedly using the fact that the degree\n2 Euler product associated to $f$ is trivial at all $q\\neq p$ such that  $q\\vert N$, and that the same is true for $g$ at primes $q\\in S_0\\backslash\\Sigma$.\nWe have mentioned this fact many times, but it is absolutely crucial and bears repeating: without this fact, we would not get any equality between\n $\\mathscr{L}_\\Sigma(r_g\\otimes\\chi, s)$ and  $\\mathscr{L}_{S_0}(r_g\\otimes\\chi, s)$ and\n Shimura's $D(s, f, \\chi)$, and so Shimura's formulae would not apply. Indeed, the functions $\\mathscr{L}_\\Sigma(r_g\\otimes\\chi, s)$ and  $\\mathscr{L}_{S_0}(r_g\\otimes\\chi, s)$ have trivial Euler factors at primes in $\\Sigma$ and $S_0$ respectively, but Shimura's L-function has trivial Euler factor at $q\\vert N_0$ if and only\n if $\\alpha_q=\\beta_q=0$ (recall that $\\chi$ is unramified at primes dividing $N_0$). If the Euler factors of $g$ at $q\\in S_0\\backslash\\Sigma$\nwere nontrivial, then we would not have $\\mathscr{L}_\\Sigma(r_g\\otimes\\chi, s) = \\mathscr{L}_{S_0}(r_g\\otimes\\chi, s)$, and if the Euler factors of \n$f$ at the primes in $S$ were nontrivial, we would not have $D(s, f, \\chi) = \\mathscr{L}_{S_0}(r_g\\otimes\\chi, s)$. \n\n\nFor the present, we concentrate on the naive L-function $D_f(\\chi, s)$, and show\nthat its special values at critical points behave well with respect to congruences. \n\n\\par Let \n$G({\\chi})$ denote the Gauss sum of $\\chi$.\nThen the quantity\n\\begin{equation}\nD_f(\\chi, s)^\\text{alg} = \\frac{D_f(\\chi, s)}{\\pi^{k-1}\\langle f, f\\rangle_N}\\cdot \\frac{G(\\overline{\\chi})}{(2\\pi i)^{s-k+1}}\n\\end{equation} is algebraic when $s=n$ is an integer in the range\n$1\\leq n\\leq k-1$\nsatisfying $(-1)^n=-\\chi(-1)$. This is well-known, see \\cite[Theorem 2.2.3]{lz16}, for the present formulation; the result goes back to\nShimura, whose method was elaborated by Schmidt \\cite{schmidt86}, \\cite{schmidt88},  and Sturm \\cite{sturm}.\n If $(-1)^n=-\\chi(-1)$ and $1\\leq n\\leq k-1$, then we say that $n$ is critical.\nWe remark that the functional equation for the primitive symmetric square L-function leads to similar algebraicity results\nfor $D_f(\\chi, s)^{\\op{alg}}$ for integer values of $s$ in the range $k\\leq s\\leq 2k-2$; we will not need these results here.\nNote that the quantity $D_f(\\chi, s)^\\text{alg}$ may be zero, even at critical values, since we are dealing with imprimitive \n$L$-functions. Furthermore, the algebraic quanitites above are not necessarily integral.  \nWe make the following assumption, to keep the notation and book-keeping simple:\n\n\\begin{itemize}\n\\item The character $\\chi=\\psi \\eta$, where $\\eta$ is a \\emph{nontrivial} even character of $p$-power conductor.\n\\item $s=n$ is an odd integer with $1\\leq n \\leq k-1$ in the algebraicity formula above. \n\\end{itemize}\n\nThe explicit formulae we will need  for interpolation and congruences originate in \\cite{shimura-holo}. They\nare cited in many different forms in  the references \\cite{schmidt86}, \\cite{schmidt88}, and \\cite{ros16}, but \neach of these references adopt slightly different normalizations and conventions, so we are forced to make a choice.\nWe have selected to follow \\cite{schmidt86}, since it is relatively easy to compare the formulae given there to those\noriginally given by Shimura, whose work remains the basic reference. \n\nThus, define\n\\begin{equation}\\theta_\\chi(z) = \\sum_j \\chi(j) \\text{exp}(2\\pi i j^2 z), \n\\end{equation} which is a modular form of weight $1\/2$ and level $4c_\\chi^2$. \nSetting $\\omega:=\\chi\\left(\\frac{-1}{\\cdot}\\right)^k$, we let $$\\Phi(z, \\chi, s) = L_{N}(\\chi^2, 2s+2-2k)E(z, s+2-2k, 1-2k, \\omega)$$ denote the \nEisenstein series as defined in \\cite[p.210]{schmidt86}.  Note that we have already imposed $4\\vert N$ as part of our assumption on the level of $f$. \n\n\n\n\nIt turns out that $\\theta_\\chi(z)\\Phi(z, \\chi, s)$ is a (non-holomorphic) modular form of weight $k$, trivial character, and level \n $N_\\chi:=\\text{lcm}(N, c_\\chi^2)$. Here we use the fact that $4\\vert N$, by construction. Recall that the level $N$ of $f$ is divisible by precisely the first power of $p$, since $f$ is assumed to be $\\mathfrak{p}$-stabilized.\nWe shall write $c_\\eta=p^{m_\\chi}=p^{m_\\eta}$ for the $p$-part of the level of $\\chi=\\psi\\eta$. Since $\\eta\\neq 1$\nwe have $m_\\chi\\neq 0$ and $N_\\chi=\nN_0c_\\psi^2p^{2m_\\chi}$ where we recall that $N_0$ is the prime to $p$-part of $N$.\n\nConsider an odd integer $n$ in the range\n$1\\leq n\\leq k-1$, and set $H_{\\overline\\chi}(n) := \\theta_{\\overline\\chi}(z)\\Phi(z,\\overline\\chi, n)$. Shimura has\n shown that $H_{\\overline\\chi}(n)$ is a nearly holomorphic modular form of level\n$N_\\chi$, weight $k$, and trivial character. In fact, one has the following formula (see equation (1.5) in \\cite{shimura-holo}):\n\\begin{equation}\\label{sturm relation}\n(4\\pi)^{-s\/2}\\Gamma(s\/2) D_f(\\chi, s) = \\langle f, \\theta_{\\overline\\chi}(z)\\Phi(z,\\overline\\chi, s)\\rangle_{N_\\chi}.\n\\end{equation}\n\n\nObserve\nnow that $f$ has level $N$, while $H_{\\overline\\chi}(n)$ satisfies a transformation property with respect to the group \n$\\Gamma_0(N_\\chi)$. According to our assumption, $\\chi$ has conductor $c_\\psi p^r$, for some $r$, and $c_\\psi$ is relatively\nprime to $N$. Thus the level of $f$ and the level of $H_{\\overline\\chi}(n)$ differ only by a power of $p$, and the primes that\ndivide the conductor $c_\\psi$  of $\\psi$. Our goal is to bring $H_{\\overline\\chi}(n)$ down to level $N$ by taking a trace, and verify\nthat we can retain\ncontrol of integrality of the Fourier coefficients. \n\nWe start by dealing with the powers of $p$. In this we reproduce the method of  Schmidt.\nLet $N_\\psi=N_0pc_\\psi^2$, and let $T_\\eta$ denote the trace\noperator that takes modular forms on $\\Gamma_0(N_\\chi)$ down to $\\Gamma_0(N_\\psi)$, normalized as in Schmidt \\cite{schmidt86}.\nWe remark here that the\ntrace operator is defined purely in terms of matrices, and can be applied to the non-holomorphic form $H_{\\overline\\chi}(n)$. \n\nIt follows from \\cite[Lemma 3.10]{CS} (which is done for weight $2$), or the calculation in the middle \nof \\cite[p.217]{schmidt86}, that the following relation is satisfied $$p^{(2m_\\chi-1)(k\/2-1)}T_\\eta\\circ W_{N_\\psi}=\nW_{N_\\chi}\\circ U_p^{2m_\\chi -1}.$$ As a result, we find that $H_{\\overline\\chi}(n)\\circ W_{N_\\chi}\\circ U_p^{2m_\\chi-1}$ is of level $N_\\psi$. \nIt follows further that for any $m\\geq m_\\chi$,\nthat $H_{\\overline\\chi}(n)\\circ W_{N_\\chi}\\circ U_p^{2m -1}$ is also a modular form of level $N_\\psi$, since the $U_p$ operator at level $N_\\chi$ is \ngiven by the same  matrices as $U_p$ at level $N_\\psi$, and $U_p$\nstabilizes the space of forms of level $N_\\psi$. Note that \\[H_{\\overline\\chi}(n)\\circ W_{N_\\chi}\\circ U_p^{2m -1}= \nH_{\\overline\\chi}(n)\\circ W_{N_\\chi}\\circ U_p^{2m_\\chi-1} \\circ U_p^{2(m-m_\\chi)-1},\\] \nand therefore from \\eqref{sturm relation} we obtain the relations\n\\begin{align*}\n(4\\pi)^{-n\/2}\\Gamma(n\/2) D_f(\\chi,n) = & \\langle f,  H_{\\overline\\chi}(n) \\rangle _{N_\\chi} \\\\\n=  &\\langle f,  H_{\\overline\\chi}(n)\\circ T_\\eta \\rangle _{N_\\psi}  \\\\\n= & \\langle f\\circ W_{N_\\psi},  H_{\\overline\\chi}(n)\\circ T_\\eta\\circ W_{N_\\psi}\\rangle_{N_\\psi}\\\\\n=& p^{-(2m_\\chi-1)(k\/2-1)}\\langle f\\circ W_{N_\\psi},  H_{\\overline\\chi}(n) \\circ W_{N_\\chi}\\circ U_p^{2m_\\chi -1}\\rangle_{N_\\psi},\n\\end{align*}\n\nand\n\n\\begin{align*}\n\\langle f\\circ W_{N_\\psi}, H_{\\overline\\chi}(n)\\circ W_{N_\\chi}\\circ U_p^{2m-1}\\rangle_{N_\\psi} \n=  &\\langle f\\circ W_{N_\\psi}, H_{\\overline\\chi}(n)\\circ W_{N_\\chi}\\circ U_p^{2m_\\chi-1} \\circ U_p^{2(m-m_\\chi)}\\rangle_{N_\\psi} \\\\  \n= & \\langle f \\circ W_{N_\\psi}\\circ (U_p^*)^{2(m-m_\\chi)}, H_{\\overline\\chi}(n)\\circ W_{N_\\chi}\\circ U_p^{2m_\\chi-1}\\rangle_{N_\\psi}  \\\\\n= & \\langle f\\circ U_p^{2(m- m_\\chi)}\\circ W_{N_\\psi}, H_{\\overline\\chi}(n)\\circ W_{N_\\chi} \\circ U_p^{2m_\\chi-1}\\rangle _{N_\\psi}\\\\\n= & \\alpha_p^{2(m-m_\\chi)} \\langle f\\circ W_{N_\\psi}, H_{\\overline\\chi}(n)\\circ W_{N_\\chi}\\circ U_p^{2m_\\chi-1}\\rangle _{N_\\psi}.\n\\end{align*}\n\nIn the second calculation above, we have used the fact that adjoint of $U_p$ at level $N_\\psi$ is given by $U_p^* = W_{N_\\psi}\\circ U_p\\circ W_{N_\\psi}$. \nPutting together the strings of equalities above, we conclude that\n \\begin{equation}\n \\label{petersson-formula}\n \\begin{split}\n&\\frac{\\Gamma(n\/2)}{(4\\pi)^{n\/2}}p^{(2m_\\chi-1)(k\/2-1)} D_f(\\chi,n) \\\\\n= & \\alpha_p^{2(m_\\chi-m)} \\langle f, H_{\\overline{\\chi}}(n)\\circ W_{N_\\chi}\\circ U_p^{2m-1}\\circ W_{N_\\psi}\\rangle_{N_\\psi}\n\\end{split}\n\\end{equation}\nfor any $m\\geq m_\\chi$. \n\n\\par We will see below that the nearly holomorphic modular form on the right hand side of the formula above may be replaced with\na \\emph{holomorphic} form $\\mathcal{H}_{\\overline\\chi}(n)$ of level $N_\\psi$, without changing the value of the inner product. Furthermore, we shall\nsee that $\\mathcal{H}_{\\overline\\chi}(n)$ has $\\mathfrak{p}$-integral Fourier coefficients. Assuming this, we shall take the (twisted) \ntrace of $\\mathcal{H}_{\\overline\\chi}(n)$ down to level $N$ to get our final result. This trace is much easier to deal with since $N_\\psi$ and and $N$ differ\nonly by primes away from $p$. Let $t_\\psi$ denote the trace operator from level $N_\\psi$ to level $N$. Let $W_{c_\\psi^2}$ denote the Atkin-Lehner \noperator  acting on modular forms of level $N_\\psi = Nc_\\psi^2$,\nas defined in \\cite{al}, page 138, just before Lemma 8. \nWe note that $(N, c^2_\\psi) =1$, so this operator is indeed defined. \nDefine\n$T_\\psi: S_k(N_\\psi) \\rightarrow S_k(N)$ by \n$$T_\\psi: h\\mapsto  h\\circ W_{c_\\psi^2}\\circ t_\\psi.$$ \n The key lemma is the following. It states\nthat the trace operator $T_\\psi$ preserves integrality. \n\n\\begin{Lemma}\n\\label{tame-trace}\nSuppose $\\phi$ is a holomorphic modular form of level $N_\\psi$ and weight $k$. Suppose that $\\phi$ has Fourier\nexpansion $\\phi = \\sum a(n, \\phi)q^n$ with  $\\mathfrak{p}$-integral coefficients $a(n, \\phi)$. Then $T_\\psi(\\phi) $ has $\\mathfrak{p}$-integral Fourier\ncoeffiicients as well. \n\\end{Lemma}\n\n\\begin{proof} The fact that $W_{c_\\psi^2}$ preserves integrality is Theorem A.1 in the appendix by Conrad to \\cite{pras09}.\nAs for $t_\\psi$, the proof is more or less standard, so we merely sketch the argument. \nThe trace is given by $\\phi\\mapsto \\sum_\\gamma\\phi\\circ\\gamma$, where\n$\\gamma$ runs over a set of coset representatives of $\\Gamma_0(N_\\psi)\\backslash\\Gamma_0(N)$. By definition of $N$ we have \n$\\Gamma_0(N)\\subset \\Gamma_0(p)$.  Thus the cusp $s=\\gamma(\\infty)$ is one of Hida's `unramified' cusps, and it is well-known that if $\\phi$ has\nintegral $q$-expansion at $\\infty$ then it has integral $q$-expansion at $s$. The key point is that both $\\infty$ and $s$ reduce modulo $\\mathfrak{p}$ \nto points on the same component of the special fibre of $X_0(N_\\psi)$, and thus if a modular form vanishes identically in a formal neighbourhood of one\ncusp, it must vanish on the whole component, and hence the expansion is zero about the other cusp as well.\nThus $t_\\psi$ preserves integrality as well. The reader may consult\n\\cite{hida86}, Section 1, or \n\\cite{hida88}, page 11, for a fuller discussion.\n\\end{proof}\n\nOur next goal is therefore to analyze the form $H_{\\overline{\\chi}}(n)\\circ W_{N_\\chi}\\circ U_p^{2m-1}\\circ W_{N_\\psi}$ \nand show how to replace it with something holomorphic and integral. This computation will occupy the next section. \n\\subsection{Holomorphic and ordinary projectors}\n\n The classical method of going from a nearly holomorphic form to something holomorphic, and which is adopted\nin \\cite{CS}, \\cite{schmidt86}, \\cite{schmidt88}, is to \npass from $H_\\chi$ to its so-called  holomorphic projection. This is a bit complicated, since the formulae giving the holomorphic \nprojection of a nearly holomorphic form involve\nfactorials and binomial coefficients, and one cannot easily control the denominators. This is why the results of \n\\cite{schmidt86}, \\cite{schmidt88} are only stated up to some unspecified rational constant.  \n\nOne of the main contributions of this paper is a solution to this problem, using $p$-adic methods. In the case we have at hand, the form $f$\nis \\emph{ordinary}, and we can replace the nearly holomorphic form with a certain \\emph{ordinary} projection, without losing\nany information. This has the significant advantage that the ordinary\nprojector is denominator-free. In view of Hida's control theorems for ordinary forms, the ordinary projection is automatically holomorphic.\nWe shall follow this alternative path, but to complete the journey, we have to make a computation of Fourier coefficients. \n\nWe have to compute the Fourier expansion of $H_{{\\chi}}(n)\\circ W_{N_\\chi}$. Since f $H_{{\\chi}}(n)$ is a product of an\nnearly holomorphic Eisenstein series  $\\Phi(z, \\chi, n)$ of weight $k-1\/2$ and a theta series $\\theta_\\chi$ of weight $1\/2$, we\nwork out the expansions of these two first, starting with the Eisenstein series.\n\nFollowing \\cite{schmidt86}, page 213,and Shimura, \\cite{shimura-holo}, Section 3, page 86,  let us write \n$$\\Phi\\left(\\frac{-1}{N_{\\chi} z}, \\chi, n\\right) \\cdot (\\sqrt{N_\\chi}z)^{1\/2-k}= \\sum_{j=0}^{(n-1)\/2}\\sum_{\\nu=0}^{\\infty} (4\\pi y)^{-j} d_{j, \\nu} q^\\nu.$$\n\n\\begin{Lemma}\n\\label{eisen-fourier} If $\\chi$ is ramified at $p$, the quantities $\\frac{\\Gamma((n+1)\/2)}{\\pi^{(1+n)\/2}} p^{m_\\chi(3-2k+2n)\/2}d_{j,\\nu}$ are algebraic\nand $\\mathfrak{p}$-integral.\n\\end{Lemma}\n\n\\begin{proof} The formulae for the $d_{j,\\nu}$ may be deduced from those on pages 212-213 of \\cite{schmidt86}, whose $n$ is our $\\nu$, and whose\n$m$ is our $n$. We remark that there appears to be a small mistake in\n in the power of $i$ in the formula for $\\tau_n$ on the top of page 213; by comparision with the formula on page 225 of  \\cite{sturm}, \n it should be $i^{-k+1\/2}$.\n\nFor $\\nu>0$,  one obtains\n$$d_{j, \\nu} = (-2i)^{(k-1\/2)}\\cdot\\pi^{\\frac{n+1}{2}}\\cdot \\nu^{\\frac{n-1}{2}} \\cdot B_j \\cdot {\\frac{n+1}{2}\\choose j}\\cdot N_\\chi^{(2k-2n-3)\/4}\\cdot L_{N_\\chi}(n+1-k,\\omega_\\nu)\\cdot\\beta(\\nu, n+2 -2k),$$\nwhere $B_j=\\frac{\\Gamma(n\/2+1-k+j)}{\\Gamma(\\frac{n+1}{2})\\cdot\\Gamma(n\/2+1-k)}\\in\\mathbb{Q}$. The definition of $\\beta$ may be \nfound in \\cite{schmidt86}, page 212. As mentioned above, our formula has a factor of $(-i)^{k-1\/2}$, while  \\cite{schmidt86}\nhas $(-1)^{k-1}$; our formula here agrees with the one in the later paper \\cite{schmidt88}, page 614 (where the normalization and \nnotations are rather different). Note also that $\\beta$ depends on $\\chi$. The character $\\omega_\\nu$ is defined on page 212\nof \\cite{schmidt86}, and $\\omega$ on page 206. \n\nAs for the constant term, one has $d_{j,0}=0$ unless $j=(n-1)\/2$, in which case one has \n$$d_{(n-1)\/2, 0} = (-2i)^{(k-1\/2)}\\cdot\\pi^{\\frac{m+1}{2}}\\cdot B_j \\cdot N_\\chi^{(2k-2n-3)\/4}\\cdot L_{N_\\chi}(2n+2-2k,\\omega^2)$$\n\nIt is clear from the formula\nfor $B_j$ that $\\Gamma((n+1\/2))B_j$ is a rational integer, and one knows from properties of Kubota-Leopoldt $p$-adic L-functions\nthat $L(\\omega_\\nu, n+1-k)$ is $\\mathfrak{p}$-integral once the character $\\omega_\\nu$ has conductor divisible by $p$. The result\nfollows upon clearing the powers of $p$ coming from $N_\\chi = N_0c_\\psi^2p^{2m_\\chi}$. \n  \\end{proof}\n\n\nOne has now to compute the Fourier expansion of the quantity $\\theta_\\chi(-1\/N_\\chi z)\\cdot (\\sqrt{N_\\chi} z)^{-1\/2}$. \nHere the exponent comes from the fact that $\\theta_\\chi$ is a form of weight $1\/2$. The requisite formula may be found\nin \\cite{shimura-half}, Proposition 2.1, and we record the result here. \nRecall the notations: $N=N_0p$ is an integer divisible by precisely\nthe first power of $p$, and $\\chi$ is a character of conductor $c_\\psi p^{m_\\chi}$. Recall that $c_\\psi$ satisfies $(c_\\psi, 2Mp)=1$. \nThe integer $N_\\chi$ is given by $N_\\chi= N_0c_\\psi^2p^{2m_\\chi}$ if $\\chi$ is ramified. Furthermore,\n$N$ is divisible by $4$. We let $N'= N_\\chi\/4c_\\chi^2=N_0\/4$. \n\n\n\\begin{Lemma} \n\\label{theta-fourier}\nSuppose that $\\chi$ is ramified. \nWe have $\\theta_{\\chi}(-1\/N_\\chi z)\\cdot (\\sqrt{N_\\chi} z)^{-1\/2} = \\theta_{\\overline{\\chi}}(N'z)\\cdot \n\\frac{g(\\chi)}{ \\sqrt{c_\\psi p^{m_\\chi}}}\\cdot i^{3\/2}\\cdot (N^\\prime)^{1\/4}$.\n\\end{Lemma}\n\n\\begin{proof} See \\cite{shimura-half}, Proposition 2.2.\n \\end{proof}\n\n\\begin{Remark} It is clear from the formulae for the $d_{j,\\nu}$in terms of Dirichlet L-functions\n that the forms $\\Phi$ occur in analytic families -- simply replace the Dirichlet L-functions with the Kubota-Leopoldt \n versions. Since this is obviously\nso for the theta functions $\\theta_\\chi$, one guesses immediately that some kind of $p$-adic L-functions should exist, simply\nby taking a suitable pairing with $\\theta_\\chi\\Phi(z, \\chi, n)$. \n\\end{Remark}\n\n\nRecall that we have put $\\chi = \\psi\\eta$, where $\\psi$ has conductor $c_\\psi$, and $\\eta$ has $p$-power \nconductor $p^{m_\\chi}$, with $m_\\chi=m_\\chi$. For the purposes of $p$-adic L-functions, we must regard the character\n$\\psi$ as fixed, and let $\\eta$ vary. \n\n\n\\begin{Corollary} \n\\label{first-itegrality-formula}\nSuppose that $\\eta$ is ramified. Then \n$$\\tilde{H}_\\chi(n)= \\frac{\\Gamma((n+1)\/2)}{\\pi^{(1+n)\/2}} p^{m_\\chi(3-2k+2n)\/2} \\cdot \\frac{\\sqrt{c_\\psi p^{m_\\chi}}}{g(\\chi)} \\cdot H_{\\chi}(n)\\circ W_{N_\\chi}$$\nis a nearly holomorphic form of level $N_{\\chi}$ with $\\mathfrak{p}$-integral Fourier coefficients.\n\\end{Corollary}\n\n\\begin{proof} This is obvious, from the formulae above. \n\\end{proof}\n\n\\begin{Remark} We have not given any formula for the case where $\\eta$ is trivial, although it may be done just as above. The \nexact formulae are slightly different, and we will not need them. Since we are assuming the existence of imprimitive $p$-adic\nL-functions, it suffices, for the purpose of integrality, to show that almost all the values are integral.\n\\end{Remark}\n\nNow we want to pass from $\\tilde{H}_\\chi(n)$ to something holomorphic, while preserving integrality. \nThus the following proposition represents a key contribution of the present work.\n\n\\begin{Proposition}\\label{ordinary holomorphic proj} Let $e$ denote Hida's ordinary projection operator, acting on $M_k(N_\\chi, {\\mathcal O})\\otimes\\bar{\\Q}$. Then \n$\\tilde{H}_\\chi(n)^{\\text{hol}}\\circ e\\in M_k(N_\\psi)\\otimes\\bar{\\Q}$ has ${\\mathfrak p}$-integral Fourier coefficients and level $N_\\psi$. Here $\\tilde{H}_\\chi(n)^{\\text{hol}}$ denotes the holomorphic\nprojection of $\\tilde{H}_\\chi(n)$. \n\\end{Proposition}\n\n\n\n\\begin{proof} The easiest way to prove the Proposition is to use the geometric theory of nearly ordinary \nand nearly holomorphic modular forms, as developed by Urban \\cite{urban12}. A resum\u00e9 of Urban's work\nas adapted to this setting may be found \nin \\cite{ros16}. However, we give a proof along classical lines, for the convenience of the reader.  \nWrite $\\tilde{H}_\\chi(n) = \\tilde{H}_\\chi(n)^{\\text{hol}} + \\sum h_i$, where each $h_i$ is in the image of a Maass-Shimura\ndifferential operator. It follows from the explicit formulae for the differential operators and the formulae for the Fourier\ncoefficients of  $\\tilde{H}_\\chi(n)$ that $\\tilde{H}_\\chi(n)^{\\text{hol}}$ and each $h_i$ has algebraic Fourier coefficients. \nLet $m$ denote a positive integer, divisible by $p-1$, and consider \n$\\tilde{H}_\\chi(n)\\circ U_p^{2m-1}$. Then\none has $$\\tilde{H}_\\chi(n) \\circ U_p^{2m-1} = \\tilde{H}_\\chi(n)^\\text{hol}\\circ U_p^{2m-1} + \\sum  h_i\\circ U_p^{2m-1}.$$ It is well known that $U_p$ multiplies\neach $h_i$ by a power of $p$ (For example, see \\cite{ros16}), formula above Lemma 2.3) Thus, the quantity on the right converges to $\\tilde{H}_\\chi(n)^{\\text{hol}}\\circ e$, as $m$\nincreases, the convergence being assured by the existence of the ordinary projector.\nOn the other hand, the quantity on the left is integral. This proves the proposition.  The fact that the level comes down to $N_\\psi$ arises from the fact\nnoted above that $U_p^m$ already brings the level down to level $N_\\psi$ for any $m\\geq m_\\chi$, as noted in the course of proving (\\ref{petersson-formula}). \n\\end{proof}\n\n\nFinally, we take the trace of the integral and holomorphic form $\\tilde{H}_\\chi(n)^{\\text{hol}}\\circ e$ all the way down to level $N$. Since $H_\\chi(n)$ is not necessarily cuspidal,\nwe shall write $M_k$ to denote spaces of all modular forms. With this notation, we \ndefine\n\\begin{equation}\n\\label{hchi}\n\\mathcal{H}_\\chi(n) = T_\\psi(\\tilde{H}_\\chi(n)^{\\text{hol}}\\circ e)\\in M_k(N, {\\mathcal O})\\otimes\\bar{\\Q}\n\\end{equation}\nand\n\\begin{equation}\n\\label{hchim}\n\\mathcal{H}_\\chi^m(n) = T_\\psi(\\tilde{H}_\\chi(n)^{\\text{hol}}\\circ U_p^{2m-1})\\in M_k(N, {\\mathcal O})\\otimes\\bar{\\Q}.\n\\end{equation}\n\nThen Lemma \\ref{tame-trace}, and the proof of Proposition \\ref{ordinary holomorphic proj},  give\n\n\\begin{Proposition} The modular form $\\mathcal{H}_\\chi(n) $ has $\\mathfrak{p}$-integral Fourier coefficients.\nThe modular forms $\\mathcal{H}_\\chi^m(n)$ have $\\mathfrak{p}$-integral Fourier coefficients for all $m$ sufficiently large.\n\\end{Proposition}\n\nIn the next section, we shall see how to compute the inner product of $f$ with  $\\mathcal{H}_\\chi(n)$ and derive integrality properties for the\nspecial values of $D_f(\\chi, s)$. \n\n\\subsection{Algebraic and analytic inner products}\n\nOur goal is to use (\\ref{petersson-formula}) to describe the imprimitive $p$-adic L-function, and show that it behaves \nwell with respect to congruences. In\norder to do this, we need to normalize our L-functions so that the algebraic parts are $\\mathfrak{p}$-integral. We accomplish this by combining\nthe inner product formula with a certain algebraic incarnation of the Petersson inner product formula due to Hida. \n\nLet $S_k(N, \\mathbb{Z})$ denote the space of modular\nforms of weight $k$ with rational integral Fourier coefficients. If $R$ is any ring, set $S_k(N, R)= S_k(\\Gamma_0(N), \\mathbb{Z})\\otimes R$.\n Let ${\\mathbf{T}}$ denote the ring generated by\nthe Hecke operators $T_q, U_q, U_p$ acting on $S_k(N, \\mathcal{O})$, and set ${\\mathbf{T}}(R)={\\mathbf{T}}\\otimes R$. \n Recall that our convention is that Hecke operators act on the \n\\emph{right}.\n\nThen the eigenform $f$ determines a ring homomorphism $\\mathbf{T}\\rightarrow \\mathcal{O}$, sending a Hecke-operator\n$T\\in \\mathbf{T}$ to the $T$-eigenvalue of $f$. \nDefine $\\mathcal{P}_f$ to denote the kernel of this homorphism. There is a unique maximal ideal $\\mathfrak{m}$ \nof $\\mathbf{T}$ that contains $\\mathcal{P}_f$ and the maximal ideal of $\\mathcal{O}$. In a sense, the maximal \nideal $\\mathfrak{m}$ determines a residual Galois representation, which we shall assume throughout to be absolutely irreducible.\n\\par There \nis a canonical duality of $\\mathbf{T}_\\mathfrak{m}$-modules \nbetween $S_k(N, \\mathcal{O})_\\mathfrak{m}$ and $\\mathbf{T}_\\mathfrak{m}$ defined by the form\n\\[S_k(N, \\mathcal{O})_\\mathfrak{m}\\times \\mathbf{T}_\\mathfrak{m}\\rightarrow \\mathcal{O}\\]$(s, t)\\mapsto a(1, s|t)\\in \\mathcal{O}$ which identifies \n$\\mathbf{T}_\\mathfrak{m}$ with $\\text{Hom}_{\\mathcal{O}}(S_k(N, \\mathcal{O})_\\mathfrak{m}, \\mathcal{O})$. Here\n$s \\in S_k(N,\\mathcal{O})$ (being an $\\mathcal{O}$-linear combination of elements in $S_k(N,\\mathbb{Z})$) is given by the Fourier expansion $s = \\sum a(n, s) q^n$, with $a(n,s)\\in \\mathcal{O}$.\n\nTo proceed further, we assume that the Galois representation associated to $f$ at $\\mathfrak{m}$ is irreducible,\nordinary and $p$-distinguished. It is well-known\nthat under these conditions that  $\\mathbf{T}_\\mathfrak{m}$ is Gorenstein, and isomorphic as a \n$\\mathbf{T}_\\mathfrak{m}$-module to $\\text{Hom}(\\mathbf{T}_\\mathfrak{m}, \\mathcal{O})$, as left modules.\nA proof may be found in \\cite[Theorem 2.1 and Corollary 2 (p. 482)]{wil95}.\n\n\\par Thus, the space $S_k(N, \\mathcal{O})_\\mathfrak{m}$ is equipped with both a left and right action of Hecke operators; one coming from the classically defined slash action of Hecke operators on modular forms, and the other the abstract left action obtained from the \nabstract Gorenstein isomorphism. In fact, the left action of ${\\mathbf{T}}_{\\mathfrak{m}}$ on $\\op{Hom}({\\mathbf{T}}_{\\mathfrak{m}}, \\mathcal{O})$ coincides with the usual right action of ${\\mathbf{T}}_{\\mathfrak{m}}$ on $S_k(N, \\mathcal{O})_{\\mathfrak{m}}$. Indeed, the isomorphism \n\\[{\\mathbf{T}}_{\\mathfrak{m}}\\xrightarrow{\\sim} \\op{Hom}({\\mathbf{T}}_{\\mathfrak{m}}, \\mathcal{O})\\] is an isomorphism of ${\\mathbf{T}}_{\\mathfrak{m}}$-modules, so the two actions are seen to coincide. One deduces that there is a duality pairing\n\\begin{equation}\n\\label{integral-pairing}\n(\\;\\cdot, \\cdot)_N: S_k(N, \\mathcal{O})_\\mathfrak{m}\\times S_k(N,\\mathcal{O})_\\mathfrak{m} \\rightarrow \\mathcal{O},\n\\end{equation} \nwhich satisfies the equivariance condition $(f_1\\vert t, f_2) = (f_1, f_2\\vert t)$.\nThis is Hida's algebraic inner product (see \\cite{hida_1993}, Chapters 7 and 8). \nUnlike the usual Petersson product, it is linear in both variables,\n and the Hecke operators are self-adjoint. \n\nNow let $f\\in S(N, \\mathcal{O})$ denote our fixed eigenform, and consider the function \n\\begin{equation}\n\\label{alg-pairing}\n\\phi_f: v \\mapsto (f, v)_N,\n\\end{equation}\n for $v\\in S(N, \\mathcal{O})_\\mathfrak{m}$.\n  Let $f^{\\perp}\\subset S(N, \\mathcal{O})_\\mathfrak{m}$ denote the kernel of $\\phi_f$, and let $\\eta_f = (f, f)_N = \\phi_f(f)$. \n  We would like to say \n  that the number $\\eta_f$ is nonzero. This is not true in general, but it will be true under the assumptions we have made in Section \n  \\ref{assumptions-and-definitions}. Let $K$ denote the fraction field of $\\mathcal{O}$.\n  \n  \\begin{Lemma}\\label{Tm is a field} Let $\\mathcal{P}=\\mathcal{P}_f$ denote the kernel\n  of the homomorphism $\\mathbf{T}_\\mathfrak{m}({\\mathcal O})\\rightarrow \\mathcal{O}$ associated to $f$. Then, there is an isomorphism\n  $\\mathbf{T}_\\mathfrak{m}({\\mathcal O})_{\\mathcal{P}}\\simeq K$.\n  \\end{Lemma}\n  \n  \\begin{proof} Setting $S:=\\mathbf{T}_\\mathfrak{m}({\\mathcal O})\\otimes\\mathbb{Q}_p$, we note that $S$ is a finite-dimensional algebra over $K$. Hence, $S$ is\n  the product of local rings $R_i$, each of which is finite dimensional over $K$. The ring $R_i$ corresponds to a height one prime ideal \n  of $\\mathbf{T}_\\mathfrak{m}({\\mathcal O})$. The localization of $\\mathbf{T}_\\mathfrak{m}({\\mathcal O})$ at $\\mathcal{P}$ is equal to one of the rings $R=R_i$. Thus, we are to show that $R$ is a field. \n The subalgebra $\\mathbf{T}'$ of  $\\mathbf{T}_\\mathfrak{m}({\\mathcal O})\\otimes\\mathbb{Q}$ generated by the Hecke\n  operators prime to the level is semisimple, and hence the product of fields $K_i\\simeq K$, each corresponding to a newform of level $N$. Thus, $K\\subset R$ is the image\n  of $\\mathbf{T}'$ in $R$. Note that the summands $K_i$ are not necessarily in bijection with the summands $R_j$, since $K_i$ could potentially be contained in multiple $R_j$s. We have a homomorphism $\\mathbf{T}'\\rightarrow K\\hookrightarrow R$ with kernel\n  $\\mathcal{P}'$. By construction, the ideal $\\mathcal{P}$ lies above $\\mathcal{P}'$.\n    \n\\par Consider the subspace of forms in $S(N, {\\mathcal O})_\\mathfrak{m}\\otimes\\mathbb{Q}_p$ annihilated by the ideal $\\mathcal{P}'$. \n  By duality, it suffices to show that this subspace is $1$-dimensional over $K$. To achieve this, recall that \n  the newform associated to $f$ is the form $g_0$ at level $M_0$, which differs from $N$\n  only at the prime $p$, and at primes $q\\in\\Sigma$. Note that if $q\\in\\Sigma$ and $q\\neq p$, then $U_qf=0$. \n  \n  \\par The argument follows by adding one prime at a time to the level. Let $q\\in\\Sigma$, and let\n  $N_q = M_0q^{e_q}$, where $q^{e_q}$ is the largest power of $q$ dividing $N$.  If $q=p$, let $N_p=M=M_0p$.\n  Then consider the space $S_q$ given as follows.\n  \\begin{itemize}\n  \\item  If $q=p$, and $g_0$ has level divisible by $p$, then $S_q$ is generated by $g_0=g$. \n  \\item If $q=p$, and $g_0$ has level prime to $p$,\n  then $S_p$ is spanned by $g_0(z)$ and $g_0(pz)$. \n  \\item If $q\\neq p$ and $e_q=1$, so $g_0$ has level exactly divisible by $q$, $S_q$ is spanned by the  $g_0(z), g_0(qz)$.\n  \\item If $q\\neq p$ and $e(q)=2$, so $g_0$ has level prime to $q$, $S_q$ is spanned by \\[\\{g_0(z), g_0(qz), g_0(q^2z)\\}.\\]\n  \\end{itemize}\n   Each of these spaces\n  is stable under the Hecke operator $U_q$, and is annihilated by $\\mathcal{P}'$. The eigenvalues of $U_q$ \n  are given as follows.\n  \n  \\begin{itemize}\n      \\item In the first case, $U_p$ has the eigenvalue \n  $\\alpha_p$, which is a $\\mathfrak{p}$-adic unit.\n  \\item In the second, the eigenvalues are $\\alpha_p$, $\\beta_p$, and $\\beta_p$ is a non-unit.\n  \\item In the third, the eigenvalues are $\\alpha_p=\\pm 1$ and $0$.\n  \\item In the fourth, we have $\\alpha_p, \\beta_p, 0$, and \n  $\\alpha_p\\beta_p=q$, so both these numbers are units.\n  \\end{itemize} \n We claim that, in each case, the localization of $S_q$ at $\\mathfrak{m}$ has dimension $1$. In the case when $q=p$, the localization of $S_q$ at $\\mathfrak{m}$ is one-dimensional. This is because $U_p$ is not $\\mathfrak{m}$ and one of the two eigenvalues $\\alpha_p$ and $\\beta_p$ is a $p$-adic unit and the other is not. On the other hand, in the case when $q\\neq p$,\n  $U_q\\in \\mathfrak{m}$. Since, $U_q$ has a unique non-unit eigenvalue ($q\\neq p)$ it follows that the localization of $S_q$ at $\\mathfrak{m}$ has dimension $1$.\n  \\par An iteration of this argument over the primes $q$, using the fact that the level raising operators commute with Hecke operators away from the level, and replacing\n  the form $g_0$ with the $1$-dimensional space produced in the previous step, implies that our space is $1$-dimensional. In greater detail, express $\\Sigma$ as $\\{q_1,\\dots, q_N\\}$ and for $m\\leq N$, set $\\Sigma_m:=\\{q_1, \\dots, q_m\\}$ and $N_m$ be the largest divisor of $N$ which is divisible by $M$ and the primes in $\\Sigma_m$. Assume that the subspace of $S_k(N_m, \\mathcal{O})_{\\mathfrak{m}}\\otimes \\mathbb{Q}_p$ which is annihilated by $\\mathcal{P}'$ is one-dimensional over $K$, and let $g_m(z)$ be a generator of this one-dimensional space. Then, apply the same argument as above to $g_m(z)$ in place of $g_0(z)$, to prove that the subspace of $S_k(N_{m+1}, \\mathcal{O})_{\\mathfrak{m}}\\otimes \\mathbb{Q}_p$ which is annihilated by $\\mathcal{P}'$ is one-dimensional over $K$. This inductive argument shows that the subspace of $S_k(N, \\mathcal{O})_{\\mathfrak{m}}\\otimes \\mathbb{Q}_p$ which is annihilated by $\\mathcal{P}'$, and the result follows from this.\n\\end{proof}\n  \n \n\\begin{Corollary} The quantity $\\eta_f = (f, f)_N$ is is non-zero.\n\\end{Corollary}\n\n\\begin{proof} It suffices to prove the corollary upon extending the pairing to $S(N, {\\mathcal O})_\\mathfrak{m}\\otimes\\mathbb{Q}\\cong\\oplus R_i$. Let $(\\cdot, \\cdot)_{i,j}$ be the restriction of the pairing $(\\cdot , \\cdot)_N$ to $R_i\\times R_j$. It follows from Hecke-equivariance that this pairing $(\\cdot, \\cdot)_{i,j}$ is $0$ if $i\\neq j$. Since the pairing $(\\cdot, \\cdot)_N$ is non-degenerate, it follows that \n\\[(\\cdot, \\cdot)_{i,i}: R_i\\times R_i\\rightarrow K\\] is non-zero. On the other hand, since Lemma \\ref{Tm is a field} gives $R_i\\simeq K$, it follows from $K$-linearity that $(x,x)\\neq 0$ for all $x\\in R_i$ such that $x\\neq 0$. Note that since $f$ is an eigenform, $f\\in R_i$ for some $i$, and hence, we have that $(f,f)_N\\neq 0$.\n  \\end{proof}\n \n To continue, let $e$ denote a fixed generator of the rank-$1$ $\\mathbf{T}_\\frak{m}$-module $S(N, \\mathcal{O})_\\mathfrak{m}$,\nand consider the number $\\phi_f(e)= (f, e) \\in\\mathcal{O}$. Then \nany element of $S(N, \\mathcal{O})_\\mathfrak{m}$ is of the form $t\\cdot e$ for $t\\in \\mathbf{T}_\\mathfrak{m}$, so the Hecke equivariance of the pairing shows that \\[(f, t\\cdot e)_N=(f\\vert t, e)_N = a(1,  f\\vert t)(f, e)_N.\\] In particular, it follows that $f^\\perp$ is the submodule \n$\\mathcal{P}S(N, \\mathcal{O})_\\mathfrak{m}$, and \\[S(N, \\mathcal{O})_\\mathfrak{m}\/f^\\perp\\cong (\\mathbf{T}_\\mathfrak{m}\\otimes \n\\mathcal{O})\/\\mathcal{P}(\\mathbf{T}_\\mathfrak{m}\\otimes \n\\mathcal{O})\\cong \\mathcal{O},\\]\nwhere $\\mathcal{P}$ is the kernel of the canonical homomorphism \n$\\mathbf{T}_\\mathfrak{m}({\\mathcal O})\\rightarrow \\mathcal{O}$ associated to $f$. Since $(f, f)_N$ is nonzero, we find that \n$f\\notin \\mathcal{P}S(N, \\mathcal{O})_\\mathfrak{m}$,  and that the function $\\phi_f$ is determined by the nonzero number \n$\\eta_f = (f, f)_N\\in \\mathcal{O}$.\n\nNext we need to compare the algebraic pairing defined above to the usual Petersson inner product. Thus given a modular form $v(z)=\\sum a_nq^n\\in S(N, \\mathbf{C})$,\ndefine $v^c(z)=\\sum \\overline{a}_n q^n$, where the bar denotes complex conjugation. We define a modified Petersson product on $S(N, \\mathbf{C})$ by setting\n\\begin{equation}\n\\label{modified-petersson}\n\\{v, w\\}_N = \\langle v, w^c\\vert W_N\\rangle_N\n\\end{equation}\nwhere the pairing on the right is the Petersson product. One sees from the definition that $\\{\\cdot, \\cdot\\}_N$ is $\\mathbf{C}$-linear in both \nvariables, and that it satisfies $\\{v\\vert t, w\\}_N= \\{v, w\\vert t\\}_N$, for any Hecke operator $t$, just like the algebraic pairing defined above. \n\nRecall that $\\mathcal{O}$ is the completion of the ring of integers of a number field at a prime $\\mathfrak{p}$ \ncorresponding to an embedding in to\n$\\mathbf{C}_p$. Making an identification of $\\mathbf{C}$ with $\\mathbf{C}_p$, we \nfind that the space $S(N, \\mathcal{O})_{\\mathfrak m}$ is equipped with\ntwo $\\mathbf{C}_p$-valued pairings $(\\cdot, \\cdot)_N$ and $\\{\\cdot, \\cdot\\}_N$. Each pairing is bilinear, and renders the Hecke operators self-adjoint. \nJust as in the algebraic case, we have a function $\\phi_f^\\infty: S(N, \\mathcal{O})\\rightarrow \\mathbf{C}_p$ defined by $v \\mapsto \\{f, v\\}_N$,\nand the adjointness implies that the kernel of $\\phi_f^\\infty$ is the submodule $\\mathcal{P}S(N, \\mathcal{O})_\\mathfrak{m}$. Thus we have two different\n$\\mathbf{C}_p$-valued functions on the rank 1 $\\mathcal{O}$-module $S(N, \\mathcal{O})_\\mathfrak{m}\/\\mathcal{P}S(N, \\mathcal{O})_\\mathfrak{m}$,\nand to compare them, it suffices to evaluate on any given element, say on $f$ itself. One is therefore led to consider\n$\\{f, f\\}_N = \\langle f, f^c\\vert W_N\\rangle_N$, in terms of the usual Petersson product. It is not clear from the definition that this number\nis nonzero; that it is so follows from the same argument that was used in the algebraic case above.\n\n\\begin{Definition}\n\\label{invariant-period}\nDefine a period associated to $f$ and the level $N$ via $\\Omega_{N} = \\frac{\\{f, f\\}_N}{(f, f)_N}$. \n\\end{Definition}\n\nAs stated, this definition depends on the level $N$. We would like to claim\nthat in fact $\\Omega_{N}$ is independent of $N$, and depends only on the \n$p$-stabilized newform $g$, up to a ${\\mathfrak p}$-adic unit. More precisely, we would \nlike to assert that $$\\Omega_N = \\text{unit}\\cdot  \\Omega_{M}$$ where $M=M_0p$.\nHere $\\Omega_M$ is defined by the same prescription as before:\n$$\\Omega_M =\\frac{\\{g, g\\}_M}{(g, g)_M}$$\nwhere the pairings at level $M$ are derived from the Gorenstein condition and the modified\nPetersson product at level $M$. This is Hida's canonical period. The construction is easier\nin this case, since multiplicity one at level $M$ is automatic.\n\nUnfortunately, we cannot quite prove this claim for general weight $k$. \nThe case of weight 2 is known -- this is due\nto Diamond, see \\cite[Theorem 4.2]{dpnas},\nand relies on Ihara's lemma. While there are various\nversions of Ihara's lemma known for weight $k> 2$, the specific version variant\nneeded here does not seem to be available.\n\nThus, we will state the precise variant of Ihara's lemma that we need, and make some remarks\nabout what is known and what is required. \nWe will then prove the independence of the period from the \nauxiliary level under the assumption that a suitable Ihara-type lemma holds.\n\nTo set the framework, fix a prime \n$p\\geq 5$, and consider integers\n$A$, $B$ (the levels), together with an auxiliary odd prime $q\\neq p$. \nWe assume that $A\\vert B$, and that one of \nthe two following conditions holds:\n\\begin{enumerate}\n    \\item $A =q^2B$, and $(B,q)=1$, or\n    \\item $A=qB$ and $B$ is divisible by precisely the first power of $q$. \n\\end{enumerate}\nLet $\\mathbf{{\\mathbf{T}}}_A$ and ${\\mathbf{T}}_B$ denote the Hecke rings generated by all the Hecke\noperators, including $U_q$ of $T_q$, at levels $A$ and $B$ respectively. Recall that we assume\ntrivial nebetype character, so the group in question is $\\Gamma_0$. \nLet $S(A), S(B)$ denote the lattice of cuspforms of levels $A, B$ respectively\nwhose Fourier coefficients are in ${\\mathcal O}$. Let $S(A, {\\mathbf{C}}), S(B, {\\mathbf{C}})$ denote\nthe corresponding complex vector spaces. We have Hecke-equivariant\nand ${\\mathbf{C}}$-bilinear and perfect analytic pairings\n$\\{\\cdot, \\cdot\\}_A:S(A, {\\mathbf{C}})\\times S(A, {\\mathbf{C}})\\rightarrow{\\mathbf{C}}$ and \nand $\\{\\cdot, \\cdot\\}_B: S(B, {\\mathbf{C}})\\times S(B, {\\mathbf{C}})\\rightarrow{\\mathbf{C}}$,\ndefined as above. \nThen let $L_A, L_B$ denote the lattices in $S(A, {\\mathbf{C}}), S(B, {\\mathbf{C}})$ that are\n${\\mathcal O}$-dual to $S_A, S_B$ respectively.  Namely, we have $x\\in L_A$ if and only\nif $\\{x, s\\}_A\\in{\\mathcal O}$, for all $s\\in S_A$, and similarly for $S_B, L_B$. Then\nwe have $L_A\\cong {\\mathbf{T}}_A$ as a ${\\mathbf{T}}_A$-module, and similarly $L_B\\cong {\\mathbf{T}}_B$ over\n${\\mathbf{T}}_B$. \n\nNext, we define a map $\\tau:S(A, {\\mathbf{C}})\\rightarrow S(B,{\\mathbf{C}})$, as follows.\nLet $h = h(z)\\in S(A,{\\mathbf{C}})$, where $z$ denotes a variable in the upper half plane.\nWe define $\\tau$ via \n\\[\\tau\\left(h(z)\\right) = \\begin{cases} h(z) - (U_q h)(qz) &\\text{ if }B=Aq,\\\\\nh(z) - (T_q h)(qz) + q^{k-1}h(q^2z) &\\text{ if }B=Aq^2.\n\\end{cases}\\]\nIt is clear that $\\tau\\left(S(A)\\right)\\subset S(B)$, and that the image is stable\nunder ${\\mathbf{T}}_B$. To check the stability under $U_q$, one can calculate explicitly that\n$U_q=0$ on the image. Thus $\\tau$ is a map that removes the Euler factor at $q$.\n\nNow let $h_A\\in S(A)$ be a modular form that is an eigenvector for every \nelement $t\\in{\\mathbf{T}}_A$. Let $\\mathcal{P}_A$ denote the kernel of the homomorphism $\\phi_A:{\\mathbf{T}}_A\\rightarrow {\\mathcal O}$ associated to $h_A$. \nLet ${\\mathfrak m}_A$ be the maximal ideal of ${\\mathbf{T}}_A$ corresponding\nto the inverse image of the maximal ideal of ${\\mathcal O}$, under $\\phi_A$. \nThen, $h_B=\\tau(h_A)$ is an eigenvector for ${\\mathbf{T}}_B$ (in fact, with $U_qh_B=0$). \nWe may repeat the constructions above for $h_B$, and obtain a height\none prime $\\mathcal{P}_B$ and a maximal ideal ${\\mathfrak m}_B$ inside ${\\mathbf{T}}_B$.\nThe ideals $\\mathcal{P}_A, \\mathcal{P}_B, {\\mathfrak m}_A, {\\mathfrak m}_B$ are required to satisfy additional properties, which we record below:\n\\begin{itemize}\n    \\item The localizations of ${\\mathbf{T}}_A, {\\mathbf{T}}_B$ at ${\\mathfrak m}_A, {\\mathfrak m}_B$ respectively are\n    both Gorenstein, and \n    \\item The localizations of ${\\mathbf{T}}_A, {\\mathbf{T}}_B$ at $\\mathcal{P}_A, \\mathcal{P}_B$ are fields isomorphic\n    to the fraction field $K$ of ${\\mathcal O}$.\n\\end{itemize}\nIn our case, we have presumed that ${\\mathfrak m}_A$ and ${\\mathfrak m}_B$ are such that\nthe residual representations at ${\\mathfrak m}_A$ and ${\\mathfrak m}_B$ are absolutely irreducible and $p$-distinguished. Then, the first property is satisfied. It follows from Lemma \\ref{Tm is a field} that the second condition is also satisfied.\n\nWith these assumptions in place, we can now state the Ihara-type results that we need.\n\n\\begin{hyp}\\label{hypothesis ihara} With the conditions and notations above, and any choice \nof $A, B, q$ as above, we have\n\\begin{itemize}\n    \\item (Ihara-1) We have $\\tau(L_A)\\subset L_B$, and\n    \\item (Ihara-2) $L_B\/\\tau(L_A)$ is ${\\mathcal O}$-torsion-free.\n\\end{itemize}\n\\end{hyp}\n\n\\begin{Remark} As we have already said, \nwe cannot prove these hypotheses in full generality. Thus we limit ourselves\nto some general comments\nhere about their validity. \nThe map $\\tau: S(A,{\\mathbf{C}}) \\rightarrow S(B, {\\mathbf{C}})$ is completely\nexplicit in terms of the usual degeneracy maps of modular curves. To analyze \nthe map $L_A\\rightarrow S(B, {\\mathbf{C}})$ in Ihara-1, and to check that $\\tau$ carries $L_A$\nto $L_B$, one has dualize everything. \nThis is not hard, and can be carried out without \nundue difficulty, although there are various compatibilities to check. \nFor the details,\nwe refer to the forthcoming thesis of Maletto.\n\n\\par However, Ihara-2 is much more delicate. The only known approach\nis to relate the lattices\n$L_A, L_B$ to the parabolic cohomology of the modular groups in question, and \nthen prove the corresponding results for group cohomology. This is standard\nfor weight 2 (see \\cite{wil95}). In the case of weight $k > 2$ and auxiliary level\n$q$, cohomological results due to Diamond for $p> k-2$, and to Manning-Shotton \\cite{manning2021ihara}\nfor general odd $p$, presumably imply the result, although in the case of small\n$p$, one has to be careful with the dualities, since the pairings on the coefficient\nmodules in the cohomology are not in general perfect. For all this, and the missing\ncase of auxiliary level $q^2$, which is not treated at all in the literature,\nwe refer once again to work in progress of Maletto.\n\\end{Remark}\n\nNow we return to the situation of Definition \\ref{invariant-period}. Consider\na $p$-stabilized newform $g$ of level $M$, and the oldform $f$ of level \n$N$ associated to a choice of $\\Sigma$ as before. We want to show that the periods\nat level $N$ and $M$ are equal up to a unit. This turns out to be a simple\ninductive argument, once Ihara's lemma is known. \n\n\\begin{Lemma} Suppose that the Hypotheses Ihara-1 and Ihara-2 hold, for any\n$A, B, q$. Then $\\Omega_N = u\\Omega_N$ for some $p$-adic unit $u$.\n\\end{Lemma}\n\n\n\\begin{proof} We start at level $M$, and work our way upwards, adding one\nprime at time. To spell out the induction, \nwe start with a modular form $h_A$ at level $A$, and we move up\nto level $B=Aq$ or $Aq^2$, and replace $h_A$ with the $q$-depleted form $h_B$.\nIn this situation, we are required to show that the periods of $h_A$ and $h_B$\nare equal up to a unit. \n\nIt is clear from the definition of the periods that $\\Omega_A$ at level $A$ is \ncharacterized up to unit by the properties:\n\\begin{itemize}\n    \\item $\\delta_A:=\\Omega_A^{-1}\\cdot h_A$ is contained in $L_A$,\n    \\item $L_A\/{\\mathcal O}\\delta_A$\nis torsion-free.\n\\end{itemize} Similarly, the period $\\Omega_B$ at level $B$ is characterized\nby:\n\\begin{itemize}\n    \\item $\\delta_B:=\\Omega_B^{-1}\\cdot h_B$ is contained in $L_B$,\n    \\item $L_B\/{\\mathcal O}\\delta_B$\nis torsion-free.\n\\end{itemize} Since $\\tau(h_A)=h_B$  by definition, Ihara-1 \nshows that $\\delta_B':=\\tau(\\delta_A)=\\tau(\\Omega_A^{-1}h_A)$ is contained in $L_B$. Let $u$ be such that $\\delta_B=u \\delta_B'$. We show that $u$ is a $p$-adic unit. We have that $\\delta_B'\\in L_B$ and $L_B\/\\mathcal{O} \\delta_B$ is torsion-free. Hence, $u^{-1}$ is contained in $\\mathcal{O}$. According to\nIhara-2, $L_B\/\\tau(L_A)$ is torsion-free. We have that $u^{-1}\\delta_B=\\delta_B'=\\tau(\\delta_A)\\in \\tau(L_A)$. Hence, $\\delta_B$ is contained in $\\tau(L_A)$, and we write $\\delta_B=\\tau(\\eta_A)$. Since the map $\\tau$ is injective, it follows that $u^{-1} \\eta_A=\\delta_A$. Since $L_A\/\\mathcal{O}\\delta_A$ is torsion-free, it follows that $u\\in \\mathcal{O}$. We have shown that $u\\in \\mathcal{O}$ and $u^{-1}\\in \\mathcal{O}$, so we have deduced that $u\\in \\mathcal{O}^\\times$.\nTherefore $\\Omega_A=u\\Omega_B$ for some unit $u$.\n\\end{proof}\n\n\\begin{Remark}\nTo get a nice formula at the end, and to check that our final\nformulae agree with those in \\cite{lz16}, we need to further\ncalculate further. \nWe have shown above that the ratio of the algebraic\nand analytic pairings at level $M$ and level $N$ are the same. It remains to express everything in terms of the \nnewform $g_0$ associated to $f$ and $g$. On other words, we have to bring everything down to level $M_0$. There are\ntwo cases to consider, depending on whether or not the $p$-stabilized form is new or old at $p$ (so $M=M_0p$ \nor $M=M_0$). \n\nStart with the case that $g_0$ is old at $p$. Then a further calculation (which we omit) shows that \n$\\{g, g\\}_M = E_p\\cdot \\langle g_0, g_0\\rangle_{M_0}$, where\n$E_p = \\pm p^{1-k\/2} \\alpha_p(1-\\frac{p^{k-2}}{\\alpha_p^2})(1-\\frac{p^{k-1}}{\\alpha_p^2})$, and \n$g_0$ is the newform of level $M_0$ associated\nto $f$, and $\\alpha_p$ is the unit root of the Hecke polynomial. The  factors\n$1-\\frac{p^{k-2}}{\\alpha_p^2}$ and $1-\\frac{p^{k-1}}{\\alpha_p^2}$ are units for $k>2$.  \nWhen $k=2$, the term $1-1\/\\alpha_p^2$\nmay be a non-unit; this is so precisely when $g_0$ is congruent to a $p$-new form of of level $pM_0$.\nThe number  $1-1\/\\alpha_p^2$ is the relative congruence number of Ribet. \n\nIn this situation, we define $$(g_0, g_0) = \\frac{(g, g)_{M}}{1-1\/\\alpha_p^2}.$$\nNote that $\\alpha_p\\neq \\pm 1$, by the Weil bounds.\nWe remark that it can be shown that in fact $(g_0, g_0)$ as defined above coincides with the pairing of $g_0$ with \nitself defined via a Gorenstein pairing at level $M_0$ (as opposed to the $p$-stabilized level \n$M=M_0p$). We don't need the this result, but mention it simply to justify the notation. Wes refer the reader to \\cite{wil95}, Chapter 2, Section 2, for a full discussion of relative congruence numbers in weight 2.\n\nIf $f$ is new at $p$ (and hence of weight 2), then of course $\\{f, f\\}_M = E_p \\langle g, g\\rangle_M$\nwhere $E_p=\\pm 1$ is the eigenvalue of the Fricke involution. \n\nThus, we define a canonical period $\\Omega$ associated to the newform $g_0$ corresponding to $f$ as follows:\n$$\\Omega =\\Omega_{g} = {\\Omega_M}.$$\n\nEvidently, $\\Omega_M = \\op{unit}\\cdot {\\Omega_N}$. \nWe can evaluate the period $\\Omega$ more explicitly, as follows. If $f$ is old at $p$, we have \n$$\\Omega_M = \\frac{\\{g, g\\}_M}{( g, g)_M} = E_p \\frac{\\langle g_0, g_0\\rangle_{M_0}}{( g, g)_{M}}=\n\\text{unit}\\cdot p^{1-k\/2}\\frac{\\langle g_0, g_0\\rangle_{M_0}}{( g_0, g_0)_{M_0}}.\n$$\nIf $f$ is new at $p$, so that $g=g_0$ and $M=M_0$, and we are in weight 2, we have \n$$\n\\label{period2} \\Omega_M =  \\frac{\\{g, g\\}_M}{( g, g)_M} =  E_p \\frac{\\langle g_0, g_0\\rangle_{M_0}}{( g_0, g_0)_{M_0}}\n=\\text{unit} \\cdot \\frac{\\langle g, g\\rangle_M}{( g, g)_M}\n$$\n\nA common way of expressing the above formulae is \n\\begin{equation}\n\\label{can-period}\n\\Omega = \\Omega_g = \\text{unit}\\cdot p^{1-k\/2}\\cdot \\frac{\\langle g_0, g_0\\rangle_{M_0}}{( g_0, g_0)_{M_0}}.\n\\end{equation}\n\n\\end{Remark}\n\n\\begin{Remark} We remind the reader that $g$ is the $p$-stabilized newform associated to the newform $g_0$. The numerator\nin the quotient appearing in the expression above is the usual Petersson norm of the newform $g_0$.  The quotient in the expression above is precisely \nthe period appearing in \\cite{lz16}. The additional \nfactor $p^{1-k\/2}$ which shows up when \n$k >2$ is important -- it shows up in the formulae   (\\ref{petersson-formula}), and those of Schmidt in \n\\cite{schmidt86}, \\cite{schmidt88}, where it is simply carried \naround, and the eventual result is proven only up to a rational\nconstant. As we will see, the factor appearing in our $\\Omega$\nis exactly what is needed to cancel unwanted powers of $p$ arising from (\\ref{petersson-formula}).\n\\end{Remark}\n\nIf $h\\in S(N,{\\mathcal O})_{\\mathfrak m}$ is arbitrary, then we have $\\frac{(f, h)_N}{(f, f)_N}=\n\\frac{\\{f, h\\}_N}{\\{f, f\\}_N} = \\frac{\\{f, h\\}_N}{(f, f)_N\\Omega_{N}}$.\nIn view of the independence of the period on the level, we get the following key evaluation formula, valid for any $h\\in S(N, {\\mathcal O})_{\\mathfrak m}$:\n\n\n \\begin{Proposition} \n \\label{evaluation-formula} Assume that the Ihara hyotheses are valid. Then\nwe have $(f, h)_N = \\frac{\\{f, h\\}_N}{\\Omega_{N}} =\\op{unit}\\cdot \\frac{\\{f, h\\}_N}{\\Omega_{g}}$. The quantity $ \\frac{\\{f, h\\}_N}{\\Omega_{g}}=\n\\op{unit}\\cdot  p^{1-k\/2}\\cdot\n \\frac{\\{f, h\\}_N\\cdot (g_0, g_0)_{M_0}}{\\langle g_0, g_0\\rangle_{M_0}} $ is ${\\mathfrak p}$-integral.\n \\end{Proposition}\n\n\\begin{Remark} Our next task will be to apply the machinery developed above to the case where $h=\\mathcal{H}_{\\chi}(n)$ is derived from a product \nof a theta\nseries and and Eisenstein series. However, there are two problems. First, this product is unlikely to be cuspidal, and second, it is not an element \nof $h\\in S(N, {\\mathcal O})_{\\mathfrak m}$. Some care is  therefore required. The number $\\{f, h\\}_N = \\langle f, h^c\\circ W_N\\rangle_N$ \n makes sense for any $h\\in M_k(N, {\\mathcal O})\\otimes\\bar{\\Q}$, since $f$ is cuspidal. Since the maximal ideal ${\\mathfrak m}$ corresponding to $f$ is residually irreducible,\n we find that if $e_{\\mathfrak m}$ is the idempotent in the Hecke algebra ${\\mathbf{T}}=\\oplus {\\mathbf{T}}_{{\\mathfrak m}_i}$ corresponding\nto the maximal ideal ${\\mathfrak m}$, then $h\\circ e_{\\mathfrak m}$ is cuspidal for any modular form $h$ of level $N$ and weight $k$. We claim now that\n that $\\{f, h\\circ e_{\\mathfrak m} \\}_N=\\{f, h\\}_N$. To see this, note that $f\\circ e_{\\mathfrak m} = f$, so we get $$\\langle f, h^c\\circ W_N\\rangle_N = \n \\langle f\\circ e_{\\mathfrak m},  h^c\\circ W_N\\rangle_N = \\langle f, h^c\\circ W_N\\circ e_{\\mathfrak m}^*\\rangle = \\langle f, h^c\\circ e_{\\mathfrak m}\\circ W_N\\rangle_N.$$ \n \n \n \nThus we may replace $h$ with $h\\circ e_{\\mathfrak m}$, and define $( f, h)_N = (f, h\\circ e_{\\mathfrak m})_N$ for any $h\\in M_k(N, {\\mathcal O})\\otimes\\bar{\\Q}$.\nThen the same formalism as above applies. In particular, \nwe still have $\\frac{(f, h)_N}{(f, f)_N}=\n\\frac{\\{f, h\\}_N}{\\{f, f\\}_N} = \\frac{\\{f, h\\}_N}{(f, f)_N\\Omega_{f, N}}$, and the conclusion of Proposition \\ref{evaluation-formula} applies without change.\n\\end{Remark}\n\n\\subsection{Integrality}\nIn view of the considerations above, we are led to compute the algebraic pairings $( f, \\mathcal{H}_\\chi(n))_N$ and $(f,  \\mathcal{H}_\\chi(n))_N$, with\n$\\mathcal{H}_\\chi(n)$ and $\\mathcal{H}_\\chi^m(n)$ being as defined in (\\ref{hchi}) and (\\ref{hchim}). We know already that \n$\\mathcal{H}_\\chi(n)$ and $\\mathcal{H}_\\chi^m(n)$ are integral, hence the corresponding pairings are integral as well. It remains only to\nrelate them to special values of $L$-functions. The starting point is Proposition \\ref{evaluation-formula}, which reduces the calculation\nto that of  analytic pairings $\\{f, \\mathcal{H}_\\chi(n)\\}_N$ and $\\{f,  \\mathcal{H}_\\chi(n)\\}_N$.\n\nRecall that $\\chi=\\psi\\eta$ with $\\eta$ ramified of conductor $m_\\chi$. Pick any $m\\geq m_\\chi$. \nWe start with  $\\{f, \\mathcal{H}_\\chi^m(n)\\}_N = \\langle f, T_\\psi(\\tilde{H}_\\chi(n)^{\\op{hol}}\\circ U_p^{2m-1})^c\\circ W_N\\rangle$. Since $T_\\psi$\nand complex conjugation commute, there exists a constant $C$ such that we get\n\\begin{align*}\n\\{f, \\mathcal{H}_\\chi^m(n)\\}_N &= \\langle f, T_\\psi(\\tilde{H}_\\chi(n)^{\\op{hol}}\\circ U_p^{2m-1})^c\\circ W_N\\rangle_N\\\\\n& = \\langle f\\circ W_N, \\tilde{H}_{\\overline\\chi}(n)^{\\op{hol}}\\circ U_p^{2m-1}\\circ W_{c_\\psi^2}\\circ t_\\psi \\rangle_N\\\\\n& = \\langle f\\circ W_N, \\tilde{H}_{\\overline\\chi}(n)^{\\op{hol}}\\circ U_p^{2m-1}\\circ W_{c_\\psi^2}\\rangle_{N_\\psi}\\\\\n& =C\\cdot  \\langle f\\circ W_N, {H}_{\\overline\\chi}(n)^{\\op{hol}}\\circ W_{N_\\chi}\\circ U_p^{2m-1}\\circ W_{c_\\psi^2}\\rangle_{N_\\psi}\\\\\n& = C \\cdot  \\langle f\\circ W_N, {H}_{\\overline\\chi}(n)\\circ W_{N_\\chi}\\circ U_p^{2m-1}\\circ W_{c_\\psi^2}\\rangle_{N_\\psi}\\\\\n& =  C \\cdot  \\langle f, {H}_{\\overline\\chi}(n)\\circ W_{N_\\chi}\\circ U_p^{2m-1}\\circ W_{Nc_\\psi^2}\\rangle_{N_\\psi}.\n\\end{align*}\n\nThe constant $C$ comes from the definition of $\\tilde{H}_\\chi(n)$. In the last equality, we have used the fact that \nthe matrix defined by Atkin-Lehner in \\cite{al}, bottom of page 138, giving the involution $W_N$ at level $N_\\psi$ also satisfies the definition of $W_N$\nat level $N$. This fact can also be seen from the point of view of representation theory, since the Atkin-Lehner operators\nadmit a purely local definition.\n\nRecall\nequation (\\ref{petersson-formula}), which states that\n\\begin{equation*}\n(4\\pi)^{-n\/2}\\Gamma(n\/2)p^{(2m_\\chi-1)(k\/2-1)} D_f(\\chi,n) =  \\alpha_p^{2(m_\\chi-m)} \\langle f, H_{\\overline\\chi}(n)\\circ W_{N_\\chi}\\circ U_p^{2m-1}\\circ W_{N_\\psi}\\rangle_{N_\\psi}.\n\\end{equation*}s\n\nUsing this formula, and plugging in all the definitions, we obtain\n\\begin{Corollary} \n\\label{petersson-formula-explici-riou twisted trace\nt}\nSuppose that $\\eta$ is ramified and $m\\geq_\\chi$ is any integer. \nThere exists a ${\\mathfrak p}$-adic unit $u$ depending only on $n$ and $k$ such \nthat we have $$u\\cdot \\frac{p^{1-k\/2}}{\\pi^{n}}\\left( \\frac{p^{n-1}}{\\psi(p)}\\right)^{m_\\chi} \\left(\\frac{1}{\\alpha_p^2}\\right)^{m-m_\\chi}\n\\cdot g(\\overline{\\eta})\\cdot D_f(\\chi,n) =  \n \\{ f, \\mathcal{H}^m_\\chi(n)\\}_N.$$ where $\\mathcal{H}_\\chi^m(n) = T_\\psi(\\tilde{H}_\\chi(n)^{\\text{hol}}\\circ U_p^{2m-1})\\in M_k(N, {\\mathcal O})\\otimes\\bar{\\Q}$\n has $\\mathfrak{p}$-integral coefficients, and \n $\\tilde{H}_\\chi(n)= \\frac{\\Gamma((n+1)\/2)}{\\pi^{(1+n)\/2}} p^{m_\\chi(3-2k+2n)\/2} \\cdot \\frac{\\sqrt{c_\\psi p^{m_\\chi}}}{g(\\chi)} \\cdot H_{\\chi}(n)\\circ W_{N_\\chi}$.\n\n\\end{Corollary}\n\n\\begin{proof} This is a direct computation, using Lemmas, \\ref{eisen-fourier} and \\ref{theta-fourier} \nand applying the doubling formula for the $\\Gamma$-function in the formula for the coefficients $d_{j,\\nu}$; see also Lemma 4.2 of \\cite{schmidt86}. \nOne has also to use the factorization $g(\\chi) = \\psi(p^{m_\\chi})\\eta(c_\\psi)g(\\psi)g(\\eta)$. \nThe constant $u$ collects up all the various powers of $2, i$, and other quantities prime to ${\\mathfrak p}$. \n\\end{proof}\n\n\nObserve that the formula above contains the nuisance factor $p^{1-k\/2}$, which also appears in our period. \nAssuming Hypothesis \\ref{hypothesis ihara}, so that $\\Omega_{g}=\\Omega_M = \\text{unit}\\cdot \\Omega_N$, \nwhere $g$ is the $p$-stabilized newform at level $M_0p$, \nand plugging in (\\ref{can-period}), \n we find that \n \\begin{align}\n  \\label{integrality-formula-1}\n (f, \\mathcal{H}_\\chi^m(n))_N  & =  u\\cdot p^{1-k\/2}\\cdot \\left( \\frac{p^{n-1}}{\\psi(p)}\\right)^{m_\\chi}  \\left(\\frac{1}{\\alpha_p^2}\\right)^{m-m_\\chi}\n\\cdot \\Gamma(n)\\cdot G(\\overline{\\eta})\\cdot\\frac{D_f(\\chi,n)}{\\pi^n \\Omega}  \\\\\n& =u' \\cdot \\left( \\frac{p^{n-1}}{\\psi(p)}\\right)^{m_\\chi} \\cdot  \\left(\\frac{1}{\\alpha_p^2}\\right)^{m-m_\\chi}\n\\cdot \\Gamma(n) \\cdot G(\\overline{\\eta})\\cdot\\frac{(g_0, g_0)_{M_0}}{\\pi^n \\langle g_0, g_0\\rangle_{M_0}}\\cdot D_f(\\chi,n).\n\\end{align}\nHere $u'$ is some other unit, independent of $\\chi$. \n\nFinally, we have to deal with $\\mathcal{H}_\\chi(n)\\circ e$. It is not hard to see that the twisted trace operator $T_\\psi$, which goes from level $N_\\psi$ to\nlevel $N$, commutes with the Hecke operator $U_p$, since $N_\\psi\/N$ has no common factor with $p$. There does not seem to be any particularly\npleasant way to deduce this fact from a classical perspective where the trace operator is given by matrices with rational integer entries,\nbut it is more or\nless obvious from the point of view of representation theory, since the local trace involves primes away from $p$, while $U_p$ is concentrated at $p$. \nIt is also evident it one considers modular forms as functions on test objects on moduli spaces of enhanced elliptic curves  -- $U_p$ is a sum over certain\nsubgroups of order $p$, while the trace from level $N_\\psi$ involves subgroups of order prime to $p$. Anyway, we take this fact for granted, so that\n$$\\mathcal{H}_\\chi^m(n) = T_\\psi(\\tilde{H}_\\chi(n)^{\\text{hol}}\\circ U_p^{2m-1}) = \\mathcal{H}_\\chi^m(n) =( T_\\psi(\\tilde{H}_\\chi(n)^{\\text{hol}})\\circ U_p^{2m-1}.$$\nWe may then consider an suitable increasing sequence of integers $m$, divisible by $p-1$, so that the forms $\\mathcal{H}_\\chi^m(n) \\circ U_p$ converge to \n$\\mathcal{H}_\\chi(n)\\circ e$.  The algebraic inner product is $p$-adically continuous as a function of the second variable, since it is a bounded ${\\mathcal O}$-linear functional,\nand we conclude that \n\\begin{equation}\n\\label{integrality-formula-2}\n (f, \\mathcal{H}_\\chi(n)\\circ e)_N  =\\op{unit} \\cdot \\left( \\frac{p^{n-1}}{\\psi(p)\\alpha_p^2}\\right)^{m_\\chi} \n\\cdot \\Gamma(n) \\cdot G(\\overline{\\eta})\\cdot\\frac{(g_0, g_0)_{M_0}}{\\pi^n \\langle g_0, g_0\\rangle_{M_0}}\\cdot D_f(\\chi,n).\n\\end{equation} In particular, the right-hand side is integral.\n\nObserve that the quantity on the right is (up to the unit factor) \nthe one appearing in the definition of the\n$\\psi$-twisted $p$-adic L-function of Schmidt (with $\\psi$ being fixed, and $\\eta$ varying over characters of \n$p$-power conductor; see \\cite{schmidt86}, Theorems 3 and 4, or  \\cite{lz16}, Theorem 2.3.2,\nwhere a formula for the $p$-adic L-function for $D_g(\\psi, s)$ is given. Observe that our formulation\nis slightly different than the one given in these references, owing to the the fact that we have normalized the period to give\na function that is integral, rather than simply bounded. Furthermore, our formulation incorporates the normalization factor\nof the congruence number $(g, g)_M$ alluded to in Proposition 2.3.5 of \\cite{lz16}. \n\n\n\n\n\\subsection{Level raising and congruences}\n\\label{congruence-section}\nIn view of the construction given above, it is more or less clear at this point to verify that that the $p$-adic L-functions satisfy\ngood congruences. To state the result, consider two $\\mathfrak{p}$-ordinary and $\\mathfrak{p}$-stabilized\nnewforms $g_1, g_2$, which are such that the residual \nrepresentations $\\overline{\\rho}_{g_1}$ and $\\overline\\rho_{g_2}$ are isomorphic. We assume, as always, that the nebentype\ncharacter is trivial. Let $M_0$ denote the prime-to-$p$\npart of the Artin conductor of $\\rho$. Then each $g_i$ has level $M_i$ divisible by $M=M_0p$.  Furthermore, each \n$M_i$ is divisible\nby precisely the first power of $p$, and if\n $q\\vert M_i\/M$, then $\\text{ord}_q(M_i\/M)= 1, 2$, by Lemma \\ref{dtlemma}.  \n \nIt is evident that the eigenvalues $a(q,f_i)$ are congruent modulo $\\mathfrak{p}$ for all primes $q$ away from $M_1M_2$.\nOur goal is to adjust the $g_i$ so that the Hecke eigenvalues are congruent at \\emph{all} primes. In order to apply the previous \narguments involving semisimplicity of the relevant local components of the Hecke algebra, \nwe want to ensure in fact that the eigenvalues of $T_q$ at the primes $q\\neq p$  that divide either $M_1$ or\n$M_2$ are simply equal to zero. Furthermore, both forms have to end up at the same level, and the level has to be sufficiently\nsmall as to maintain  control over the semisimplicity of the bad Hecke operators. As we have already remarked\nin the introduction,  it is clear that \nstriking out Euler factors gives forms with congruent Hecke eigenvalues, but is not\nat all clear that the the forms so-obtained actually have the same levels. \n\nWe remark also that this part of the argument \nrelies heavily on the fact that we are dealing with trivial central character. \n\nThus, fix $i$, and let $j=i+1$ modulo $2$ denote\nthe other choice. We will define two sets $\\Sigma_i, \\Sigma_j$ of primes $q$ and then take $\\Sigma:=\\Sigma_1\\cup \\Sigma_2$.\nIt suffices to just define $\\Sigma_i$, since the prescription for $\\Sigma_j$ is the same. \nConsider first the prime $2$. If $4$ divides $N_i$, we do nothing, since the $U_2$ eigenvalue is already zero. If $4\\nmid M_i$, \nwe put $2$ in $\\Sigma_i$. This increases the level by either 2 or 4. Now consider an odd prime $q$. The cases are as follows.\n\\begin{enumerate}\n\\item Suppose that $M_i$ is divisible by $q^2$. In this case, we do nothing. \n\\item Suppose that $M_i$ is divisible by precisely the first power of $q$. In this case, we place $q$ in $\\Sigma_i$. \n\\item Suppose that $q$ does not divide $M_i$. If $q$ divides $M_j$, then we put $q$ in $\\Sigma$. \n\\end{enumerate}\n\nFinally, we may enlarge both sets $\\Sigma_i$ and $\\Sigma_j$ by adding (to both!)\n finitely many other primes that do not divide either $M_1$ or $M_j$, or the conductor \nof $\\psi$ (which is assumed to be relatively prime to $2M_1M_2$). \n\nWith this choice of $\\Sigma_i, \\Sigma_j$, we replace $g_i$ (resp. $g_j$ with\n$f_i$ (resp. $f_j$)  by striking out coefficients in the Fourier expansion of $g_i$ (resp. $g_j$) at the primes dividing \n$\\Sigma_i$ (resp. $\\Sigma_j$). Then we claim that $f_i$ and $f_j$ have the same level. Most of the following proposition\nis obvious, except for the second statement. It is harder to state the proposition clearly than to prove it. \n\n\\begin{Lemma}  The following statements hold.\n\\begin{enumerate}\n\\item The sets $\\Sigma_i$ and $\\Sigma_j$ are such that $q\\in\\Sigma_i$ implies $\\text{ord}_q(N_i)\\leq 1$, and similarly for $\\Sigma_j$. \n\\item  The forms $f_i, f_j$ are of the same \nlevel $N$, where $N\/M_i$ and $N\/M_j$ are cube-free. \n\\item If $N\/M_i$ is divisible by $q^2$, then $q\\nmid M_i$, and similarly for  $M_{j}$.\n\\item If $N\/M_i$ divisible by exactly the first power of $q$, then $M_i$ is divisible by precisely\nthe first power of $q$, and similarly for the index $j$.\n\\item  Each\nof $f_i$ and $f_j$ is an eigenvector for $U_q$ with eigenvalue $0$, for any prime $q\\neq p$ which divides $N$. \n\\item Each of $f_i$ and $f_j$ is an eigenvector for the operator $U_p$, with eigenvalues $\\alpha_i(p)$ and\n$\\alpha_j(p)$ respectively, and we have $\\alpha_i(p)\\equiv\\alpha_j(p)\\pmod{\\mathfrak{p}}$. \n\\end{enumerate}\n\\end{Lemma}\n\n\n\\begin{proof} This is yet another application of Lemma \\ref{dtlemma}. Consider first the prime $2$.\n If $4\\vert M_0$, then we are making no change at $2$, and both $f_i$ and $f_i$ have \nlevel exactly divisible by $4$. If  $4\\nmid M_0$, then the $2$-part of $M_i$ and $M_j$ is bounded by $4$. \nSince we have $2\\in\\Sigma_i, \\Sigma_j$, we end up at level divisible by exactly 4. \n\n\nNow consider an odd prime $q$. If $q^2$ divides $M_0$, then each\nof the $g_i$ has the property that the $U_q$ eigenvalue is zero (since the central character is trivial) and $a(n, g_i)=0$ \nfor any $n$ divisible  by $q$ in any case. Such primes are therefore excluded from the sets $\\Sigma_i,\\Sigma_j$, since\nno adjustment is needed in this case. In this situation, the prime $q$ does not divide either of $M_i\/M_0$ or \n$M_j\/M_0$ and the $q$-part of the levels is the $q$-part of $M_0$, which is the same in either case. \n\nNext, consider the case where $q\\vert M_0$ to the first power. In this case, $q^2$ may divide $M_i$ or $M_j$ or both.\nThe list of possible cases shows that $q^3$ does not divide either $M_i$ or $M_j$. If $M_i$ is divisible by $q^2$, \nwe do nothing. If $M_i$ is divisible by exactly $q$, then $q\\in\\Sigma_i$, and we end up with level $q^2$. In either case,\nwe get a level whose $q$-part is $q^2$.  \n\nFinally, we have to deal with $q\\nmid M_0$. If $q^2$ divides $M_i$, then $q^2$ exactly divides $M_i$.\n Then we do nothing and we remain at level with $q$-part equal to $q^2$. \n If $q$ exactly divides $M_i$, then $q\\in\\Sigma_i$, and the level goes up by $q$ to \n$q^2$ once again. If $q\\nmid M_i$ then there are some sub-cases.  If $q\\vert M_j$, then $q\\in \\Sigma_i$ and $f_i$\nhas level whose $q$-part is $q^2$, again.  If $q\\nmid M_j$, then either $q$ is in both $\\Sigma_i$ and $\\Sigma_j$ or in neither. Then we get\neither no level at $q$ (and this case only occurs when $q$ is prime to everything in sight) or level $q^2$ at $q$ (in case\n$q$ divides one or both of $M_i, M_j$, or if $q$ is one of the supplementary primes that was added to both sets. \n \\end{proof}\n \n We can now state the theorems around congruences for the imprimitive symmetric square L-function, but we need to \n recall the hypotheses and notation.\n\nThus, suppose that  $g_1, g_2$ are newforms of weight $k$, and \n level $M_1, M_2$ respectively, and that ${\\mathfrak p}$ is a prime\n of $\\bar{\\Q}$ with residue characteristic $p\\geq 5$ such that  the the Fourier\n coefficients $a(q, g_i)$ satisfy the congruence $a(q, g_1)\\equiv a(q, g_2)\\pmod{{\\mathfrak p}}$, for each prime \n $q\\nmid N_1N_2p$. Suppose also that the $g_i$ are ordinary at $p$, and that each $g_i$ has trivial \n central character, and the corresponding residual representation is absolutely irreducible and $p$-distinguished.\n Let $\\alpha_{i,p}$ denote the eigenvalue of the ${\\mathfrak p}$-stabilized newform associated to $f_i$.\n  Let $\\psi$ denote an even character of conductor prime to $2pM_1M_2$, and let $\\eta$ denote a nontrivial\n Dirichlet character of $p$-power conductor. Let $\\Omega_1, \\Omega_2$ denote the canonical periods associated\n to the $g_1, g_2$ and the prime ${\\mathfrak p}$, as above. Thus $\\Omega_i = p^{1-k\/2} \\frac{(g_i, g_i)_{M_i}}{\\langle g_i, g_i \\rangle_{M_i}}$.\n  Let $n$ denote an odd integer in the range $1\\leq n\\leq k$. Let the sets $\\Sigma_1, \\Sigma_2$ be defined as before and recall that $\\Sigma=\\Sigma_1\\cup\\Sigma_2$. Let $f_1, f_2$ denote the imprimitive forms of level $N$, associated to the forms $g_1, g_2$, and the set\n  $\\Sigma$. We assume also that the coefficient ring ${\\mathcal O}$ contains the values of the character $\\psi$. \n \n \\begin{Th}\\label{special values congruence} Let the hypotheses be as above. \n Then there exist units $u_i$, depending only on $g_i$, and $n$, such that we have the congruence \n \n $$u_1 \\cdot \\left( \\frac{p^{n-1}}{\\psi(p)\\alpha_{1,p}^2}\\right)^{m_\\chi} \n\\cdot\\Gamma(n)\\cdot G(\\overline{\\eta})\\cdot \\frac{D_{f_1}(\\chi,n)}{\\pi^n\\Omega_1} \\equiv\nu_2 \\cdot \\left( \\frac{p^{n-1}}{\\psi(p)\\alpha_{2,p}^2}\\right)^{m_\\chi} \n\\cdot \\Gamma(n) \\cdot G(\\overline{\\eta})\\cdot\\frac{ D_{f_2}(\\chi,n)}{\\pi^n\\Omega_2} \\pmod{{\\mathfrak p}}.\n$$\n  \\end{Th}\n \n \\begin{proof} This follows from the continuity of the functional  $S(N, {\\mathcal O})\\rightarrow {\\mathcal O} $ given by $ x \\mapsto \n ( x, \\mathcal{H}_\\chi(n)\\circ e)_N$. \n \\end{proof}\n \n \n\n\n\\subsection{The primitive L-function and $p$-adic interpolation}\n\\label{primitive-ss-Lfunction}\nWe now write down the relationships between the primitive and variously imprimitive L-functions, and \nthe interpolation properties that characterize the $p$-adic L-functions.\nFor notational\nsimplicity, let us fix the $p$-stabilized\nnewform $g_0$, and write the level of $g_0$ as $M_0$. The corresponding $p$-stabilized newform will be denoted by $g$\nand its level shall be denoted by $M$.\nFor each prime $q$, we have associated to $g$, a complex\nrepresentation $\\pi_q$ of $\\op{GL}_2(\\mathbb{Q}_q)$ \nassociated to $g$. \nThe first task is to work out the Euler factors of \nthe symmetric square lift $\\Pi_q$ of $\\pi_q$ to $\\op{GL}_3$. This is all contained in \\cite{GJ78}, \nand is recapitulated in Section 1 of \\cite{schmidt88},\nespecially Lemmas 1.5 and 1.6,\nbut some translation is required. We notice first of all\nthat the representations $\\Pi$ and $\\Sigma$ considered by Schmidt\nare not exactly the symmetric square of \\cite{lz16}, and that his \nnormalization introduces an inverse when comparing\nwith the Euler product of Shimura considered here. The exact \nrelationship is given in the last line of page 603. For us, the\npoint is that the Euler factors of our $D_g(\\chi_0, s)$ coincide\nwith Schmidt's $L(s-k+1, \\Sigma\\otimes\\chi^{-1})$. for any \nprimitive (in our case even and non-quadratic) Dirichlet\ncharacter $\\chi_0$ with corresponding idele class character $\\chi$,\nat almost all primes. With this normalization in mind, one can\nread off the Euler factors at the bad primes for the automorphic\nrepresentation $\\Pi$ from Schmidt's Lemmas 1.5 and 1.6. To state \nthe result, let us write $\\mathscr{L}(r_{g_i}, \\chi, s)=\\prod_q\nP_\\ell(r_{g_i}, \\chi, q^{-s})$ to denote the complex L-function\nassociated to the Galois representation $r_{g_i}\\otimes\\chi$. This function\nis denoted by $L(\\op{Sym}^2 g\\otimes\\chi, s)$ in \\cite{lz16}. \nFor all but finitely primes $q$, we have \n$$P_q = \\left( (1-\\chi(q)\\alpha_q\\beta_q q^{-s})(1-\\chi(q)\\beta_q^2q^{-s})(1-\\chi(q)\\alpha_q^2q^{-s})\\right)^{-1}.$$\nSchmidt's formulae show that for any choice of the set $S_0$ of \nbad primes, that the quantity\n$$\\frac{D_f(\\chi, s)}{\\mathscr{L}(r_{g_i}, \\chi, s)}=\\prod_q P_q(\\chi, q^{-s})$$\nis a product of polynomials $P_q(\\chi, X)$ in the variables $X=q^{-s}$, \nfor $q\\in\\Sigma$,\nwhose only zeroes lie on the line $s=k-1$ (compare \\cite{lz16}, Proposition 2.1.5). \n\nNow, if $q$ is any prime distinct from $p$, we may write \n$q=\\eta_1(q)\\eta_2(q)$, where $\\eta_1:\\mathbb{Z}_p^\\times\\rightarrow \\mu_{p-1}$\nthe Teichm\\\"uller character, and $\\eta_2$ is the projection to the group\n$1+p\\mathbb{Z}$. The quantity $\\eta_2(q)^s$ makes sense for all rational \nintegers $s$. Then, making the substitution $X=\\eta_2(q)^{-s}$, where the variable\n$s$ is a rational integer, in the polynomial $P_q(\\chi, X)$ mentioned above\ngives an Iwasawa function of $s\\in \\mathbb{Z}$ whose values coincide \nwith the complex polynomial $P_q(q^{-s})$ for all integers $s$ divisible by\n$p-1$. \n\nIn the setup of $p$-adic L-functions, we have $\\chi = \\psi\\eta$, where \n$\\psi$ has conductor prime to the level, and $\\eta$ has $p$-power \nconductor. A glance at the formulae in Schmidt shows that in fact\none has $P_q(\\psi, \\eta_2(q)^s) = P_q(\\psi\\eta_1^{-s}, q^s)$ holds for\nany rational integer value of $s$. More generally, if\n$\\eta$ is any character $\\mathbb{Z}_p^\\times$ of the form $x \\mapsto\\eta'(x) x^{-s}$\nwhere $\\eta'$ has finite order and $s$ is a rational integer, then \nwe have $P_q(\\psi, \\eta(q)) = P_q(\\psi\\eta'\\eta_1(q)^{s}, q^{-s})$. Characters\nof the given form are dense in the group of continuous characters of $\\mathbb{Z}_p^\\times$,\nand if we restrict to the case where $s$ is divisible by $p-1$, then we get characters of \nthe Galois group of the cyclotomic $\\mathbb{Z}_p$-extension, by class field theory. Thus\nwe may view the polynomials $P_q$ as being elements of the completed group ring $\\Lambda\n=\\mathcal{O}[[\\mathbb{Z}_p]]$. Here we remark that we are identifying the additive group\n$\\mathbb{Z}_p$ with the multiplicative group $1+p\\mathbb{Z}_p$, via some choice\nof topological generator of the latter. \nIn general, given $\\lambda\\in\\Lambda$, and \nany continuous $\\mathbb{C}_p$-valued character $\\eta$ of $\\mathbb{Z}_p$,\nwe will write $\\lambda(\\eta)$ for the evaluation of $\\lambda$ at $\\eta$. \n\nRecall that $g$ is a $p$-stabilized newform of even weight $k$, with trivial nebentype character,\nand that $\\psi$ denotes an even non-quadratic character of conductor prime to\nthe level. As above, we write $\\eta$ to denote an even Dirichlet character of $p$-power\nconductor, and of $p$-power order (so that the tame part is trivial). Thus\n$\\eta$ may be identified with a character of $1+p\\mathbb{Z}_p$, as above. For each odd\ninteger $n$ in the range $1\\leq n\\leq k-1$, we write $\\eta_n$ for the character\nof $1+p\\mathbb{Z}_p$ given by $x\\mapsto \\eta(x)x^n$. \n\n\nThe primitive $p$-adic L-function associated to the newform $g_0$ of level $M$ \nand the representation\n$r_g\\otimes\\psi$ is an element $L^{\\op{an}}(r_g\\otimes\\psi)$ of $\\Lambda=\\mathcal{O}[[1+p\\mathbb{Z}_p]]$\ncharacterized by \n\\begin{equation}\n\\label{messy-definition}\n\\eta_n(L^{\\op{an}}(r_g\\otimes\\psi)) = \\frac{(-1)^{n-k+1}\\eta(-1)\\Gamma(n)}{{4^k}}\n\\times E_p(n, \\eta)\\frac{G(\\eta)}{(2\\pi i)^{n-k+1}}\\frac{\\mathscr{L}(r_g\\otimes \\psi\\eta_1^{-n}\\eta^{-1}, n)}{\\pi^{k-1}\\Omega_g}.\n\\end{equation}\nfor $n$ odd, $1\\leq n\\leq k-1$, and $\\eta$ even. The period\nin the formula is given by\n$$\\Omega_{g_0} =p^{1-k\/2}\\frac{(g, g)_{M_0}}{\\langle g, g\\rangle_{M_0}}.$$ The Euler factor\n$E_p(n, \\chi)$ is given by\n$$E_p(n,\\chi) = (p^{n-1}\\psi(p)^{-1}\\alpha_p^{-2})^{m_\\chi}$$\nif $\\eta$ is nontrivial and has conductor $p^{m_\\chi}> 1$. If $\\eta$ is trivial and $g$ has level prime to $p$,\nthen \n$$E_p(n, \\eta) = (1- p^{n-1}\\psi(p)^{-1}\\alpha_p^{-2})(1-\\psi(p)p^{k-1-n})\n(1-\\psi(p)\\beta_p^2p^{-n}).$$\nA similar formula holds when $k=2$ and $g$ has level divisible by the first power of $p$; we omit it here, as we\ndo not need it. \n\nObserve\nthat our formula is precisely the same as that in \\cite{lz16}, except that \nwe have scaled by the constant factor $p^{1-k\/2}( g_0, g_0)_{M_0} = \\op{unit} \\cdot (g, g)_M$. It is clear\nthat if such a function exists, then it is characterised by the validity\nof the formula above, for any infinite collection of characters of the form\n$\\eta_n$, for varying $n$ and $\\eta$. \n\nNext we want to define the various imprimitive L-functions. Let $T$ denote any set of prime numbers.\nThe $T$-imprimitive $L$-function $\\mathscr{L}_T(r_g\\otimes\\psi, s)$ is defined by the Euler product (\\ref{shimura-euler-product}).\nThen $L_T^{\\op{an}}(r_g\\otimes\\psi)$ is an element of $\\Lambda$ \ncharacterized by the analogue of (\\ref{messy-definition}) where\none replaces $\\mathscr{L}(r_g\\otimes\\psi\\eta^{-1}, n)$ with $\\mathscr{L}_T(r_g\\otimes\\psi\\eta^{-1}, n)$. \n\n\n\nThe existence of $L^{\\op{an}}(r_g\\otimes\\psi)$ and $L^{\\op{an}}_T(r_g\\otimes\\psi)$ (for $T$ being the empty set)\nwas proven by Schmidt, under our hypotheses.\nHe states in his work that very similar results were obtained by Hida, but never\npublished. Schmidt first established the existence of the imprimitive L-function\nunconditionally \\cite{schmidt86}, and then proved the existence of the primitive \nL-function under\nsome conditions which are subsumed by our hypothesis that $\\psi$ be non-quadratic\n\\cite{schmidt88}. As we have remarked, he was unable to make precise the period, and stated\nhis results with a period that was only determined up to an unknown constant\nmultiple of $(g, g)_M$. He was therefore unable to deduce integrality. Schmidt did not \ncontstruct $L^{\\op{an}}_T(r_g\\otimes\\psi)$ when $T\\neq\\emptyset$, but in fact that follows easily:\none simply multiplies the L-function for the empty set  (which exists) by the appropriate Euler factors \nfrom (\\ref{shimura-euler-product}), each of which is represented by an element of $\\Lambda$. Thus we may assume \nexistence of  $L^{\\op{an}}_T(r_g\\otimes\\psi)$: it is an element of $\\Lambda\\otimes\\mathbb{Q}$ characterized by the formula\n\\begin{equation}\n\\label{messy-definition-2}\n\\eta_n(L^{\\op{an}}(r_g\\otimes\\psi)) = \\frac{(-1)^{n-k+1}\\eta(-1)\\Gamma(n)}{{4^k}}\n\\times E_p(n, \\eta)\\frac{G(\\eta)}{(2\\pi i)^{n-k+1}}\\frac{\\mathscr{L}_T(r_g\\otimes\\psi\\eta_1^{-n}\\eta^{-1}, n)}{\\pi^{k-1}\\Omega_g}\n\\end{equation}\nfor almost all characters $\\eta$ of $1+p\\mathbb{Z}_p$. \n\nThen, the relationship between\nthe primitive and imprimitive $p$-adic L-functions is given by\n\\begin{equation}\n\\label{prim-imprim}\nL^{\\op{an}}_{S_0}(r_g\\otimes\\psi) = \\prod_{q\\in S_0} P_q\\cdot L^{\\op{an}}(r_g\\otimes\\psi).\n\\end{equation}\nA priori, both L-functions above are elements of $\\Lambda\\otimes\\mathbb{Q}$.\n\nNow assume that we have $T=S_0$, where $S_0$ is a set associated to the choice of a set $\\Sigma$ of primes\nsatisfying the conditions in the introduction of this paper. Furthermore, assume that Hypothesis \\ref{hypothesis ihara}\nholds. \n\nWe can now give the proof of the various remaining result on $p$-adic L-functions\nstated in the introduction. \n\nWe start with a simple lemma, which follows directly from the explicit\nformulae in \\cite{schmidt88}, or \nthe observation there on page 605\nthat the polynomials $P_q$ all satisfy $P(0)=1$. \n\n\\begin{Lemma}\nFor any prime $q\\in S_0$, the element $P_q\\in\\Lambda$ has $\\mu$-invariant zero.\n\\end{Lemma}\n\n\\begin{Corollary} We have $\\mu_{S_0}^{\\op{an}}=0\\iff \\mu^{\\op{an}}=0$.\n\\end{Corollary}\n\nThis lemma implies Proposition \\ref{analytic-invariants-intro},\nsimply by taking $g=g_i$, and $\\sigma^{(q)}_i$ to be the degree of the polynomial $P_q$ \nassociated above to $g_i$ at $q$, and using the fact that the \n$\\mu$-invariant of $P_Q$ is zero\nin the formula (\\ref{prim-imprim}).\n\nNext we deal with integrality properties, as stated in Theorem \\ref{integrality-thm-intro}.\nWe claim that in fact both $L^{\\op{an}}_{S_0}(r_g\\otimes\\psi) $ and $L^{\\op{an}}(r_g\\otimes\\psi)$\nlie in $\\Lambda$ are are integral. First consider the imprimitive case.\nThen it follows from the formulae (\\ref{integrality-formula-1})\nand (\\ref{integrality-formula-2}) that the right hand side of \n(\\ref{messy-definition-2}) is integral, for almost all characters $\\eta_n$. Then the \nWeierstrass preparation theorem, applied\nto $L^{\\op{an}}_{S_0}(r_g\\otimes\\psi)$, shows that the latter is an element\nof $\\Lambda$. As for the primitive case, it follows from the imprimitive case,\nand the basic relation fact that the polynomials $P_q$ are integral and\nhave $\\mu$-invariant zero. This proves Theorem \\ref{integrality-thm-intro}.\n\nFinally, we have to deal with congruences. Let $g_1, g_2$ be $p$-congruent newforms satisfying our running conditions.\nLet $S$ denote\nany set of primes including $2$, and the set of primes dividing $M_1M_2$, and let\n$S_0=S\\backslash\\{p\\}$ be as above.  We claim that we have\n\n\\begin{Proposition}\\label{p-adic LFs congruent} Let the notation be as above. Then we have\n $L_{S_0}^{\\op{an}}(r_{g_1}\\otimes\\psi\\eta) \\equiv  u L_{S_0}^{\\op{an}}(r_{g_2}\\otimes\\psi\\eta)\\pmod{{\\mathfrak p}}$, where $u$ is a $p$-adic unit and the congruence\n is that of elements in the completed group algebra ${\\mathcal O}[[{\\mathbb{Z}}_p^\\times]]]$. \n \\end{Proposition}\n \n \\begin{proof} Let $\\Sigma_1, \\Sigma_2$ denote the sets associated to $g_1, g_2$ in Section \\ref{congruence-section}. \n As we have remarked\n$\\mathscr{L}_{\\Sigma_i}(r_{g_i}\\otimes\\psi\\eta, s) = \\mathscr{L}_{S_0}(r_{g_i}\\otimes\\psi\\eta, s) $ as complex L-functions, for each $i$. \nThen the result follows from the congruence of special values in Theorem \\ref{special values congruence}, and the Weierstrass preparation theorem, since\nwe have $D_{f_i}( \\chi, s) = \\mathscr{L}_{S_0}(r_{g_i}\\otimes\\chi, s)$ as complex $L$-functions, and $i=1,2$. \n   \\end{proof}\n\nFinally, we observe that the analytic part of Theorem \\ref{intro-thm} follows immediately, since two congruent Iwasawa functions with \n$\\mu$-invariant zero necessarily have the same $\\lambda$-invariant, and that if one has $\\mu$-invariant zero, then so does the other.\n\n\\section{Imprimitive Iwasawa Invariants: the algebraic side}\\label{s 3}\n\\par Throughout, let $p\\geq 5$ be a fixed prime and $g$ be a normalized Hecke-eigencuspform of weight $k\\geq 2$ on the congruence group $\\Gamma_0(M)$. Denote the number field generated by the field of Fourier coefficients of $g$ by $L$. For each prime $q$, choose an embedding $\\iota_q:\\bar{\\mathbb{Q}}\\hookrightarrow \\bar{\\mathbb{Q}}_q$. Let $\\mathfrak{p}|p$ be the prime of $L$, such that the inclusion of $L$ in $L_\\mathfrak{p}$ is compatible with $\\iota_p$. Denote by $K$ the completion of $L$ at $\\mathfrak{p}$, and $\\mathcal{O}$ the valuation ring of $K$. Associated with $g$ is the continuous Galois representation $\\rho_g:\\op{Gal}(\\bar{\\mathbb{Q}}\/\\mathbb{Q})\\rightarrow \\op{GL}_2(K)$. Let $\\op{V}_g\\simeq K^2$ be the underlying $2$-dimensional vector space on which $\\op{Gal}(\\bar{\\mathbb{Q}}\/\\mathbb{Q})$ acts via $K$-linear automorphisms. Fix a Galois stable $\\mathcal{O}$-lattice $\\op{T}_g$ inside $\\op{V}_g$. Let $\\mathbb{F}$ be the residue field of $\\mathcal{O}$. The mod-$\\mathfrak{p}$ reduction of $\\rho_g$ is denoted by\n\\[\\bar{\\rho}_g:\\op{G}_{\\mathbb{Q}}\\rightarrow \\op{GL}_2(\\mathbb{F}),\\]and it follows from the Brauer-Nesbitt theorem that the semi-simplification of $\\bar{\\rho}_g$ is independent of the choice of lattice $\\op{T}_g$.\nThroughout, we make the following assumptions on $g$:\n\\begin{enumerate}\n    \\item $g$ is ordinary at $\\mathfrak{p}$,\n    \\item $\\bar{\\rho}_g$ is absolutely irreducible.\n\\end{enumerate}\nSince $\\bar{\\rho}_g$ is absolutely irreducible, the choice of Galois stable lattice $\\op{T}_g$ is unique.\nLetting $\\op{G}_q$ denote the Galois group $\\op{Gal}(\\bar{\\mathbb{Q}}_q\/\\mathbb{Q}_q)$, we note that the choice of embedding $\\iota_q$ prescribes an inclusion of $\\op{G}_q$ into the absolute Galois group $\\op{G}_{\\mathbb{Q}}$. Let $\\chi_{\\op{cyc}}:\\op{G}_p\\rightarrow \\mathcal{O}^\\times$ denote the $p$-adic cyclotomic character. Since $g$ is ordinary at $\\mathfrak{p}$, there is a short exact sequence \n\\[0\\rightarrow \\op{T}_g^+\\rightarrow \\op{T}_g\\rightarrow \\op{T}_g^-\\rightarrow 0\\] of $\\op{G}_p$-stable $\\mathcal{O}$-lattices such that there is an unramified characters $\\gamma_1, \\gamma_2:\\op{G}_{p}\\rightarrow \\mathcal{O}^{\\times}$ for which\n\\[\\op{T}_g^+\\simeq \\mathcal{O}(\\chi_{\\op{cyc}}^{k-1} \\gamma_1) \\text{ and } \\op{T}_g^-\\simeq \\mathcal{O}(\\gamma_2).\\]Fix a finite order even character $\\psi$ of conductor coprime to $Mp$ and conductor $c_\\psi$. Consider the lattice $\\textbf{T}_g:=\\op{Sym}^2 \\op{T}_g$ and the symmetric square representation \n\\[r_g\\otimes \\psi:=\\op{Sym}^2(\\rho_g)\\otimes \\psi:\\op{Gal}(\\bar{\\mathbb{Q}}\/\\mathbb{Q})\\rightarrow \\op{GL}_3(\\mathcal{O}).\\]\nSet $\\mathbf{V}_g:=\\textbf{T}_g\\otimes \\mathbb{Q}_p$ and $\\mathbf{A}_g:=\\mathbf{V}_g\/\\textbf{T}_g$.\nThe representation $\\textbf{T}_g$ is $\\mathfrak{p}$-ordinary, i.e., is equipped with a filtration \n\\[\\textbf{T}_f=\\mathcal{F}^0(\\textbf{T}_g)\\supset \\mathcal{F}^1(\\textbf{T}_g)\\supset \\mathcal{F}^2(\\textbf{T}_g)\\supset \\mathcal{F}^3(\\textbf{T}_g)=0.\\] For $i=1,2,3$, and unramified characters $\\delta_j$, we have that \\[\\begin{split}&\\op{gr}_0(\\textbf{T}_g)\\simeq \\mathcal{O}(\\chi_{\\op{cyc}}^{2k-2}\\delta_0),\\\\\n&\\op{gr}_1(\\textbf{T}_g)\\simeq \\mathcal{O}(\\chi_{\\op{cyc}}^{k-1}\\delta_1),\\\\\n& \\op{gr}_2(\\textbf{T}_g)\\simeq \\mathcal{O}(\\delta_2).\n\\end{split}\\]\n\n\\par With this notation in place, we consider Hecke eigencuspforms $g_1$ and $g_2$ of the same weight $k\\geq 2$ and trivial nebentype character. Setting $L$ to be the number field generated by the Fourier coefficients of $g_1$ and $g_2$, let $\\mathfrak{p}$ be the prime of $L$ above $p$ corresponding to the choice of $\\iota_p$. Assume that $\\bar{\\rho}_{g_i}$ is absolutely irreducible and that the following equivalent conditions are satisfied.\n\\begin{enumerate}\n    \\item The residual representations are isomorphic: $\\bar{\\rho}_{g_1}\\simeq \\bar{\\rho}_{g_2}$.\n    \\item For all primes $q\\neq p$ coprime to the level of $g_1$ and $g_2$, the Fourier coefficients satisfy the congruence\n    \\[a(q,g_1)\\equiv a(q,g_2)\\mod{\\varpi}.\\]\n    \\item Letting $M_i$ denote the level of $g_i$, assume that the conductor of $\\psi$ is coprime to $M_1 M_2 p$.\n\\end{enumerate}\nNote that $\\textbf{T}_{g_i}$ fits into a short exact sequence \n\\[0\\rightarrow \\textbf{T}_{g_i}^+\\rightarrow \\textbf{T}_{g_i}\\rightarrow \\textbf{T}_{g_i}^-\\rightarrow 0,\\]where $\\textbf{T}_{g_i}^+=\\mathcal{F}^1(\\textbf{T}_{g_i})$ and $\\textbf{T}_{g_i}^-=\\textbf{T}_{g_i}\/\\textbf{T}_{g_i}^+$. Set $\\mathbf{A}_i$ (resp. $\\mathbf{A}_i^{\\pm}$) to denote the $p$-divisible Galois module $\\textbf{T}_{g_i}\\otimes \\mathbb{Q}_p\/\\mathbb{Z}_p$ (resp. $\\textbf{T}_{g_i}^{\\pm}\\otimes \\mathbb{Q}_p\/\\mathbb{Z}_p$). Note that $\\mathbf{A}_i\\simeq (K\/\\mathcal{O})^d$, where $d=3$. Let $d^{\\pm}$ be the dimensions (over $K$) of the $\\pm$ eigenspaces for complex conjugation on $\\mathbf{V}_{g_i}$, we have that $d^+=2$ and $d^-=1$ and that $\\mathbf{A}_i^{\\pm}\\simeq (K\/\\mathcal{O})^{d^{\\pm}}$. Note however that the action of complex conjugation on $\\mathbf{A}_i^{\\pm}$ is not prescribed.\n\n\\par Let $\\mathbb{Q}_n$ be the subfield of $\\mathbb{Q}(\\mu_{p^{n+1}})$ degree $p^n$ and set $\\mathbb{Q}_{\\op{cyc}}:=\\bigcup_{n\\geq 0} \\mathbb{Q}_n$. Letting $\\Gamma:=\\op{Gal}(\\mathbb{Q}_{\\op{cyc}}\/\\mathbb{Q})$ fix an isomorphism $\\op{Gal}(\\mathbb{Q}_{\\op{cyc}}\/\\mathbb{Q})\\xrightarrow{\\sim} \\mathbb{Z}_p$. The extension $\\mathbb{Q}_{\\op{cyc}}$ is the cyclotomic $\\mathbb{Z}_p$-extension of $\\mathbb{Q}$. The Iwasawa algebra $\\Lambda$ is defined as the following inverse limit $\\Lambda:=\\varprojlim_n \\mathbb{Z}_p[\\op{Gal}(\\mathbb{Q}_n\/\\mathbb{Q})]$, and is isomorphic to the formal power series ring $\\mathbb{Z}_p\\llbracket T\\rrbracket$. For $i=1,2$, letting $M_i$ be the conductor of $g_i$, fix the set $S$ to consist of primes $q$ that divide $c_\\psi M_1M_2p$. In what follows, set $\\mathbf{A}_{i,\\psi}:=\\mathbf{A}_i\\otimes \\psi$. The $p$-primary Selmer group $\\op{Sel}_{p^\\infty}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})$ is defined as the kernel of the following restriction map\n\\[\n\\lambda_i:H^1\\left(\\mathbb{Q}_{S}\/\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_{i,\\psi}\\right)\\rightarrow \\bigoplus_{q\\in S}\\mathcal{H}_q(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}}).\n\\]\nHere for each prime $q\\neq p$, the local term is defined as follows\n\\[\n\\mathcal{H}_q(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}}) = \\bigoplus_{\\eta|q} H^1( {\\mathbb{Q}}_{\\op{cyc},\\eta}, \\mathbf{A}_{i,\\psi}),\n\\]\nwhere $\\mathbb{Q}_{\\op{cyc}, \\eta}$ is the union of all completions of number fields contained in $\\mathbb{Q}_{\\op{cyc}}$ at the prime $\\eta$.\nThe definition at the prime $q=p$ is more subtle, set\n\\[\n\\mathcal{H}_q(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}}) = H^1( \\mathbb{Q}_{\\op{cyc}, \\eta_p}, \\mathbf{A}_{i,\\psi})\/\\mathcal{L}_{\\eta_p}\n\\]\nwith \n\\[\n\\mathcal{L}_{\\eta_p} = \\ker\\left( H^1( \\mathbb{Q}_{\\op{cyc}, \\eta_p}, \\mathbf{A}_{i,\\psi}) \\rightarrow H^1( I_{\\eta_p}, \\mathbf{A}_{i,\\psi}^{-})\\right).\n\\]\nHere $\\eta_p$ is the unique prime of $\\mathbb{Q}_{\\op{cyc}}$ above $p$, set $I_{\\eta_p}$ denotes the inertia group at $\\eta_p$. The following is a special case of \\cite[Conjecture 4.1]{CS}.\n\\begin{Conjecture}[Coates-Schmidt]\nThe Selmer group $\\op{Sel}_{p^\\infty}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})$ is a cotorsion $\\Lambda$-module.\n\\end{Conjecture}\nNote that it is crucial that $\\psi$ is even. This conjecture has been settled by Loeffler and Zerbes in \\cite{lz16}.\n\\begin{Proposition}\nLet $\\mathbf{A}_{i,\\psi}$ be as above. The localization map $\\lambda_i$ is surjective.\n\\end{Proposition}\n\\begin{proof}\nWe let $\\textbf{T}_{i,\\psi}^*:=\\op{Hom}(\\mathbb{T}_{g_i}\\otimes \\psi, \\mu_{p^{\\infty}})$. Note that $\\bar{\\rho}_{g_i}$ is assumed to irreducible as a Galois module. It is easy to show that $H^0(\\mathbb{Q}, \\textbf{T}_{i,\\psi}^*)=0$. Since $\\op{Gal}(\\mathbb{Q}_{\\op{cyc}}\/\\mathbb{Q})$ is pro-$p$, it follows that $H^0(\\mathbb{Q}, \\textbf{T}_{i,\\psi}^*)=0$ as well and thus in particular, is finite. The result follows from \\cite[Proposition 2.1]{GV00}.\n\\end{proof}\nDenote by $S_0:=S\\backslash \\{p\\}$ and introduce the $S_0$-imprimitive Selmer group to be the Selmer group obtained by imposing conditions only at $p$.\n\n\\begin{Definition}\nThe \\emph{imprimitive Selmer group} is defined by:\n\\[\n\\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}}) = \\ker\\left( H^1\\left( \\mathbb{Q}_{S}\/\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_{i,\\psi}\\right)\\xrightarrow{\\lambda_i^p} \\mathcal{H}_p(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})\\right).\n\\]\n\\end{Definition}\nSince the map defining the Selmer group is surjective, it follows that\n\\begin{equation}\n\\label{quotient}\n\\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})\/\\op{Sel}_{p^\\infty}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})\\simeq \\bigoplus_{q\\in S_0}\\mathcal{H}_q(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}}).\n\\end{equation}\n\\begin{Lemma}\\label{local mu is 0}\nLet $q\\neq p$, then $\\mathcal{H}_q(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})$ is a cofinitely generated and cotorsion $\\Lambda$-module with $\\mu$-invariant equal to $0$. \n\\end{Lemma}\n\\begin{proof}\nIt suffices to show that $\\mathcal{H}_q(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})$ is a cofinitely generated as a $\\mathbb{Z}_p$-module, or equivalently, the $p$-torsion subgroup $\\mathcal{H}_q(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})[\\mathfrak{p}]$ is finite. Consider the short exact sequence\n\\[0\\rightarrow \\bigoplus_{\\eta|q} \\frac{H^0(\\mathbb{Q}_{\\op{cyc}, \\eta}, \\mathbf{A}_{i,\\psi})}{p\\left( H^0(\\mathbb{Q}_{\\op{cyc}, \\eta}, \\mathbf{A}_{i,\\psi})\\right)}\\rightarrow \\bigoplus_{\\eta|q} H^1( \\mathbb{Q}_{\\op{cyc}, \\eta}, \\mathbf{A}_{i,\\psi}[\\mathfrak{p}]) \\rightarrow \\mathcal{H}_q(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})[\\mathfrak{p}]\\rightarrow 0.\\]\nThe set of primes $\\eta|q$ of $\\mathbb{Q}_{\\op{cyc}}$ is finite and so is $ H^1( \\mathbb{Q}_{\\op{cyc}, \\eta}, \\mathbf{A}_{i,\\psi}[\\mathfrak{p}])$. The result follows.\n\\end{proof}\nLet $\\sigma_i^{(q)}$ denote the $\\mathbb{Z}_p$-corank of $\\mathcal{H}_q(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})$ for $q\\in S_0$.\nSet $\\lambda^{S_0}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})$ to be the $\\lambda$-invariant of $\\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})$. It follows from the structure theory of $\\Lambda$-modules that \n\\[\\lambda^{S_0}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})=\\op{corank}_{\\mathbb{Z}_p} \\left(\\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})\\right).\\]\nIt follows from $\\eqref{quotient}$ that the following relation is satisfied:\n\\begin{equation}\n\\label{relating im primitive and classical lambda invariant}\n\\lambda^{S_0}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})=\n\\lambda(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}}) + \\sum_{q\\in S_0} \\sigma_i^{(q)}.\n\\end{equation}\nAnalogous to the classical and imprimitive Selmer group, we also define the \\emph{reduced} classical and imprimitive Selmer groups which we denote by $\\op{Sel}_{p^\\infty}(\\mathbf{A}_{i,\\psi}[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})$ and $\\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{i,\\psi}[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})$, respectively. \n\\par For $q\\in S_0$ set\n\\[\n\\mathcal{H}_q(\\mathbf{A}_{i,\\psi}[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}}) := \\prod_{q^\\prime|q} H^1\\left( {\\mathbb{Q}_{\\op{cyc}}}_{,q^\\prime}, \\mathbf{A}_{i,\\psi}[\\mathfrak{p}]\\right),\n\\]\nand for $q=p$, set\n\\[\\mathcal{H}_q(\\mathbf{A}_{i,\\psi}[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}}):=\\bigoplus_{q'|q} H^1(K_{\\infty, q'}, \\mathbf{A}_{i,\\psi}[\\mathfrak{p}])\/\\overline{L}_{q'},\\] where \n\\[\\overline{L}_{q'}:=\\ker\\left( H^1\\left( {\\mathbb{Q}_{\\op{cyc}}}_{,q^\\prime}, \\mathbf{A}_{i,\\psi}[\\mathfrak{p}]\\right) \\rightarrow H^1\\left( I_{q^\\prime}, \\mathbf{A}_{i,\\psi}^-[\\mathfrak{p}]\\right)\\right).\\]\n\\begin{Definition}\nThe reduced imprimitive Selmer group is defined as follows\n\\[\\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}}):=\\op{ker} \\left(H^1\\left( \\mathbb{Q}_{S}\/\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_{i,\\psi}\\right)\\xrightarrow{\\overline{\\theta}_0}  \\mathcal{H}_p(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})\\right)\\]\n\\end{Definition}\n\n\n\\begin{Proposition}\n\\label{Kim 2.10}\nFor $i=1,2$, we have a natural isomorphism\n\\[\n\\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{i,\\psi}[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}}) \\simeq \\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})[\\mathfrak{p}].\n\\]\n\\end{Proposition}\n\\begin{proof}\nWe consider the diagram relating the two Selmer groups\n\\[\n\\begin{tikzcd}[column sep = small, row sep = large]\n0\\arrow{r} & \\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}}) \\arrow{r}  \\arrow{d}{f} & H^1(\\mathbb{Q}_{S}\/\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_{i,\\psi}[\\mathfrak{p}]) \\arrow{r} \\arrow{d}{g} & \\op{im} \\overline{\\theta}_0 \\arrow{r} \\arrow{d}{h} & 0\\\\\n0\\arrow{r} & \\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}}) [\\mathfrak{p}] \\arrow{r} & H^1(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_{i,\\psi})[\\mathfrak{p}]  \\arrow{r}  &\\left(\\op{im} \\theta_0\\right)[\\mathfrak{p}]\\arrow{r}  & 0,\n\\end{tikzcd}\\]\nwhere the vertical maps are induced by the Kummer sequence. Note that \\[H^0(K,\\mathbf{A}_{i,\\psi}[\\mathfrak{p}])=H^0(\\mathbb{Q}_{\\op{cyc}},\\mathbf{A}_{i,\\psi}[\\mathfrak{p}])^{\\Gamma}=0.\\]\nSince $\\mathbb{Q}_{\\op{cyc}}\/\\mathbb{Q}$ is a pro-$p$ extension, we deduce that $H^0(\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_{i,\\psi})=0$ and therefore $g$ is injective.\nOn the other hand, it clear that $g$ is surjective.\n\nIt only remains to show that $h$ is injective.\nFor $q=p$, denote by $\\iota_q$ the natural map\n\\[\\iota_q: \\mathcal{H}_q(\\mathbf{A}_{i,\\psi}[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})\\rightarrow \\mathcal{H}_q(\\mathbf{A}_{i, \\psi}\/\\mathbb{Q}_{\\op{cyc}})[\\mathfrak{p}].\\]\nConsider the commutative square with injective horizontal maps\n\\[\n\\begin{tikzcd}[column sep = small, row sep = large]\n & \\mathcal{H}_q(\\mathbf{A}_{i, \\psi}[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}}) \\arrow{r}  \\arrow{d}{\\iota_q} & \\bigoplus_{q'|q} H^1\\left( I_{q^\\prime}, \\mathbf{A}_{i, \\psi}^-[\\mathfrak{p}]\\right) \\arrow{d}{j_q} \\\\\n & \\mathcal{H}_q(\\mathbf{A}_{i, \\psi}\/\\mathbb{Q}_{\\op{cyc}})[\\mathfrak{p}]\\arrow{r} & \\bigoplus_{q'|q} H^1\\left( I_{q^\\prime}, \\mathbf{A}_{i, \\psi}^-\\right)[\\mathfrak{p}].\n\\end{tikzcd}\\]\nSince $\\mathbf{A}_{i, \\psi}^-$ is unramified at all primes $q\\in S_p$, it follows that $H^0(I_{q'}, \\mathbf{A}_{i, \\psi}^-)=\\mathbf{A}_{i, \\psi}^-$ is divisible.\nThe kernel of the map \n\\[\\iota_q: H^1(I_{q'}, \\mathbf{A}_{i, \\psi}^-[\\mathfrak{p}])\\rightarrow H^1(I_{q'}, \\mathbf{A}_{i, \\psi}^-)[\\mathfrak{p}]\\] is $H^0(I_{q'}, \\mathbf{A}_{i, \\psi}^-)\/p=0$.\n\\end{proof}\n\n\\begin{Lemma}\nThe isomorphism $ \\mathbf{A}_{1,\\psi}[\\mathfrak{p}]\\simeq  \\mathbf{A}_{2,\\psi}[\\mathfrak{p}]$ of Galois modules induces an isomorphism of residual Selmer groups \n\\[\n\\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{1,\\psi}[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})\\simeq \\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{2,\\psi}[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}}).\n\\]\n\\end{Lemma}\n\\begin{proof}\nNote that the $\\op{G}_{\\mathbb{Q}_p}$-action on $\\mathbf{A}_{i, \\psi}^+[\\mathfrak{p}]$ is ramified and that on $\\mathbf{A}_{i, \\psi}^-[\\mathfrak{p}]$ is via an unramified character. Let $\\Phi:\\mathbf{A}_{1,\\psi}[\\mathfrak{p}]\\xrightarrow{\\sim} \\mathbf{A}_{2,\\psi}[\\mathfrak{p}]$ be a choice of isomorphism of Galois modules, it is easy to see that $\\Phi$ induces an isomorphism \n\\[\\Phi:\\mathbf{A}_{1,\\psi}^+[\\mathfrak{p}]\\xrightarrow{\\sim} \\mathbf{A}_{2,\\psi}^+[\\mathfrak{p}].\\] As a result, we have an isomorphism of $\\op{G}_{\\mathbb{Q}_p}$-modules $\\mathbf{A}_{1,\\psi}^-[\\mathfrak{p}]\\simeq \\mathbf{A}_{2,\\psi}^-[\\mathfrak{p}]$.\nClearly, $\\Phi$ induces an isomorphism $H^1(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_{1,\\psi}[\\mathfrak{p}])\\xrightarrow{\\sim} H^1(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_{2,\\psi}[\\mathfrak{p}])$.\nIt suffices to show that for $q\\in \\Sigma$, the isomorphism $\\Phi:\\mathbf{A}_{1,\\psi}[\\mathfrak{p}]\\xrightarrow{\\sim} \\mathbf{A}_{2,\\psi}[\\mathfrak{p}]$ induces an isomorphism \n\\[\n\\mathcal{H}_q(\\mathbb{Q}_{\\op{cyc}},\\mathbf{A}_{1,\\psi}[\\mathfrak{p}])\\xrightarrow{\\sim} \\mathcal{H}_q(\\mathbb{Q}_{\\op{cyc}},\\mathbf{A}_{2,\\psi}[\\mathfrak{p}]).\n\\]\nThis is clear for $q\\neq p$.\nFor $q=p$, this follows from the fact that $\\Phi$ induces an isomorphism $\\mathbf{A}_{1,\\psi}^-[\\mathfrak{p}]\\xrightarrow{\\sim} \\mathbf{A}_{2,\\psi}^-[\\mathfrak{p}]$.\n\\end{proof} \n\n\\begin{Corollary}\n\\label{p-torsion of Sigma_0 fine selmer are iso}\nThe isomorphism $\\mathbf{A}_{1,\\psi}[\\mathfrak{p}]\\simeq \\mathbf{A}_{2,\\psi}[\\mathfrak{p}]$ of Galois modules induces an isomorphism \\[\\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{1,\\psi}\/\\mathbb{Q}_{\\op{cyc}})[\\mathfrak{p}]\\simeq \\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{2,\\psi}\/\\mathbb{Q}_{\\op{cyc}})[\\mathfrak{p}].\\]\n\\end{Corollary}\n\n\n\\begin{Lemma}\n\\label{lemma: the two mus are the same}\nThe $\\mu$-invariant of the Selmer group $\\op{Sel}_{p^\\infty}(\\mathbf{A}_{i, \\psi}\/\\mathbb{Q}_{\\op{cyc}})$ coincides with that of $\\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{i, \\psi}\/\\mathbb{Q}_{\\op{cyc}})$, i.e.,\n\\[\\mu(\\mathbf{A}_{i, \\psi}\/\\mathbb{Q}_{\\op{cyc}})=\\mu^{S_0}(\\mathbf{A}_{i, \\psi}\/\\mathbb{Q}_{\\op{cyc}}).\\]\n\\end{Lemma}\n\\begin{proof}\nThe result follows from Lemma \\ref{local mu is 0}, which states that $\\mathcal{H}_q(\\mathbf{A}_{i, \\psi}\/\\mathbb{Q}_{\\op{cyc}})$ has $\\mu=0$ for $q\\neq p$.\n\\end{proof}\n\\begin{Proposition}\\label{prop 3.10}\nLet $p\\geq 5$ be a prime and $g_1$ and $g_2$ be $p$-ordinary Hecke eigencuspforms with trivial nebentype character. Let $N_i$ be the level of $g_i$. Recall that $\\mathbf{A}_{i, \\psi}$ is the $p$-divisible symmetric square representation associated with $g_i$. Let $S_0$ be a finite set of primes containing those dividing $N_1N_2p$. Then, we have that \n\\[\\mu(\\mathbf{A}_{1,\\psi}\/\\mathbb{Q}_{\\op{cyc}})=0\\Leftrightarrow \\mu(\\mathbf{A}_{2,\\psi}\/\\mathbb{Q}_{\\op{cyc}})=0.\\]Moreover, if these $\\mu$-invariants are $0$, then the imprimitive $\\lambda$-invariants coincide, i.e.,\n\\[\\lambda^{S_0}(\\mathbf{A}_{1,\\psi}\/\\mathbb{Q}_{\\op{cyc}})=\\lambda^{S_0}(\\mathbf{A}_{2,\\psi}\/\\mathbb{Q}_{\\op{cyc}}).\\] This relationship translates to the following relationship between $\\lambda$-invariants\n\\[\\lambda(\\mathbf{A}_{2,\\psi}\/\\mathbb{Q}_{\\op{cyc}})-\\lambda(\\mathbf{A}_{1,\\psi}\/\\mathbb{Q}_{\\op{cyc}})=\\sum_{q\\in S_0} \\left(\\sigma_q^{(1)}-\\sigma_q^{(2)}\\right),\\]\nwhere $\\sigma_q^{(i)}$ is the $\\mathbb{Z}_p$-corank of $\\mathcal{H}_q(\\mathbf{A}_{i, \\psi}\/\\mathbb{Q}_{\\op{cyc}})$. \n\\end{Proposition}\n\\begin{proof}\nSet $M_i$ to denote $\\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{i, \\psi}\/\\mathbb{Q}_{\\op{cyc}})$.\nLemma \\ref{lemma: the two mus are the same} asserts that the $\\mu$-invariant of $M_i$ coincides with $\\mu(\\mathbf{A}_{i, \\psi}\/\\mathbb{Q}_{\\op{cyc}})$.\nTherefore, $\\mu(\\mathbf{A}_{i, \\psi}\/\\mathbb{Q}_{\\op{cyc}})=0$ if and only if $M_i$ is cofinitely generated as a $\\mathbb{Z}_p$-module.\nNote that $M_i$ is cofinitely generated as a $\\mathbb{Z}_p$-module if and only if $M_i[\\mathfrak{p}]$ has finite cardinality.\nCorollary \\ref{p-torsion of Sigma_0 fine selmer are iso} asserts that $M_1[\\mathfrak{p}]\\simeq M_2[\\mathfrak{p}]$; thus,\n\\[\\mu(\\mathbf{A}_{1,\\psi}\/\\mathbb{Q}_{\\op{cyc}})=0\\Leftrightarrow \\mu(\\mathbf{A}_{2,\\psi}\/\\mathbb{Q}_{\\op{cyc}})=0.\\]\n\\par Assume that $M_1[\\mathfrak{p}]$ (or equivalently $M_2[\\mathfrak{p}]$) is finite.\nIt follows from \\cite[Proposition 2.5]{GV00} that $M_i$ has no proper $\\Lambda$-submodules of finite index.\nIt is an easy exercise to show that $M_i$ therefore is a cofree $\\mathbb{Z}_p$-module.\nTherefore, $M_i\\simeq (\\mathbb{Q}_p\/\\mathbb{Z}_p)^{\\lambda_i}$, where $\\lambda_{i}:=\\lambda^{S_0}(\\mathbf{A}_{i, \\psi}\/\\mathbb{Q}_{\\op{cyc}})$.\nAs a result,\n\\[\\lambda^{S_0}(\\mathbf{A}_{i, \\psi}\/\\mathbb{Q}_{\\op{cyc}})=\\op{dim}_{\\mathbb{F}_p} M_i[\\mathfrak{p}],\\] and the isomorphism $M_1[\\mathfrak{p}]\\simeq M_2[\\mathfrak{p}]$ implies that \n\\[\\lambda^{S_0}(\\mathbf{A}_{1,\\psi}\/\\mathbb{Q}_{\\op{cyc}})=\\lambda^{S_0}(\\mathbf{A}_{2,\\psi}\/\\mathbb{Q}_{\\op{cyc}}).\\]\n\n\\end{proof}\n\n\\par Let $g_1$ and $g_2$ be $\\mathfrak{p}$-congruent Hecke eigencuspforms satisfying the conditions stated in the introduction. We denote by $\\mu^{\\op{alg}}_{S_0}(r_{g_i}\\otimes \\psi)$ and $\\lambda^{\\op{alg}}_{S_0}(r_{g_i}\\otimes \\psi)$ to denote the Iwasawa invariants of the imprimitive Selmer group $\\op{Sel}_{p^\\infty}^{S_0}(\\mathbf{A}_{i,\\psi}\/\\mathbb{Q}_{\\op{cyc}})$ obtained by dropping conditions at the primes $q\\in S_0$. Denote by $\\mu^{\\op{an}}_{S_0}(r_{g_i}\\otimes \\psi)$ and $\\lambda^{\\op{an}}_{S_0}(r_{g_i}\\otimes \\psi)$ the Iwasawa invariants of the $S_0$-imprimitive $p$-adic L-function $L^{\\op{an}}_{S_0}(r_{g_i}\\otimes \\psi)$ obtained by dropping Euler factors at primes $q\\in \\Sigma$.\n\n\\begin{Proposition}\\label{boring prop}\nLet $g_1$ and $g_2$ be as above and $\\ast\\in \\{\\op{an}, \\op{alg}\\}$. Then, $\\mu^{\\ast}(r_{g_i}\\otimes \\psi)=\\mu^{\\ast}_{S_0}(r_{g_i}\\otimes \\psi)$ for $i=1,2$ and \n\\[\\lambda^{\\ast}_{S_0}(r_{g_i}\\otimes \\psi)=\\lambda^{\\ast}(r_{g_i}\\otimes \\psi)+\\sum_{q\\in S_0} \\sigma_i^{(q)}.\\]\n\\end{Proposition}\n\n\\begin{proof}\nWhen $\\ast=\\op{alg}$, the result follows from Lemma \\ref{local mu is 0} and \\eqref{relating im primitive and classical lambda invariant}. On the other hand, when $\\ast=\\op{an}$, the result follows from \\cite[Proposition 2.4]{GV00}.\n\\end{proof}\n\n\\begin{Th}\nLet $g_1$ and $g_2$ satisfy the conditions stated in the introduction. Suppose the relations $\\mu^{\\op{alg}}(r_{g_1}\\otimes \\psi)=\\mu^{\\op{an}}(r_{g_1}\\otimes \\psi)=0$ and $\\lambda^{\\op{alg}}(r_{g_1}\\otimes \\psi)=\\lambda^{\\op{an}}(r_{g_1}\\otimes \\psi)$ hold. Then, we have further equalities $\\mu^{\\op{alg}}(r_{g_2}\\otimes \\psi)=\\mu^{\\op{an}}(r_{g_2}\\otimes \\psi)=0$ and $\\lambda^{\\op{alg}}(r_{g_2}\\otimes \\psi)=\\lambda^{\\op{an}}(r_{g_2}\\otimes \\psi)$.\n\\end{Th}\n\n\\begin{proof}\nLet $\\ast\\in \\{\\op{an}, \\op{alg}\\}$, it follows from Proposition \\ref{boring prop} that the equalities \\[\\mu^{\\op{alg}}(r_{g_i}\\otimes \\psi)=\\mu^{\\op{an}}(r_{g_i}\\otimes \\psi)=0\\] and \\[\\lambda^{\\op{alg}}(r_{g_i}\\otimes \\psi)=\\lambda^{\\op{an}}(r_{g_i}\\otimes \\psi)\\] hold if and only if the relations \\[\\mu_{S_0}^{\\op{alg}}(r_{g_i}\\otimes \\psi)=\\mu_{S_0}^{\\op{an}}(r_{g_i}\\otimes \\psi)=0\\] and $\\lambda_{S_0}^{\\op{alg}}(r_{g_i}\\otimes \\psi)=\\lambda_{S_0}^{\\op{an}}(r_{g_i}\\otimes \\psi)$ hold. Therefore, by assumption, these relations hold for $i=1$, and we are to deduce them for $i=2$. Proposition \\ref{p-adic LFs congruent} asserts that there is a $p$-adic unit $u$ such that \n\\[L^{\\op{an}}_{S_0}(r_{g_1}\\otimes \\psi)\\equiv u L^{\\op{an}}_{S_0}(r_{g_2}\\otimes \\psi)\\mod{\\mathfrak{p}}.\\]\n\nFrom the above congruence, we find that \\[\\mu^{\\op{an}}_{S_0}(r_{g_1}\\otimes \\psi)=0\\Leftrightarrow \\mu^{\\op{an}}_{S_0}(r_{g_2}\\otimes \\psi)=0\\] and if these $\\mu$-invariants vanish, then, \\[\\lambda^{\\op{an}}_{S_0}(r_{g_1}\\otimes \\psi)= \\lambda^{\\op{an}}_{S_0}(r_{g_2}\\otimes \\psi).\\]On the other hand, the same assertion for $\\ast=\\op{an}$ replaced by $\\ast=\\op{alg}$ holds by Proposition \\ref{prop 3.10}. Therefore, the result follows.\n\\end{proof}\n\n\\section{A Criterion for the Vanishing of the $\\mu$-invariant}\\label{s 4}\n\\par Throughout, $f$ is a Hecke eigencuspform of weight $k\\geq 2$ and $r_f$ the symmetric square representation associated to $f$. In this section, we study the relationship between the fine Selmer group associated to the symmetric square representation, and the residual representation. We thus establish a criterion for the vanishing of the $\\mu$-invariant of $\\op{Sel}_{p^\\infty}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})$ purely in terms of the residual representation $\\mathbf{A}_f[\\mathfrak{p}]$.\n\\par Let $L$ be a number field contained in $\\mathbb{Q}_S$. For any abelian group $N$ equipped with a continuous $\\op{Gal}(\\bar{\\mathbb{Q}}\/\\mathbb{Q})$-action, prime number $q$ and index $i=0,1,2$, set\n\\[K_q^i(N\/L):=\\bigoplus_{q'|q} H^i(L_{q'}, N).\\]Over the infinite extension $\\mathbb{Q}_{\\op{cyc}}$, set\n\\[K_q^i(N\/\\mathbb{Q}_{\\op{cyc}}):=\\varinjlim_L K_q^i(N\/L),\\]where the inductive limit is taken with respect to restriction maps over all number fields $L$ contained in $\\mathbb{Q}_{\\op{cyc}}$. The fine Selmer group is the kernel of the restriction map\n\\[\\op{R}\\left(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}}\\right):=\\op{ker}\\left\\{H^1\\left(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_f\\right)\\longrightarrow \\bigoplus_{q\\in S} K_q^1(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}}) \\right\\}.\\]\nFor $i=0,1,2$, define the compact $\\op{G}$-modules \n\\[\\mathcal{Z}^i\\left(\\textbf{T}_f\/\\mathbb{Q}_{\\op{cyc}}\\right):=\\varprojlim_L H^i(\\mathbb{Q}_S\/L, \\textbf{T}_f)\\]where the projective limit is taken over corestriction maps as $L$ ranges over all number fields contained in $\\mathbb{Q}_{\\op{cyc}}$. The Poitou Tate sequence for $\\textbf{T}_f$ over $\\mathbb{Q}_{\\op{cyc}}$ breaks up into short exact sequences\n\\begin{equation}\\label{sesPT}\\begin{split}\n    0&\\rightarrow H^0(\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_f)\\rightarrow \\bigoplus_{q\\in S_L} K_q^0(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})\\rightarrow \\mathcal{Z}^2(\\textbf{T}_f\/\\mathbb{Q}_{\\op{cyc}})^{\\vee}\\\\\n    & \\op{R}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})\\rightarrow 0,\\\\\n    0&\\rightarrow \\op{R}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})\\rightarrow H^1(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_f)\\rightarrow \\bigoplus_{q\\in S} K_q^1(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})\\\\\n    & \\mathcal{Z}^1(\\textbf{T}_f\/\\mathbb{Q}_{\\op{cyc}})^{\\vee}\\rightarrow H^2(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_f)\\rightarrow 0.\n\\end{split}\\end{equation}\nLet $\\op{Y}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})$ be the Pontryagin dual for the fine Selmer group $\\op{R}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})$.\n\\begin{Lemma}\\label{torsionconditions}\n\nThe following statements are equivalent\n\\begin{enumerate}\n    \\item\\label{one} $\\op{Y}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})$ is $\\Lambda$-torsion,\n    \\item\\label{two} $\\mathcal{Z}^2(\\textbf{T}_f\/\\mathbb{Q}_{\\op{cyc}})$ is $\\Lambda$-torsion.\n\\end{enumerate}\n\\end{Lemma}\nFurthermore, if the above statements hold, then\\[\\mu\\left(\\op{Y}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})\\right)=0\\Leftrightarrow \\mu\\left(\\mathcal{Z}^2(\\textbf{T}_f\/\\mathbb{Q}_{\\op{cyc}})\\right)=0.\\]\n\\begin{proof}\nLet $U_q$ and $A_q$ denote the Pontryagin duals of $K_q^0(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})$ and $H^0(\\mathbb{Q}_{\\infty,q},\\mathbf{A}_f)$ respectively. From $\\eqref{sesPT}$ we arrive at the exact sequence\n\\[0\\rightarrow Y(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})\\rightarrow \\mathcal{Z}^2(\\textbf{T}_f\/\\mathbb{Q}_{\\op{cyc}})\\rightarrow \\bigoplus_{q\\in S} U_q.\\] Since $q$ is finitely decomposed in the cyclotomic $\\mathbb{Z}_p$-extension $\\mathbb{Q}_{\\op{cyc}}$, it follows that $U_q$ is finitely generated as a $\\mathbb{Z}_p$-module and hence, it follows that $U_q$ is torsion as a $\\Lambda$-module. Therefore, $\\eqref{one}$ and $\\eqref{two}$ are equivalent. Since $U_q$ is finitely generated as a $\\mathbb{Z}_p$-module, it follows that \n\\[\\mu\\left(\\op{Y}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})\\right)=0\\Leftrightarrow \\mu\\left(\\mathcal{Z}^2(\\textbf{T}_f\/\\mathbb{Q}_{\\op{cyc}})\\right)=0.\\]\n\n\\end{proof}\n\n\\begin{Lemma}\\label{globalECLemma}Let $M$ be a finite dimensional $\\mathbb{F}_p$-vector space on which $\\op{G}_{\\mathbb{Q},S}$ acts. Then, we have the following relation\n\\[\\begin{split}&\\operatorname{corank}_{\\Omega}H^1(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}}, M)-\\operatorname{corank}_{\\Omega}H^2(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}}, M)\\\\=&\\dim M-\\dim H^0(\\mathbb{R}, M).\\end{split}\\]\n\\end{Lemma}\n\\begin{proof}\n\\par It follows from \\cite[Proposition 1.6]{howson2002euler} that the $\\Omega$-corank of a module $N$ may be calculated via the following formula\n\\[\\operatorname{corank}_{\\Omega} N=\\sum_{j\\geq 0} (-1)^j \\dim H^j(\\Gamma, N).\\] Since $M$ is finite-dimensional vector space over $\\mathbb{F}_p$. We have that\n\\[\\begin{split}&\\sum_{i\\geq 0}(-1)^i \\operatorname{corank}_{\\Omega}H^{i}\\left(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}},M\\right)\\\\=&\\sum_{i,j\\geq 0}(-1)^{i+j+1}\\dim H^j(\\Gamma, H^{i}(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}},M))\\\\=&\\sum_{i\\geq 0}(-1)^{i+1}\\dim  H^{i}(\\mathbb{Q}_S\/\\mathbb{Q},M)\n\\\\=&\\dim M-\\dim H^0(\\mathbb{R}, M).\\end{split}\\]\nThe last equality follows from the global Euler-characteristic formula.\n\\end{proof}\n\n\\begin{Lemma}\\label{localEClemma}\nLet $q$ be a finite prime and $M$ an $\\mathbb{F}_p[\\op{G}_q]$-module which is also finite dimensional as an $\\mathbb{F}_p$-vector-space. \nWe have that \n\\[\\operatorname{corank}_{\\Omega}K_q^1(M\/\\mathbb{Q}_{\\op{cyc}})=\\begin{cases}[K_q:\\mathbb{Q}_p]\\dim M\\text{ if }q|p\\\\\n0\\text{ if }q\\nmid p.\\end{cases}\\]\n\\end{Lemma}\n\\begin{proof}\nFor a choice of a prime $w|q$ of $\\mathbb{Q}_{\\op{cyc}}$, let $\\Gamma_w:=\\op{Gal}(\\mathbb{Q}_{\\infty,w}\/\\mathbb{Q}_q)$. By an argument similar to the proof of \\ref{globalECLemma}, we obtain the following relation\n\\[\\begin{split}&\\sum_{i\\geq 0}(-1)^{i+1}\\operatorname{corank}_{\\Omega(\\Gamma_w)}H^i(\\mathbb{Q}_{\\infty,w}, M)\\\\=&\\begin{cases}\\dim M& q|p\\\\\n0& q\\nmid p.\\end{cases}\\end{split}\\]\nSince $\\op{Gal}(\\bar{\\mathbb{Q}}_{\\infty,w}\/\\mathbb{Q}_{\\infty,w})$ has $p$-cohomological dimension $\\leq 1$, the cohomology groups $H^i(\\mathbb{Q}_{\\infty,w},M)=0$ for $i>1$. Since $q$ is finitely split in $\\mathbb{Q}_{\\op{cyc}}$, it follows that\n\\[\\operatorname{corank}_{\\Omega(\\Gamma_w)} H^0(\\mathbb{Q}_{\\infty,w},M)=0.\\] One deduces that \n\\[\\begin{split}&\\operatorname{corank}_{\\Omega(\\Gamma_w)}H^1(\\mathbb{Q}_{\\infty,w}, M)\\\\=&\\begin{cases}\\dim M& q|p\\\\\n0& q\\nmid p.\\end{cases}\\end{split}\\]\nOn the other hand, \n\\[K^1_q(M\/\\mathbb{Q}_{\\op{cyc}})^{\\vee}=\\op{Ind}_{\\Gamma_w}^{\\Gamma}\\left(H^1(\\mathbb{Q}_{\\infty,w}, M)^{\\vee}\\right)=\\Omega\\otimes_{\\Omega(\\Gamma_w)}H^1(\\mathbb{Q}_{\\infty,w}, M)^{\\vee} ,\\]\nand as a result, \n\\[\\operatorname{corank}_{\\Omega} K^1_q(M\/\\mathbb{Q}_{\\op{cyc}})=\\operatorname{corank}_{\\Omega(\\Gamma_w)}H^1(K_{\\infty,w}, M).\\]The assertion of the Lemma follows.\n\\end{proof}\n\nThe following is an easy consequence Lemmas \\ref{globalECLemma} and \\ref{localEClemma}.\n\\begin{Corollary}\\label{balancedCor}\nAssume that $f$ has good ordinary reduction at $p$. Then, we have the following relation\n\\[\\begin{split}&\\operatorname{corank}_{\\Omega}H^1(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_f[\\mathfrak{p}])-\\operatorname{corank}_{\\Omega}H^2(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_f[\\mathfrak{p}])\\\\=&\\operatorname{corank}_{\\Omega}\\left(\\bigoplus_{q\\in S} K^1_q(\\mathbf{A}_f^-[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})\\right)\\\\=& 1.\\end{split}\\]\n\\end{Corollary}\n\n\\begin{Lemma}\\label{muzeroH2}\nAssume that the conditions of Lemma $\\ref{torsionconditions}$ are satisfied. Then the following are equivalent\n\\begin{enumerate}\n    \\item $\\mu\\left(Y(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})\\right)=0$\n    \\item $H^2(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_f[\\mathfrak{p}])$ is a cotorsion $\\Omega$-module.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nLetting $E^{i,j}:=E^j\\left(H^i(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}},\\mathbf{A}_f[\\mathfrak{p}]\\right)^{\\vee})$, the Iwasawa cohomology group \\[\\mathcal{Z}^{2}(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})\\] is related to the cohomology groups $H^i\\left(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}},\\mathbf{A}_f[\\mathfrak{p}]\\right)^{\\vee}$ via Jannsen's spectral sequence\n\\[E_2^{i,j}\\Rightarrow \\mathcal{Z}^{i+j}(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}}).\\]The cohomology groups $H^i(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}},\\mathbf{A}_f[\\mathfrak{p}])$ are cofinitely generated as $\\Omega$-modules. As a consequence, for $j>0$, $E^j\\left(H^i(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}},\\mathbf{A}_f[\\mathfrak{p}])^{\\vee}\\right)$ is $\\Omega$-torsion. If condition (2) is satisfied, $E^{2,0}$ is $\\Omega$-torsion. From Jansen's spectral sequence, condition $(2)$ is equivalent to the assertion that $\\mathcal{Z}^2(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})$ is $\\Omega$-torsion. From\n\\[0\\rightarrow \\textbf{T}_f\\xrightarrow{p} \\textbf{T}_f\\rightarrow \\mathbf{A}_f[\\mathfrak{p}]\\rightarrow 0,\\] we obtain the following\n\\[\\mathcal{Z}^2(\\textbf{T}_f\/\\mathbb{Q}_{\\op{cyc}})\\xrightarrow{p}\\mathcal{Z}^2(\\textbf{T}_f\/\\mathbb{Q}_{\\op{cyc}})\\rightarrow \\mathcal{Z}^2(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})\\rightarrow \\mathcal{Z}^3(\\textbf{T}_f\/\\mathbb{Q}_{\\op{cyc}})=0.\\]As a result,\n\\[\\mathcal{Z}^2(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})\\simeq \\mathcal{Z}^2(\\textbf{T}_f\/\\mathbb{Q}_{\\op{cyc}})\/p\\mathcal{Z}^2(\\textbf{T}_f\/\\mathbb{Q}_{\\op{cyc}})\\]is $\\Omega$-torsion if and only if the $\\mu$ invariant of $\\mathcal{Z}^2(\\textbf{T}_f\/\\mathbb{Q}_{\\op{cyc}})$ is zero. Therefore, condition (2) is equivalent to condition (1).\n\\end{proof}\nThe $\\mathfrak{p}^n$-Selmer group is \n\\[\\op{Sel}_{p^\\infty} (\\mathbf{A}_f[\\mathfrak{p}^n]\/\\mathbb{Q}_{\\op{cyc}}):=\\ker\\left\\{H^1(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_f[\\mathfrak{p}^n])\\rightarrow \\bigoplus_{w\\in S(\\mathbb{Q}_{\\op{cyc}})} H^1( \\mathbb{Q}_{\\infty,w}, D_w[\\mathfrak{p}^n])\\right\\},\\] where $D_w=\\mathbf{A}_f$ (resp. $D_w=\\mathbf{A}_f^-$) if $w\\nmid p$ (resp. $w\\mid p$).\nSet \\[\\operatorname{S}^*(\\mathbf{A}_f[\\mathfrak{p}^n]\/\\mathbb{Q}_{\\op{cyc}}):=\\varprojlim_L \\operatorname{Sel}(\\mathbf{A}_f[\\mathfrak{p}^n]\/L),\\]where the projective limit is taken over number fields $L\\subset \\mathbb{Q}_{\\op{cyc}}$ with respect to corestriction maps.\n\n\\begin{Lemma}\\label{Sstarinjection}\nWe have an injection \\[\\operatorname{S}^*(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})\\hookrightarrow \\operatorname{Hom}_{\\Omega}(\\op{Sel}_{p^\\infty} (\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})^{\\vee},\\Omega).\\]\n\\end{Lemma}\n\\begin{proof}\nThe proof is identical to that of \\cite[Lemma 5.5]{lim2018fine}.\n\\end{proof}\n\\begin{Lemma}\\label{muequalszeroOmega}\nThe following conditions are equivalent\n\\begin{enumerate}\n    \\item $\\mu\\left(\\op{Sel}_{p^\\infty}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})\\right)=0$,\n    \\item $\\operatorname{Sel}(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})$ is $\\Omega$-cotorsion.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nStandard arguments show that the kernel and cokernel of the natural map\n\\[\\operatorname{Sel}(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})\\rightarrow \\operatorname{Sel}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})[\\mathfrak{p}]\\] are finite, and the result follows from this.\n\\end{proof}\n\\begin{Th}\\label{muzeroconditions}\nThe following statements are equivalent:\n\\begin{enumerate}\n    \\item $\\mu\\left(\\op{Sel}_{p^\\infty}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})\\right)=0$,\n    \\item the group $H^2(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}},\\mathbf{A}_f[\\mathfrak{p}])$ is $\\Omega$-torsion and there is a short exact sequence\n    \\[0\\rightarrow \\op{Sel}_{p^\\infty}(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})\\rightarrow H^1(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}},\\mathbf{A}_f[\\mathfrak{p}])\\rightarrow \\bigoplus_{w\\in S(\\mathbb{Q}_{\\op{cyc}})} H^1(K_{\\infty,w}, D_w[\\mathfrak{p}])\\rightarrow 0.\\]\n    \\item The group $H^2(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}},\\mathbf{A}_f[\\mathfrak{p}])$ is $\\Omega$-torsion and $\\operatorname{S}^*(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})=0$.\n\\end{enumerate}\n\\end{Th}\n\\begin{proof}\nWe begin by showing that conditions $(2)$ and $(3)$ are equivalent. Assume condition $(2)$. Then from the Poitou-Tate sequence, $\\operatorname{S}^*(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})^{\\vee}$ injects into $H^2(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}},\\mathbf{A}_f[\\mathfrak{p}])$ and hence according to Lemma \\ref{muzeroH2}, is $\\Omega$-torsion. It follows from Lemma $\\ref{Sstarinjection}$ that $\\operatorname{S}^*(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})^{\\vee}=0$. Condition $(3)$ therefore follows from $(2)$. On the other hand, condition $(2)$ is a direct consequence of condition $(3)$ and the Poitou-Tate sequence.\n\\par In order to complete the proof, it suffices to show that conditions $(1)$ and $(3)$ are equivalent. Suppose that condition $(1)$ holds. Then being a quotient of $X(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})$ the dual fine Selmer group $Y(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})$ also has zero $\\mu$-invariant. By Lemma $\\ref{muzeroH2}$, it follows that $H^2(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}},\\mathbf{A}_f[\\mathfrak{p}])$ is $\\Omega$-torsion. From the Poitou-Tate sequence, \n\\[\\begin{split}&\\operatorname{corank}_{\\Omega}\\operatorname{Sel}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})+\\operatorname{corank}_{\\Omega}\\left(\\bigoplus_{q\\in S} K_q^1(D_q[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})\\right)\\\\=&\\operatorname{corank}_{\\Omega} H^1(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}}, \\mathbf{A}_f[\\mathfrak{p}])+\\operatorname{corank}_{\\Omega}\\operatorname{S}^*(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}}).\\end{split}\\]It follows from Corollory $\\ref{balancedCor}$ that \n\\[\\operatorname{corank}_{\\Omega} H^1(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}}), \\mathbf{A}_f[\\mathfrak{p}])=\\operatorname{corank}_{\\Omega}\\left(\\bigoplus_{q\\in S} K_q^1(\\mathcal{S}(D_q)_p\/\\mathbb{Q}_{\\op{cyc}})\\right)\\] and as a result, \n\\[\\operatorname{corank}_{\\Omega}\\operatorname{S}^*(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})=\\operatorname{corank}_{\\Omega}\\operatorname{Sel}(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}}).\\] Since $\\mu\\left(\\operatorname{Sel}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})\\right)=0$ it follows from Lemma $\\ref{muequalszeroOmega}$ that $\\operatorname{Sel}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})$ cotorsion over $\\Omega$. It follows that $\\operatorname{Sel}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})$ is $\\Omega$-cotorsion and as a result, $\\operatorname{S}^*(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})$ is $\\Omega$-cotorsion. It follows from Lemma $\\ref{Sstarinjection}$ that $\\operatorname{S}^*(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})=0$. Therefore, Poitou-Tate gives rise to a short exact sequence\n    \\[0\\rightarrow \\op{Sel}_{p^\\infty}(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})\\rightarrow H^1(\\mathbb{Q}_S\/\\mathbb{Q}_{\\op{cyc}},\\mathbf{A}_f[\\mathfrak{p}])\\rightarrow \\bigoplus_{w\\in S(\\mathbb{Q}_{\\op{cyc}})} H^1(\\mathbb{Q}_{\\infty,w}, D_w[\\mathfrak{p}])\\rightarrow 0.\\]\n    On the other hand, if condition (2) is satisfied, it from Corollory $\\ref{balancedCor}$ that $\\operatorname{Sel}(\\mathbf{A}_f[\\mathfrak{p}]\/\\mathbb{Q}_{\\op{cyc}})$ is $\\Omega$-cotorsion. It follows from Lemma $\\ref{muequalszeroOmega}$ that \\[\\mu\\left(X(\\mathbf{A}_f\/\\mathbb{Q}_{\\op{cyc}})\\right)=0.\\] This completes the proof of the Theorem.\n\\end{proof}\n\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\citet{gunn65} predicted that  Ly$\\alpha$ absorption would give rise to a sudden drop of \ncontinuum flux at wavelengths shorter than 1216 $\\AA$ if a tiny amount of neutral hydrogen is present along the line of sight. \nThe dramatic clearing of the Gunn-Peterson trough \nfrom the observation of quasars at $z\\sim6$ demonstrates \nthat hydrogen in the Universe is highly ionized at $z\\lesssim6$ \\citep{becker01,fan01,fan06}. \nPolarization signals from the comic microwave background (CMB) also suggest that \na large fraction of hydrogen may already be ionized by $z \\sim 10-12$ \\citep{komatsu11,planck-collaboration13}.\nYet, the detailed processes on how reionization has occurred remain unclear.\n\nIn the standard $\\Lambda$CDM universe, dwarf galaxies form early \\citep[e.g.,][]{somerville03}\nand could dominate the budget of hydrogen ionizing photons at the epoch of reionization. \nPhotons that escape from the porous interstellar medium \\citep[ISM,][]{clarke02}, \ndriven by supernova (SN) explosions \\citep{mckee77}, \nto the intergalactic medium (IGM) create \\mbox{{\\sc H ii}}\\ bubbles, which expand as more stars form.\nThe eventual percolation of \\mbox{{\\sc H ii}}\\ bubbles would mark the end of the cosmological reionization\n\\citep[e.g.,][]{gnedin00,mcquinn07,shin08}. \nThis stellar reionization scenario has been studied extensively, both (semi-) analytically \\citep[e.g.][]{madau99,miralda-escude00,barkana01,bianchi01,cen03,wyithe03,somerville03,bolton07,wyithe07,kuhlen12,robertson13} and \nnumerically \\citep[e.g.][]{gnedin00,razoumov02,ciardi03,fujita03,trac07,gnedin08,wise09,razoumov10,yajima11,Paardekooper13}.\nIt appears that dwarf galaxies are the most plausible source of the ionizing photons,\nprovided that the escape fraction is significant ($\\mbox{$f_{\\rm esc}$} >10 \\%$).\nActive galactic nuclei also contribute to ionizing photons in both the ultraviolet (UV) and X-ray bands but\nare generally believed to be sub-dominant to stellar sources \n\\citep{haehnelt01,wyithe03,schirber03,faucher-giguere08a,cowie09,willott10,fontanot14}.\nThe strong accretion shock present in massive halos ($\\mbox{${M}_{\\rm vir}$} \\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} 10^{10.5}\\, \\mbox{${M}_\\odot$}$) \nmay also produce a non-negligible amount of hydrogen ionizing photons in the vicinity of the galactic gaseous disk \\citep{dopita11}.\n\nThe major uncertainty in the dwarf galaxy-driven reionization picture is the escape fraction of ionizing photons.  \nObservationally, this is difficult to probe, because the hydrogen ionizing photons escaping from \ndwarf galaxies will get easily absorbed by the IGM during reionization ($z\\gtrsim7$).\nBesides, it requires a large sample of galaxies to obtain a statistically significant estimate of the\nescape fraction ($f_{\\rm esc}$). Nevertheless, it is worth noting that galaxies at higher \nredshift often exhibit a larger relative escape fraction ($f_{\\rm esc}^{\\rm rel}$), which is defined as the ratio of \nthe escape fraction at 900$\\AA$ and 1500$\\AA$, than their low-$z$ counterparts \\citep{siana10}. \nObservations of star-forming galaxies at $z\\lesssim1$ indicate that the relative escape fraction is only \na few percent \\citep{leitherer95,deharveng01,malkan03,siana07,cowie09,bridge10,siana10}. \nThe only exception reported so far is Haro 11, which shows  $f_{\\rm esc}\\sim 4-10\\%$ \\citep{bergvall06}.\nOn the other hand,  a non-negligible fraction ($\\sim10\\%$) of star-forming galaxies at $z\\sim3$ reveals \na high escape of  $f_{\\rm esc}^{\\rm rel} \\ge  0.5$ \\citep{shapley06,iwata09,nestor11,nestor13,cooke14}.\nFor typical Lyman break galaxies at $z\\sim3$ in which 20--25\\% of UV photons are escaping \\citep{reddy08},\nthe relative fraction corresponds to a high escape fraction of $\\mbox{$f_{\\rm esc}$}\\sim0.1$.\nGiven that galaxies are more actively star forming at high redshift \\citep[e.g.][]{bouwens12a,dunlop13},\nit has been suggested that there may be a correlation between star formation rate and \\mbox{$f_{\\rm esc}$}, \nand possibly evolving \\mbox{$f_{\\rm esc}$}\\ with redshift \\citep[][]{kuhlen12}. \n\nPredicting the escape fraction in theory is also a very challenging task.\nThis is essentially because there is little understanding on the structure of the ISM at high-$z$ dwarf galaxies. \nNumerical simulations are perhaps the most suited to investigate this subject, \nbut different subgrid prescriptions and\/or finite resolution often lead to different conclusions. \nUsing an adaptive mesh refinement (AMR) code, ART \\citep{kravtsov97},  with SN-driven energy \nfeedback, \\citet{gnedin08} claim that the angle-averaged escape fraction increases with galaxy mass \nfrom $10^{-5}$ to a few percents in the range $10^{10} \\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} M_{\\rm gal} \\le 4\\times10^{11}$.\nThey attributed this trend to the fact that more massive galaxies have smaller gas-to-stellar scale-height than \nlower mass galaxies in their simulations. On the other hand, \\citet{razoumov10} argue based on cosmological \nTreeSPH simulations \\citep{sommer-larsen03} that more than 60\\% of the \nhydrogen ionizing photons escape from dwarf galaxies in dark matter halos of $M_{\\rm halo}=10^8-10^9\\mbox{${M}_\\odot$}$. \nMore massive halos of $10^{11}\\mbox{${M}_\\odot$}$ are predicted to have a considerably smaller \\mbox{$f_{\\rm esc}$}\\ ($\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 10\\%$). \nA similar conclusion is reached by \\citet{yajima11}.  It should be noted, however, that resolution could \npotentially be an issue in these two studies in the sense that their resolution of a few hundreds to \nthousands of parsec is unable to resolve most star-forming regions and hence capture obscuring \ncolumn densities and a porous ISM. \\citet{wise09} performed cosmological radiation hydrodynamic \nsimulations employing very high resolution (0.1 pc), and found that the neutral hydrogen column \ndensity varies over the solid angles from $N_{\\rm HI}\\sim 10^{16}\\, {\\rm cm^{-2}}$ \nto $10^{22}\\, {\\rm cm^{-2}}$ with the aid of SN explosions and photo-ionization.\nBecause of the porous ISM, a high \\mbox{$f_{\\rm esc}$}\\ of $\\sim40\\%$ is achieved   \nin small halos of $M_{\\rm halo}=10^{7} - 10^{9.5} \\mbox{${M}_\\odot$}$. \n\\citet{wise14} show that an even higher fraction ($\\sim 50\\%$) of hydrogen \nionizing photons escapes from minihalos of $M_{\\rm halo}=10^{6.25} - 10^{7} \\mbox{${M}_\\odot$}$.\n\n\nAnother potentially important source of ionizing radiation is runaway OB stars that are dynamically \ndisplaced from their birthplace. The runaway OB stars are normally defined by their peculiar motion \n\\citep[$v_{\\rm pec} \\ge 30\\, {\\rm km\\,s^{-1}}$,][]{blaauw61}, and roughly $30\\%$ of OB stars are \nclassified as runaways in the Milky Way \\citep{stone91,hoogerwerf01,tetzlaff11}.\nAlthough the fraction is still uncertain, their peculiar speed of $\\left<v_{\\rm pec}\\right>\\sim 40\\,{\\rm km\\,s^{-1}}$ means \nthat the runaway OB stars can, in principle, travel away from the birthplace by $\\sim$200 pc in 5 Myrs,\nmaking them an attractive source for the ionizing photons. \nThe runaway OB stars are thought to originate from a three-body interaction with other stars in a young \ncluster \\citep{leonard88}, and\/or from a SN explosion of a companion in a binary system \\citep{blaauw61}.\n\\citet{conroy12} evaluated the impact of the inclusion of runaway OB stars on \\mbox{$f_{\\rm esc}$}\\ using a simple analytic argument, \nand concluded that the runaway OB stars may enhance \\mbox{$f_{\\rm esc}$}\\ by a factor of up to $\\sim4.5$ in halos with  \n$M_{\\rm halo}=10^8-10^9\\mbox{${M}_\\odot$}$. \n\nThe aim of this study is to investigate the importance of the aforementioned two processes \nby measuring the escape fraction from high-resolution cosmological radiation hydrodynamics simulations. \nFirst, given that modeling the SN explosion as thermal energy\n is well known to have the artificial radiative cooling problem \\citep[e.g.][]{katz92,slyz05}, \nwe expect that the role of the SN is likely to be underestimated in some cosmological simulations \\citep[e.g.][]{gnedin08}. \nWith a new physically based SN feedback model that captures all stages of the Sedov explosion from \nthe free expansion to the snowplow phase, we study the connection between the escape of ionizing photons and feedback processes \nin dwarf galaxies. Second, we extend the idea by \\citet{conroy12}, and quantify \nthe impact from the runaway OB stars on reionization in a more realistic environment.\n\nWe first describe the details of our cosmological radiation hydrodynamics simulations \nincluding the implementation of runaway OB stars in Section~2.  \nWe present the feedback-regulated evolution of the escape fraction and the impact of the inclusion \nof runaway OB stars in Section~3. We summarize and discuss our findings in Section~4. \nOur new mechanical feedback from SN explosions  is detailed in Appendix.\n\n\n\n\n\\section{Method}\n\n\\subsection{Hydrodynamics code}\nWe make use of the Eulerian adaptive mesh refinement code, {\\sc ramses} \\citep[][ver. 3.07]{teyssier02}, to investigate \nthe escape of ionizing radiation from high-$z$ galaxies. \n{\\sc ramses} is based on the fully threaded oct-tree structure \\citep{khokhlov98}, \nand uses the second-order Godunov scheme to solve Euler equations.\nThe hydrodynamic states reconstructed at the cell interface are limited using the MinMod method,\nand then advanced using the Harten-Lax-van Leer contact wave Riemann solver \\citep[HLLC,][]{toro94}.\nWe adopt a typical Courant number of 0.8. The poisson equation is solved using the adaptive particle-mesh method.\nGas can effectively cool down to $10^4$ K by atomic and metal cooling \\citep{sutherland93}.\nBelow $10^4$ K, metal fine-structure transitions, such as {\\sc [CII]} 158$\\mu m$, can further lower \nthe temperature down to 10 K, as in \\citet{rosen95}. We set the initial metallicity to $2\\times10^{-5}$, \nas primordial SNe can quickly enrich metals in mini-halos of mass $10^7\\,\\mbox{${M}_\\odot$}$ \\citep[e.g.,][]{whalen08}, \nwhich our simulations cannot resolve properly.\n\nWe use the multi-group radiative transfer (RT) module developed by \\citet{rosdahl13} \nto compute the photoionization by stars. \nThe module solves the moment equations for three photon packets ({\\sc Hii}, {\\sc Heii}, and {\\sc Heiii} ionizing photons) \nusing a first-order Godunov method with M1 closure for the Eddington tensor. \nWe adopt the Harten-Lax-van Leer \\citep[HLL,][]{harten83} intercell flux function. \nIonizing photons from each star are taken into consideration in every fine step.\nNote that an advantage of  the moment-based RT is that it is not limited by the number of sources.\nThe production rate of the ionizing photon varies with time for a given initial mass function \\citep[IMF,][see also \\citealt{rosdahl13}]{leitherer99}. The majority of the ionizing photons are released in $\\sim$ 5 Myr of stellar age.\nWe adopt the production rate equivalent to that of Kroupa IMF \\citep{kroupa01} \nfrom the {\\sc Starburst99} library \\citep{leitherer99}\\footnote{Note that we use the Chabrier IMF to \nestimate the frequency of SN explosions. We choose the number of ionizing  photons equivalent to that of the Kroupa IMF, \nbecause the models with the Chabrier IMF is not yet available in the {\\sc Starburst99} \\citep{leitherer99}}.\nThe radiation is coupled with gas via photo-ionization and photo-heating,\nand a set of non-equilibrium chemistry equations for {\\sc Hii}, {\\sc Heii}, and {\\sc Heiii} \nare solved similarly as in \\citet{anninos97}. We assume that photons emitted by recombination are \nimmediately absorbed by nearby atoms (case B).\nThe speed of light is reduced for the speed-up of the simulations by 0.01 \\citep[e.g.][]{gnedin01}.\nThis is justifiable because we are mainly interested in {\\it the flux} of escaping photons at the virial sphere.\n\n\n\n\\begin{table}\n   \\caption{Summary of cosmological simulations}\n      \\label{table1}\n      \\centering\n   \\begin{tabular}{@{}cccccccc} \n   \\hline\n   \\hline\nModel &  SNII & RT & Run-       & $\\Delta x_{\\rm min}$ & ${m_{\\rm star,min}}$ &  $m_{\\rm dm}$  \\\\\n            &           &       & aways     &  [pc]                              &    [$\\mbox{${M}_\\odot$}$]                & [$10^5\\,\\mbox{${M}_\\odot$}$]  \\\\\n\\hline\nFR & $\\checkmark$ &  $\\checkmark$ & -- &  4.2 & 49 &  1.6 \\\\\nFRU &$\\checkmark$ & $\\checkmark$ & $\\checkmark$ & 4.2 & 49 & 1.6  \\\\\n\\hline\n   \\end{tabular}\n\\end{table}\n\n\n\\begin{figure}\n   \\centering\n   \\includegraphics[width=7.5cm]{fig1.eps}\n   \\caption{Dark matter halo mass function from the zoomed-in region of the $\\textsf{FR}$\\ run at $z=7$.\n   Comparison with \\citet{jenkins01} mass function at the same epoch indicates that our simulated volume \n   represents the average region of the universe. }\n   \\label{fig:mf}\n\\end{figure}\n\n\n\\subsection{Cosmological Simulations}\n\nWe carry out cosmological simulations to investigate \nthe escape fraction in realistic environments. For this purpose, we generate the initial condition \nusing the {\\sc music} software \\citep{hahn11}, with the WMAP7 cosmological parameters \\citep{komatsu11}:\n$(\\Omega_{\\rm m}, \\Omega_{\\Lambda}, \\Omega_{\\rm b}, h, \\sigma_8, n_s  = 0.272, 0.728, 0.045, 0.702, 0.82, 0.96)$.\nA large volume of $(25\\,{\\rm Mpc} \\, h^{-1})^3$ is employed to include the effect of the large-scale tidal field. \nTo achieve high mass resolution, we first run dark matter-only simulations with 256$^3$ particles, \nand identify a rectangular region of $3.8\\times4.8\\times9.6$ Mpc (comoving)\nthat encloses two dark matter halos of $\\simeq 1.5\\times 10^{11} \\mbox{${M}_\\odot$}$ at $z=3$.\nThen, we further refine the mass distribution of the zoomed-in region, such that the mass of a dark matter particle \nis $m_{\\rm dm}=1.6\\times10^5\\,\\mbox{${M}_\\odot$}$, which corresponds to 2048$^3$ particles in effect.\nDespite that we purposely select the region in which two massive dark matter halos are present at $z=3$,\na comparison with the number of dark matter halos per volume predicted by \\citet{jenkins01} shows that \nour simulated box represents an average region of the universe at $z=7$  (Figure~\\ref{fig:mf}). \n\n\\begin{figure}\n   \\centering\n      \\includegraphics[width=8.6cm]{fig2.eps}\n      \\caption{Expansion of the \\mbox{{\\sc H ii}}\\ bubble in a cosmological simulation ($\\textsf{FR}$). Three panels show the evolution of \n      the density-weighted fraction of ionized hydrogen of the zoomed-in region. The horizontal size of the figure is \n      9.5 Mpc (comoving).}\n            \\label{fig:hii}\n\\end{figure}\n\n\nThe level of the root grid in the zoomed-in region is 11, consistent with the dark matter resolution.\nFurther 12 levels of refinement are triggered if the dark matter plus baryon mass in a cell exceeds \n8 times the mass of a dark matter particle. We keep the minimum physical size of a cell to \n $\\Delta x_{\\rm min}=25\\,{\\rm Mpc} \\, h^{-1}\/ 2^{23} = 4.2\\,{\\rm pc}$ over the entire redshift. However, this refinement \ncriterion is not optimized to resolve the structure of the ISM, unless extremely high mass resolution is adopted. \nFor example, for a gas cell of $n_{\\rm H}=10\\, {\\rm cm^{-3}}$, the criterion will come into play only if the size of \nthe cell is larger than $\\sim$ 160 pc.  In order to better resolve the structure of the ISM, \nwe enforce a cell with $n_{\\rm H}\\ge 1\\, {\\rm cm^{-3}}$ to be resolved on $8 \\Delta x_{\\rm min}=34\\,{\\rm pc}$.\nIn a similar context, we apply more aggressive refinement criterion for the star-forming gas \nin such a way that gas with $n_{\\rm H}=100\\, {\\rm cm^{-3}}$ ($800\\,{\\rm cm^{-3}}$) is always \nresolved on a 8.5 pc (4.2 pc) cell.\nWe adopt very high stellar mass resolution of $\\approx 49\\,\\mbox{${M}_\\odot$}$.\nThis means that a star particle with the minimum mass will produce a single SN event for the Chabrier IMF.\n\n\n\nWe run two sets of cosmological simulations, $\\textsf{FR}$\\ and $\\textsf{FRU}$, with the identical initial condition down to $z=7$.\nBoth runs include star formation, metallicity-dependent radiative cooling \\citep{sutherland93,rosen95}, \nthermal stellar winds, mechanical feedback from SN explosions, and photoionization by stellar radiation.\nThe runaway OB stars are included only in the $\\textsf{FRU}$\\ run. In Figure~\\ref{fig:hii}, we show an example of the \ngrowth of \\mbox{{\\sc H ii}}\\ bubbles in the $\\textsf{FR}$\\ run. Our simulated region is nearly ionized at $z=7$.\n\n\n\nDark matter (sub) halos are identified using the {\\sc Amiga} halo finder \\citep[{\\sc Ahf},][]{gill04,knollmann09}.\n{\\sc Ahf} first constructs the adaptive meshes based on the particle distribution, finds the density minima,\nand determines physical quantities based on a virial overdensity ($\\Delta_{\\rm vir}$).\nGravitationally unbound particles are removed iteratively if they move faster than the local escape velocity \nduring this procedure. The virial radius is defined such that \nthe mass enclosed within the virial sphere is the virial overdensity times the critical density of the universe times the volume, \ni.e. $\\mbox{${M}_{\\rm vir}$}(z) = \\Delta_{\\rm vir}(z) \\rho_{\\rm crit}(z) 4 \\pi r_{\\rm vir}^3 \/ 3$.\nWe take $\\Delta_{\\rm vir}=177$, appropriate for a $\\Lambda$-dominated universe at $z>6$ \\citep{bryan98}.\nThis results in 796, 443, and 183 dark matter halos of mass $\\mbox{${M}_{\\rm vir}$}\\ge10^{8}\\,\\mbox{${M}_\\odot$}$ immune to the contamination \nby coarse dark matter particles ($m_{\\rm dm} > 1.6\\times10^{5}\\,\\mbox{${M}_\\odot$}$) at $z=7$, 9, and 11, respectively. \n\n\n\n\\subsection{Star Formation and Feedback}\n\nStars form in a very dense, compact molecular core. \nInfrared extinction maps of nearby interstellar cores indicate that their size ranges from 0.01 to 0.4 pc\n\\citep[e.g.][]{alves07,konyves10}, which is difficult to resolve in current cosmological simulations.\nNevertheless, studies of gravitational collapse in converging flows \\citep{gong11} seem to suggest that \na gravitationally bound cloud is likely to experience runaway collapse no matter how the collapse is initiated.  \nIn a similar spirit, we assume that stars would form in a cell if the following conditions are met simultaneously\n\\citep[e.g.][]{cen92}:\n\\begin{itemize}\n\\itemsep0em\n\\item[1.] the flow is convergent ($\\vec{\\nabla}\\cdot \\rho {\\vec  v} <0$) ,\n\\item[2.] the cooling time is shorter than the dynamical time, \n\\item[3.] the gas is Jeans unstable,  and\n\\item[4.] the number density of hydrogen exceeds the threshold density $n_{\\rm th}={\\rm 100 \\,cm^{-3}}$.\n\\end{itemize}\nThe last condition is motivated by the density of  a Larson-Penston profile \\citep{larson69,penston69} at $0.5\\Delta x$,\n $\\rho_{\\rm LP}\\approx8.86 c_s^2 \/ \\pi\\,G\\,\\Delta x^2$, where $c_s$ is the sound speed and $\\Delta x$ is \n the size of the most refined cell. \n Star particles are created based on the Schmidt law \\citep[][]{schmidt59}, \n $ \\dot{\\rho}_{\\star} = \\epsilon_{\\rm ff} \\, \\rho_{\\rm gas} \\, \/ \\, t_{\\rm ff} $,  assuming that 2\\% of the star-forming \n gas ($\\epsilon_{\\rm ff}$)  is converted into stars per its free-fall time ($t_{\\rm ff}$) \\citep{krumholz07,kennicutt98}.  \n The mass of each star particle is determined as $m_\\star=\\alpha\\, N_p \\rho_{\\rm th} \\, \\Delta x_{\\rm min}^3 $, \n where $\\rho_{\\rm th}$ is the threshold density for star formation, \n$\\Delta x_{\\rm min}$ is the size of the most refined cell,  and $\\alpha$ is a parameter that \ncontrols the minimum mass of a star particle. $N_p$ is the number of star particles to be formed in a cell,\nwhich is drawn from a Poisson random distribution, $P(N_p) = (\\lambda ^{N_p} \/ N_p! ) \\exp\\left(-\\lambda\\right)$.\nHere the Poissonian mean ($\\lambda$) is computed as \n$\\lambda \\equiv \\epsilon_{\\rm ff} \\left({\\rho\\Delta x^3}\/{m_{\\rm \\star,min}}\\right) \\left( {\\Delta t_{\\rm sim}}\/{t_{\\rm ff}}\\right), $\nwhere $\\Delta t_{\\rm sim}$ is the simulation time step, and $m_{\\rm \\star,min}$ is the minimum stellar mass (i.e. $N_p=1$).\n\nWe describe the SN feedback using a new physical model which captures \nthe SN explosion at all stages from the early free expansion to the final momentum-conserving snowplow phase.\nBriefly, we deposit radial momentum to the cells affected by supernova feedback, conserving energy appropriately.\nThe amount of input momentum is determined by the stage the blast wave is in, which in turn is dependent upon the \nphysical condition  (density and metallicity) of the gas being swept up and simulation resolution. \nThe virtue of our scheme is that an approximately (within 20\\%) correct  amount of momentum is imparted to the \nsurrounding gas regardless of the resolution. Thus, this prescription should be useful to cosmological simulations,\nespecially those with finite resolution that potentially suffer from the artificial radiative cooling. \nThe details of our implementation and a simple test are included in the Appendix.\n\nThe frequency of a SN per solar mass is estimated assuming the Chabrier IMF \\citep{chabrier03}.\nFor the simple stellar population with a low- (high-) mass cut-off of 0.1 (100) \\mbox{${M}_\\odot$}, \nthe total mass fraction between 8 to 100 \\mbox{${M}_\\odot$}\\ is 0.317, and the mean SN progenitor mass is 15.2 \\mbox{${M}_\\odot$}\\ on the zero-age main sequence.\nAt the time of the explosion, we also deposit newly processed metals into the surrounding. \nThe mass fraction of newly synthesized metals in stellar ejecta is taken to be 0.05 following \\citet{arnett96}.\nA star particle is assumed to undergo the SN phase after the main sequence lifetime of the mean SN progenitor \\citep[10 Myr,][]{schaller92}. As discussed in \\citet{slyz05}, allowing for the delay between the star formation and explosion \n(i.e. stellar lifetimes) is crucial to the formation of hot bubble in the ISM. \nWe find that the physically based SN feedback employed in this study drives stronger galactic winds \nthan the runs with thermal feedback or kinetic feedback that are valid only under certain conditions\n \\citep[][see below]{dubois08}. \nStellar winds from massive stars are modeled as thermal input, based on \\citet{leitherer99}.\n\n\n\n\n\\subsection{Runaway OB Stars}\n\nOur implementation of runaway OB stars is largely motivated by \\citet{tetzlaff11},\nwho compiled candidates of runaway stars younger than 50 Myr for the 7663  {\\it Hipparcos} sample. \nBy correcting the solar motion and Galactic rotation, they found that the peculiar space velocity of the stars \nmay be decomposed into two Maxwellian distributions intersecting at 28 ${\\rm km\\,s^{-1}}$.\nAssuming that each Maxwellian distribution represents a kinematically distinctive population, \nthey estimated the fraction of the runaways to be $\\sim 27.7\\%\\pm 1.9$ for the sample \nwith full kinematic information. The dispersion of the Maxwellian distribution is \nmeasured as 24.4 ${\\rm km\\, s^{-1}}$ for the high-velocity group.\n\nSince either runaway OB stars formed through the explosion of a SN in a binary or \nthose dynamically ejected in a cluster are not resolved in our simulations, \nwe crudely approximate this by splitting a star particle into a normal (70\\% in mass) \nand a runaway particle (30 \\%) at the time of star formation. While the initial velocity of the normal star is chosen \nas the velocity of the birth cloud, we add a velocity drawn from the Maxwellian distribution \non top of the motion of the birth cloud for runaway particles. To do so, we generate the distribution following \nthe Maxwellian with the dispersion of $\\sigma_v = 24.4\\,{\\rm km\\,s^{-1}}$ and the minimum space \nvelocity of $v_{\\rm 3D}=28 \\,{\\rm km\\,s^{-1}}$ using the rejection method \\citep{press92}. The direction of the \nrunaway motion is chosen randomly for simplicity. A similar approach is taken by \\citet{ceverino09} to \nstudy the formation of disk galaxies in a cosmological context.\n\n\n\n\n\n\\subsection{Estimation of Escape Fraction}\n\nThe fraction of escaping ionizing photons ($f_{\\rm esc}$) is measured by comparing the photon flux at the virial radius \nand the photon production rate from young massive stars.\nSince the speed of light is finite, there is a small delay in time between the photons produced by the stars and \nthe photons escaping at the virial sphere. In order to take this into account, we use the photon production rate at earlier time ($t-r_{\\rm vir}\/c'$), \nwhere $c'$ is the reduced speed of light used in the simulations. The escape fraction is then computed as\n\\begin{equation}\nf_{\\rm esc}(t) \\equiv \\frac{\\int d\\Omega\\, \\vec{F}_{\\rm ion}(t) \\cdot \\hat{r} ~\\Theta(\\vec{F}_{\\rm ion}\\cdot \\hat{r})}{\\int dm_* \\, \\dot{N}_{\\rm ion} (t-r_{\\rm vir}\/c')},\n\\label{eq:fesc}\n\\end{equation}\nwhere $\\vec{F}_{\\rm ion}$ is the ionizing photon flux, $d\\Omega$ is the solid angle, $m_*$ is the mass of each star particle,\n$\\dot{N}_{\\rm ion}(t)$ is the photon production rate of a simple stellar population of age $t$ per solar mass, \nand $\\Theta$ is the Heaviside step function. Here, we approximate the delay time  to be a constant, $r_{\\rm vir}\/c'$, \nfor each halo assuming that the central source is point-like. \nSince only outflowing photons are considered in Equation~\\ref{eq:fesc},\nwe find that  a minor fraction ($\\sim 5\\%$) of galaxies exhibit $f_{\\rm esc}$ greater than 1.\nThis happens mostly when there is little absorbers left in the halo after disruptive SN explosions.\nIn this case, we randomly assign $f_{\\rm esc}$ between 0.9 and 1.0.\nWe confirm that the photon production rate-averaged escape fraction, which is the most important quantity in this study, \nis little affected by this choice even if the net flux is used, and thus we decide to take a simpler method. \n\nDust can also affect the determination of the escape of the hydrogen ionizing photons. \nHowever, given that our simulated galaxies are very \nmetal-poor ($0.002-0.05\\,Z_{\\odot}$) and galaxies with lower metallicity have a progressively lower amount of \ndust \\citep{lisenfeld98,engelbracht08,galametz11,fisher13},  it is unlikely that dust decreases the escape \nfraction substantially. Thus, we neglect the absorption of hydrogen ionizing photons by dust in this study. \n\n\n\n\n\n\\section{Results}\n\\subsection{Feedback-regulated Escape of Ionizing Photons}\n\n\n\\begin{figure}\n   \\centering\n            \\includegraphics[width=8.5cm]{fig3.eps}\n   \\caption{The baryon-to-star conversion efficiency at $z=7$ from the $\\textsf{FR}$\\ (blue) and the $\\textsf{FRU}$\\ (orange) runs. \n   Only central galaxies are shown. The cosmic mean    ($\\Omega_{\\rm b}\/\\Omega_{\\rm m}=0.165$) is \n   shown as a black solid line. Also included as a star is the stellar fraction measured from \n   the NutFB simulation \\citep{kimm11b}.\n   Our mechanical feedback from SN explosions is more effective \n   at regulating star formation, compared with previous studies injecting thermal or kinetic energy (see the text).\n   }\n   \\label{fig:mstar}\n\\end{figure}\n\n\\begin{figure*}\n   \\centering\n      \\includegraphics[width=17cm]{fig4_1.eps}\n      \\includegraphics[width=18cm]{fig4_2.eps}\n      \\includegraphics[width=17cm]{fig4_3.eps}\n      \\caption{Evolution of the escape fraction (\\mbox{$f_{\\rm esc}$}) and specific star formation rate (sSFR) in two massive \n      halos from the $\\textsf{FR}$\\ run. Black solid lines in the top and bottom panels indicate the escape fraction measured \n      at the virial radius at each snapshot as a function of the age of the universe. We denote the logarithmic stellar mass \n      at different times by orange text.\n      Black dashed lines correspond to the photon number-weighted average of \\mbox{$f_{\\rm esc}$}\\ by that time (\\mbox{$\\left<f_{\\rm esc}\\right>$}). \n      Blue shaded regions display the sSFR in ${\\rm Gyr^{-1}}$. One can see that there is a delay between \n      the peak in \\mbox{$f_{\\rm esc}$}\\ and sSFR due to the \n      delay in the onset of the strong outflow. The middle panels show \n      an example of this delay identified in the top panel (a,b). The projected density of gas and the fraction of \n      ionized hydrogen are shown in both cases, as indicated in each panel. Interestingly, the volume filling \n      fraction of the neutral hydrogen within 0.2 \\mbox{${R}_{\\rm vir}$}\\ is found to be 25\\% large in the snapshot (b), indicating that \n      \\mbox{$f_{\\rm esc}$}\\ depends not only by the volume-filling, circumgalactic neutral gas, but also dense star\n      forming gas.  We do not display the physical quantities if $M_{\\rm vir}\\le10^8\\,\\mbox{${M}_\\odot$}$.\n      }\n            \\label{fig:ex}\n\\end{figure*}\n\n\\begin{figure*}\n   \\centering\n            \\includegraphics[width=8.2cm]{fig5_1.eps}\n                  \\includegraphics[width=8.2cm]{fig5_2.eps}\n   \\caption{ \n   {\\it Left}: Escape fraction measured at the virial radius at three different redshifts from the $\\textsf{FR}$\\ run.\n   Different redshifts are shown as different colors and symbols, as indicated in the legend. \n   To increase the statistical significance, we combine the results \n   from seven consecutive snapshots for each redshift. Solid lines indicate the median, and error bars show \n   the interquartile range. Although there is a large scatter,  more than 50\\% of the galaxies reveal $\\mbox{$f_{\\rm esc}$} \\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} 10\\%$.\n   {\\it Right:} Photons escaping per second through the virial sphere.  \n   }\n   \\label{fig:fesc_stat}\n\\end{figure*}\n\nCosmological hydrodynamics simulations often suffer from the artificial over-cooling problem \nin forming disk galaxies \\citep[e.g.][]{kimm11b,hummels12}, mainly because the energy from SN \nexplosions is radiated away before it is properly transferred to momentum due to inadequate resolution\nof the multi-phase ISM.  This directly affects the escape of ionizing photons. Motivated by this challenge, \nwe have implemented a SN feedback scheme that reasonably approximates the Sedov blast waves \nfrom the free expansion to snowplow stages. In Figure~\\ref{fig:mstar}, we present the baryon-to-star \nconversion efficiency ($f_{\\star}\\equiv M_{\\rm star}\/(\\Omega_{\\rm b} M_{\\rm vir}\/\\Omega_{\\rm m}$)\nof the central galaxies in dark matter halos at $z=7$ from the $\\textsf{FR}$\\ run.  \nIt shows that our new physically motivated SN feedback is very effective at suppressing star formation. \nFor example, the most massive halo with $M_{\\rm vir}\\sim 3\\times10^{10}\\,\\mbox{${M}_\\odot$}$ at $z=7$ shows $f_{\\star}\\approx0.08$. \nAlthough the direct comparison may be difficult due to a different initial condition used, \nit is worth noting that the conversion efficiency is about a factor of 7 smaller than \nthat found in the {\\sc NutFB} run \\citep[][see Fig.13]{kimm11b}, shown as a star in Figure~\\ref{fig:mstar}.  \nWe note that the momentum input from SN explosions used in the {\\sc NutFB} run is a factor of $3-4$ smaller \ncompared with that at the end of the cooling phase \\citep[see Appendix,][]{blondin98}.\nFor lower mass halos, the conversion efficiency is found to be even lower, \nreaching $M_{\\rm star} \/ M_{\\rm vir} \\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 0.01 \\, \\Omega_{\\rm b} \/ \\Omega_{\\rm m} $ at $M_{\\rm vir} \\sim 10^9\\,\\mbox{${M}_\\odot$}$.\nIt is also interesting to note that the conversion \nefficiency at $M_{\\rm vir}\\ge 10^{10}\\mbox{${M}_\\odot$}$ also agrees reasonably well within error bars with the \nsemi-analytic results obtained to reproduce the observed stellar mass function, star formation rate, and \ncosmic star formation rate density  \\citep[e.g.,][Figure~7]{behroozi13}.\nAs the feedback becomes more effective and fewer stars are formed, the stellar metallicity of these high-$z$ galaxies \nwould be lower. \nWe find that the most massive galaxy in our $z=7$ sample ($M_{\\rm star}=4\\times10^8\\,\\mbox{${M}_\\odot$}$) \nhas a stellar metallicity of 0.05 $Z_{\\rm \\odot}$. \nThis is at least factor of 2--3 smaller than the prediction by \\citet{finlator11} at the same epoch.  \n\\citet{kimm13} also investigated UV properties of $z=7$ galaxies of stellar mass\n$5\\times10^8 - 3\\times10^{10}\\,\\mbox{${M}_\\odot$}$ using a SN energy-based feedback scheme,\nand found that stellar metallicities are generally higher than those found in the $\\textsf{FR}$\\ run. \n\\citet{kimm13} found that the stellar metallicity for galaxies of mass $4\\times 10^{8}\\mbox{${M}_\\odot$}$\nfalls in the range of $0.1-0.5Z_{\\rm \\odot}$. \nThe gas metallicities ($Z_{\\rm gas}$) are also different in the two simulations.\nThe gas metallicity of the ISM within $2.56$ kpc for the $4\\times 10^{8}\\mbox{${M}_\\odot$}$ galaxies\nis $0.083Z_{\\rm \\odot}$ in the FR run, which is about a factor of 3 lower, on average, \nthan that of \\citet{kimm13}  ($Z_{\\rm gas}=0.1-0.7Z_{\\rm \\odot}$).\nThese comparisons lead us to conclude that our physically based feedback scheme is effective in \nalleviating the overcooling problem.\n\nOne may wonder whether stars form inefficiently in these small haloes \n($10^8\\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} M_{\\rm vir} \\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 10^9\\,\\mbox{${M}_\\odot$}$) \nbecause gas accretion is suppressed due to the ionizing background radiation \\citep{shapiro94,thoul96,gnedin00b,dijkstra04,sobacchi13,noh14}. \nHowever, this is unlikely the case, given that galaxies in the atomic cooling halos \nare fed mainly by dense filaments and satellites at high redshift \\citep[e.g.,][]{powell11}, \nwhich are self-shielded from the background radiation \\citep{faucher-giguere10,rosdahl12}.\nEven in the absence of the self-shielding, \\citet{geen13} find no clear sign that reionization suppresses\nstar formation in such halos at $z>6$. \\citet{wise14} also show that the fraction of baryons \nin a $10^8$-$10^9\\,\\mbox{${M}_\\odot$}$ halo is reduced only by less than a factor of two compared with the cosmic mean in \ntheir cosmological radiation hydrodynamics simulations with thermal supernova feedback and reionization.\nIndeed, we confirm that our mechanical supernova feedback is \nprimarily responsible for the low conversion efficiency by directly comparing the stellar mass of the dwarf galaxies \nbetween the simulations with and without ionizing radiation (see the Appendix).\n\nWe now present the time evolution of star formation rate and ionizing photon escape fraction\nof two randomly chosen relatively massive galaxies in Figure~\\ref{fig:ex}.\nThe plot corroborates that the feedback from stars governs the evolution of galaxies. \nThe top and bottom panels show the evolution of specific star formation rate \n(sSFR$ \\equiv \\dot{M}_{\\rm star}\/M_{\\rm star}$) and instantaneous \\mbox{$f_{\\rm esc}$}\\ of the central galaxy  \nin dark matter halos of mass $3\\times10^{10}$ and $10^{10}\\,\\mbox{${M}_\\odot$}$, respectively.  \nThe SFR is computed by averaging the mass of newly formed stars over 3 Myr.\nIt is evident that star formation is episodic on a time scale of $10-30$ Myr with both\nthe frequency and oscillation amplitude decreasing with increasing stellar mass. \nThis means that SN explosions effectively \ncontrol the growth and disruption of star-forming clouds.\nWhen the galaxies are small ($t_{\\rm H} \\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 0.5\\, {\\rm Gyr}$), the explosions even completely \nshut down the star formation across the galaxies, as stars form only in a few dense clouds. \nDuring these quiet periods, \\mbox{$f_{\\rm esc}$}\\ is kept high ($\\mbox{$f_{\\rm esc}$} \\lower.5ex\\hbox{$\\; \\buildrel > \\over \\sim \\;$} 0.2$). On the other hand, massive \ngalaxies contain  many star-forming clumps,  as can be seen in the projected density plot (middle row).\nThe fact that the episodic star formation history becomes more smooth at late times indicates that \nthese clumps are not entirely susceptible, but somewhat resilient to the SN explosions \narising from neighboring star clusters.\n\nMore importantly, we find that there is a time delay between the peak of \\mbox{$f_{\\rm esc}$}\\ and sSFR. This is \nessentially because massive stars with $M\\approx15\\,\\mbox{${M}_\\odot$}$ explode $\\sim$10 Myr after their \nbirth in our simulation. Let us suppose a dense cloud that just begins to form stars. \nSince the gas flow is usually convergent in these regions, the density of the gas will rise with time, and \nso does the SFR. This means that more and more massive stars will explode as time goes on.\nOnce enough SNe that can significantly redistribute the birth cloud go off, \nSFR will begin to drop, and \\mbox{$f_{\\rm esc}$}\\ will increase. Note that the increase in the number \nof SNe continues even after the peak of SFR, as massive stars live $\\sim$10 Myr.\nOnce the massive stars formed at the peak of SFR evolve off, star formation \nwill be further suppressed as a result of the destruction of the star-forming clouds, and strong \noutflows are likely to be produced, thus maximizing \\mbox{$f_{\\rm esc}$}.\nTherefore, the time delay stems from the interplay between the build-up of a non-coeval star cluster  \nand subsequent SN explosions after the lifetime of the massive stars ($\\sim$ 10 Myr).\nThe projected density distributions of gas at two snapshots, \none of which displays the peak in sSFR (a) and the other shows the peak in \\mbox{$f_{\\rm esc}$}\\ (b), \nsubstantiates that it is indeed the strong outflow that elevates \\mbox{$f_{\\rm esc}$}\\ (middle row).\nWhen sSFR is at the peak value, the central galaxy appears relatively quiet (panel-(a)), whereas  \nstrong outflows are seen when \\mbox{$f_{\\rm esc}$}\\ is highest and sSFR drops rapidly (panel-(b)).\nAs one can read from the figure, this mis-match of SFR and \\mbox{$f_{\\rm esc}$}\\ means that \na large amount of ionizing photons at the peak of SF are absorbed by their birth clouds.\nAlthough \\mbox{$f_{\\rm esc}$}\\ is high in the early time ($t_{\\rm H} \\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 0.5\\,{\\rm Gyr}$), the photon number-weighted \nmean \\mbox{$f_{\\rm esc}$}\\ (dashed lines) stays at around $10\\%$ level in these two examples. \n\n\n\\begin{figure*}\n   \\centering\n            \\includegraphics[width=8.6cm]{fig6_1.eps}\n                        \\includegraphics[width=8.6cm]{fig6_2.eps}\n   \\caption{\n   {\\it Left:} Photon production rate-weighted escape fraction, $\\mbox{$\\left<f_{\\rm esc}\\right>$}$, \n   averaged over the age of the universe ($t_{\\rm H}$) in the $\\textsf{FR}$\\ run. \n   The effective escape fraction in different halo mass bins is shown as different color codings, as indicated in the legend.\n   We also display the photon rate-averaged escape fraction of the whole sample at each \n   snapshot ($\\mbox{$\\left<f_{\\rm esc}\\right>$}(t)$) (black dotted line), as opposed to the time-averaged quantities (solid and dashed lines). \n  We find the effective escape fraction to be $\\sim$10\\%, regardless of the halo mass and redshift. \n   Altogether, 11.4\\% of the photons produced until $z=7$ have escaped from halos of $\\mbox{${M}_{\\rm vir}$}\\ge10^8\\,\\mbox{${M}_\\odot$}$.\n   {\\it Right:} Relative contribution of halos of different mass ranges to the \n   total number of ionizing photons measured at the virial radius. The contribution is computed by taking into \n   account the cumulative number of photons produced and the cumulative number of photons escaped from \n   halos of relevant mass range until $t\\le t_{\\rm H}$.  \n   \n   \\label{fig:fesc_wei}\n\\end{figure*}\n\nWe present  statistical results of the escape fraction in Figure~\\ref{fig:fesc_stat}.\nSince there are a limited number of galaxies in our simulated volume and \\mbox{$f_{\\rm esc}$}\\ varies \nsignificantly on $\\sim$10 Myrs, we compute the median and interquartile range of \\mbox{$f_{\\rm esc}$}\\ by combining \nthe results from seven consecutive snapshots spanning 21 Myrs.  Several features can be gleaned from this figure.\nFirst, although there is a considerable scatter, high-$z$ galaxies exhibit a high \\mbox{$f_{\\rm esc}$}\\  on the order of 10\\%,\nwhich is normally required by semi-analytic calculations of reionization to ionize the universe \nby $z\\sim6$ \\citep{wyithe07,shull12,robertson13}.\nSecond, there is a hint that photons can escape more easily in the galaxies hosted by lower mass halos.\nWe attribute this to the fact that feedback from stars efficiently destroys a few star-forming clouds that are \nresponsible for the total SF in smaller halos, as opposed to larger ones in which young massive stars are \nburied in many star-forming clouds that are relatively resilient to the SN feedback arising \nfrom neighboring star clusters.\nAs shown in the top and bottom panels of Figure~\\ref{fig:ex},\nwhen galaxies are small, the entire star formation can be suppressed due to the energetic outflows driven by \nSN explosions.\nThird, we find that \\mbox{$f_{\\rm esc}$}\\ is slightly higher at lower redshift for a given halo mass, consistent with \\citet{Paardekooper13}.\nThis is essentially because the mean density of the gas is smaller at lower redshift, and the impact from SNe becomes \nmore effective.\n\n\n\nNote that high \\mbox{$f_{\\rm esc}$}\\ does not necessarily mean that more photons would leave their host halo. \nStar clusters older than $\\sim$ 5 Myr would not contribute \nsignificantly to the total ionizing photon budget even if their \\mbox{$f_{\\rm esc}$}\\ is 1. The more relevant quantity for  \nreionization should take into account the photon production rate, and we find that the (weak) redshift \ndependence of \\mbox{$f_{\\rm esc}$}\\ disappears when the photon escape rate is plotted (right panel in Figure~\\ref{fig:fesc_stat}).\nSince the instantaneous measurement of \\mbox{$f_{\\rm esc}$}\\ could be misleading,\nwe also present the photon production rate-weighted, time-averaged escape fraction, \n$\\mbox{$\\left<f_{\\rm esc}\\right>$} (\\le t_{\\rm H}) \\equiv  \\int_0^{t_{\\rm H}} \\dot{N}_{\\rm ion}(t) f_{\\rm esc}(t) dt \/  \\int_0^{t_{\\rm H}}  \\dot{N}_{\\rm ion}(t) dt,$\nin Figure~\\ref{fig:fesc_wei} (left panel). \nThis is a better quantity to be used for the semi-analytic calculations \nof reionization than \\mbox{$f_{\\rm esc}$}\\ from Figure~\\ref{fig:fesc_stat}.\nOverall, we find that the time-averaged escape fraction at $z=7$ is around $\\sim$ 10\\%, \nregardless of the halo mass in the range considered.\nAlso included as the black dotted line in Figure~\\ref{fig:fesc_wei} is the photon production rate-weighted average of \\mbox{$f_{\\rm esc}$}\\ \nof all the samples at different times ($\\mbox{$\\left<f_{\\rm esc}\\right>$}(t)$). Again, the value is found to fluctuate around 10\\%, \nbut no clear sign of redshift dependence is detected.   \n\nThe relative contributions from halos of different masses to the total escaping ionizing photons are \ncompared in Figure~\\ref{fig:fesc_wei} (right panel).\nAs the small structures form first in the $\\Lambda$CDM universe, the small halos of mass \n$\\mbox{${M}_{\\rm vir}$} \\le 10^{8.5}\\,\\mbox{${M}_\\odot$}$ dominate down to $z\\sim9$. \nMore massive halos and galaxies emerge later, and their cumulative contribution \nbecomes comparable with that of the smallest halos ($\\mbox{${M}_{\\rm vir}$} \\le 10^{8.5}\\,\\mbox{${M}_\\odot$}$) by $z=7$. \nIn our simulations, 14 most massive halos supply more ionizing photons than 556 smallest halos with $\\mbox{${M}_{\\rm vir}$} \\le 10^{8.5}\\,\\mbox{${M}_\\odot$}$ at $z=7$.\nThis is mainly because $f_{\\star}$ is much higher in the more massive halos than \nin the small halos, while the effective escape fraction is similar.\nThe typical number of escaping photons per second in halos with $\\mbox{${M}_{\\rm vir}$}\\sim10^{8.5}\\,\\mbox{${M}_\\odot$}$ is \n$f_{\\rm esc}\\,\\dot{N}_{\\rm ion}\\sim10^{49}\\,{\\rm s^{-1}}$, whereas the number can increase up to \n$f_{\\rm esc}\\,\\dot{N}_{\\rm ion}\\sim10^{52}\\,{\\rm s^{-1}}$ in the most massive halos ($\\mbox{${M}_{\\rm vir}$} > 10^{10}\\,\\mbox{${M}_\\odot$}$)\n(Figure~\\ref{fig:fesc_stat}, right panel). \nNotice, however, that this does not necessarily translate to their relative role to the reionization of the universe.\nSmall halos at high redshift may make a more significant contribution\nto the Thompson optical depth \\citep{wyithe07,shull12,kuhlen12,robertson13}.\n\n\nIt is noted that the recombination timescale corresponding to the mean \ndensity of the universe at $z\\sim10$ ($n_{\\rm H}\\sim10^{-3}\\,{\\rm cm^{-3}}$) is relatively long \n($\\sim$ 50--100 Myr)\\footnote{Given that gas accretion is mostly filamentary \n\\citep[e.g.][]{ocvirk08,dekel09,kimm11,stewart11a}, the actual density of the gas that occupies \nmost of the volume in the halo is likely to be even lower than the mean density of the universe, \nand the recombination timescale could be longer.}, and thus the halo gas around a galaxy \nmay be kept partially ionized even though it is irradiated by the galaxy intermittently.\nFigure~\\ref{fig:ex} (the second panel in the middle row) indeed shows that a large fraction of the \nIGM in the vicinity of the central galaxy is largely ionized despite the fact that instantaneous \\mbox{$f_{\\rm esc}$}\\ is low. \nAlthough we do not include the whole distribution \nof the ionized hydrogen inside the halo, we confirm that the halo gas between 2 kpc and 12 kpc (virial radius) \nis fully ionized apart from the small region taken by cold filamentary gas.\nIn fact, the volume filling fraction of the neutral hydrogen ($f_{\\rm v}$) \ninside $0.2\\,\\mbox{${R}_{\\rm vir}$}$ ($\\sim$2.3 kpc) is found to be $\\sim$ 25\\% larger in the snapshot (b) ($f_{\\rm v}\\approx0.04$) \nthan that in the snapshot (a), suggesting that dense star-forming gas plays a more important role \nin determining the escape fraction than volume-filling diffuse neutral gas. \n\n\\begin{figure}\n   \\centering\n            \\includegraphics[width=8cm]{fig7.eps}\n   \\caption{ Effective optical depth in the Lyman continuum ($\\tau_{\\rm eff}$) by the gas in the vicinity of each star \n   ($<$100 pc) in galaxies with a low escape fraction ($f_{\\rm esc} < 0.1$) at $z\\sim8$ from the $\\textsf{FR}$\\ run. We cast 768 rays \n   uniformly distributed across the sky for individual star particles and combine the absorption of Lyman continuum \n   by neutral hydrogen at the distance of 100 pc from each star to obtain the effective optical depth. Different color codings \n   display the distribution in different halo mass bins, as indicated in the legend.  The dashed lines indicate the \n   photon production rate-weighted average of the effective optical depth. \n  Again, we combine the results from seven consecutive snapshots to increase the sample size.\n   We find that $\\tau_{\\rm eff,100pc}$ is generally \n   large (2 -- 4) for the galaxies with the low escape fraction, indicating that the nearby gas alone could reduce the \n   number of ionizing photons by 7 -- 45. This demonstrates that the ISM should be properly \n   resolved to better understand the escape of ionizing photons.\n    }\n   \\label{fig:tau}\n\\end{figure}\n\n\nFigure~\\ref{fig:tau} demonstrates the importance of resolving the ISM in predicting the escape \nof ionizing photons. In order to estimate the optical depth by neutral hydrogen in the vicinity of \neach star particle ($<$ 100 pc), we spawn 768 rays per particle using the {\\sc Healpix} algorithm \\citep{gorski05}.\nEach ray carries the spectral energy distribution determined by the age and mass of the star particle \\citep{leitherer99}. \nAs the ray propagates, we compute the absorption of the Lyman continuum by neutral hydrogen as, \n$F_{\\rm abs} (\\nu)  = F_{\\rm int} (\\nu) \\exp{\\left[-\\tau_{\\rm HI} (\\nu)\\right]}$,\nwhere  $\\tau_{\\rm HI}$ ($=N_{\\rm HI} \\sigma_{\\rm HI}$) is the optical depth and $\\sigma_{\\rm HI}$ is the hydrogen ionization\ncross section \\citep{osterbrock06}\nWe then combine the attenuated spectral energy distributions propagated out to 100 pc from each star particle, \nand measure the remaining number of ionizing photons ($N_{\\rm ion,tot}^{\\rm final}$) per galaxy. \nThis is compared with the initial number of ionizing photons ($N_{\\rm ion,tot}^{\\rm int}$) to obtain the effective \noptical depth as $\\tau_{\\rm eff, 100pc} \\equiv \\ln \\left(N_{\\rm ion,tot}^{\\rm int} \/ N_{\\rm ion,tot}^{\\rm final} \\right)$.\nFigure~\\ref{fig:tau} shows the distribution of the effective optical depth by the nearby gas for the galaxies with a \nlow escape fraction ($\\mbox{$f_{\\rm esc}$} < 0.1$) at $z\\sim8$. We find that $\\tau_{\\rm eff,100pc}$ shows a wide distribution ranging from \n0.01 to $\\sim$ 100, with the photon production rate-weighted averages of $\\tau_{\\rm eff,100pc}=$ 3.8 and 1.9 for less \n($10^8 < \\mbox{${M}_{\\rm vir}$} \\le 10^9\\,\\mbox{${M}_\\odot$}$) and more massive ($10^9 < \\mbox{${M}_{\\rm vir}$} \\le 10^{10.5}\\,\\mbox{${M}_\\odot$}$) halo groups, respectively. \nThis indicates that the number of escaping photons is reduced by a factor of $7-45$ due to the gas near young stars \nin galaxies with the small \\mbox{$f_{\\rm esc}$}.  In this regard, one may find it reconcilable that \nresults from cosmological simulations with limited resolutions \\citep[e.g.,][]{fujita03,razoumov10,yajima11} \noften give discrepant results.\n\nTo summarize, we find that there is a time delay between the peak of star formation activity and the escape fraction\ndue to the delay in the onset of effective feedback processes that can blow birth clouds away. \nBecause of the delay, only 11.4 \\% of the ionizing photons could escape from their host halos \nwhen photon production rate-averaged over all halos at different redshifts, despite the fact that \nthe instantaneous \\mbox{$f_{\\rm esc}$}\\ could reach a very high value temporarily. Halos of different masses \n($8\\le \\log \\mbox{${M}_{\\rm vir}$}\\le10.5$) contribute comparably per logarithmic mass interval to reionization, and \na photon production rate-averaged escape fraction ($\\mbox{$\\left<f_{\\rm esc}\\right>$}(t)$) shows a weak dependence on redshift \nin the range examined \\citep[c.f.,][]{kuhlen12}.\n\n\n\n\n\\begin{figure}\n   \\centering\n   \\includegraphics[width=8.6cm]{fig8.eps}\n   \\caption{Difference in environment where runaway and non-runaway stars younger than 5 Myr are located.\n   Approximately $2\\times10^5$ stars from the most massive galaxy at $z=7$ are used to plot the histograms. \n   It can be seen that runaway stars tend to be located in less dense regions than non-runaway stars.\n }\n   \\label{fig:nH_runaway}\n\\end{figure}\n\n\\begin{figure}\n   \\centering\n   \\includegraphics[width=7.5cm]{fig9.eps}\n   \\caption{Comparison of the temperature distribution in the run without (top, $\\textsf{FR}$) and with \n   runaway OB stars (bottom, $\\textsf{FRU}$) at $z=10.2$. The white bar measures 100 kpc (proper).\n   The $\\textsf{FRU}$\\ run shows bigger hot bubbles (30\\%) with $T\\ge10^5\\,K$ than the $\\textsf{FR}$\\ run,\n   suggesting that runway OB stars affect the regulation of star formation.\n }\n   \\label{fig:tem}\n\\end{figure}\n\n\n\n\\subsection{Escape Fraction Enhanced by Runaway OB Stars}\n\n\\begin{figure*}\n   \\centering\n   \\includegraphics[width=8.1cm]{fig10_1.eps}\n   \\includegraphics[width=8.5cm]{fig10_2.eps}\n   \\caption{Impact of the inclusion of runaway OB stars on the escape fraction. {\\it Left:} Instantaneous escape fraction measured \n   at the virial radius. Different color codings display different redshifts, as indicated in the legend.\n   The median \\mbox{$f_{\\rm esc}$}\\ from the $\\textsf{FRU}$\\ run (with runaway OB stars) and the $\\textsf{FR}$\\ run are shown as solid and dotted lines, respectively. \n   The shaded regions mark the interquartile range of \\mbox{$f_{\\rm esc}$}\\ from the $\\textsf{FRU}$\\ run. It can be seen that runaway OB stars tend to \n   increase the escape probability of ionizing photons. \n {\\it Right:}  Photon production rate-weighted escape fraction, $\\mbox{$\\left<f_{\\rm esc}\\right>$}$, \n averaged over the \n age of the universe ($t_{\\rm H}$). The black lines include the whole sample of the simulation, \n while the results in different halo mass bins are presented as dashed lines with different colors. \n The solid and dashed lines show the time-averaged $\\mbox{$\\left<f_{\\rm esc}\\right>$}$, while the dotted line \n shows a measurement of $\\mbox{$\\left<f_{\\rm esc}\\right>$}$ for all halos at each snapshot. \n The time-averaged escape fraction of $\\mbox{$\\left<f_{\\rm esc}\\right>$}$ measured at $z=7$ \n is 13.8\\% in this simulation. We find that the inclusion of runaway OB stars \n increases the escape of ionizing photons by 22\\% by $z=7$, compared with that from the $\\textsf{FR}$\\ run.\n }\n   \\label{fig:fesc_runaway}\n\\end{figure*}\n\nIonizing photons can not only escape from their birth clouds by destroying them through feedback processes,\nbut also emerge from runaway OB stars displaced from the birth clouds.\nIf we take the typical velocity of the runaway OB stars  \n$\\sim\\,40\\,{\\rm km\\,s^{-1}}$ \\citep{stone91,hoogerwerf01,tetzlaff11}, \nthey could travel a distance of $\\sim$ 200 pc in 5 Myr. \n\\citet{conroy12} examined the possible ramification of the inclusion of the runaway OB stars \nusing a simple analytic formulation, and concluded  that \\mbox{$f_{\\rm esc}$}\\ can be enhanced by a factor of \nup to 4.5 from $\\mbox{$f_{\\rm esc}$}\\approx0.02-0.04$ to $\\mbox{$f_{\\rm esc}$}\\approx0.06-0.18$ \nin halos of mass $10^8 \\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} \\mbox{${M}_{\\rm vir}$} \\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 10^{9}\\,\\mbox{${M}_\\odot$}$.\nGiven the complexity of the ISM dynamics \\citep[e.g.][]{mckee07}, \nit would seem prudent to examine this issue in greater details in realistic environments.\nTo do so, we have performed a twin cosmological simulation of \nthe $\\textsf{FR}$\\ run by designating 30\\% of mass in each stellar particle as a separate runaway particle \nand dynamically follow their motion.\n\n\nFigure~\\ref{fig:nH_runaway} shows an example of the difference in environment \nbetween runaway and non-runaway particles in a galaxy in a $3\\times10^{10}\\,\\mbox{${M}_\\odot$}$ halo at $z=7$. \nAt this redshift, the central galaxy shows $\\mbox{$f_{\\rm esc}$}=0.14$.\nThe average hydrogen number density for runaways younger than 5 Myr ($n_{\\rm H}\\sim130\\,{\\rm cm^{-3}}$) is found to be \nroughly 20 times smaller than that of non-runaways ($n_{\\rm H}\\sim3000\\,{\\rm cm^{-3}}$).\nGiven that these stars will explode in the next 5--10 Myrs, the fact that the local density of some runaway OB stars \nis smaller than non-runaways suggests that the impact from SN explosions will be enhanced.\nIndeed, we find that the stellar mass of the galaxies in halos of mass $\\mbox{${M}_{\\rm vir}$}\\gtrsim10^9\\,\\mbox{${M}_\\odot$}$ is smaller by a factor \nof 1.7 on average, compared with that from the $\\textsf{FR}$\\ run (see Figure~\\ref{fig:mstar}). \nFor galaxies in smaller halos, there is no clear hint that the runaway OB stars help suppress the star formation.\nThis is partly because runaway OB stars can not only provide energy but also distribute metals more efficiently, \nwhich can increase the cooling rate in halos. Comparison of the temperature distribution between the \ntwo runs further substantiates the claim that runaway OB stars help regulate the star formation (Figure~\\ref{fig:tem}). \nThe volume of $T\\ge10^{5}\\,{\\rm K}$ gas inside the zoomed-in region in the $\\textsf{FRU}$\\ run ($\\approx$ 7 kpc$^3$, physical)\nis 30\\% larger than that in the $\\textsf{FR}$\\ run.\n\nThe left panel in Figure~\\ref{fig:fesc_runaway} shows the instantaneous \\mbox{$f_{\\rm esc}$}\\ measured \nat three different redshifts from the $\\textsf{FRU}$\\ run. Again, less massive galaxies tend to exhibit a \nhigher \\mbox{$f_{\\rm esc}$}, which can be attributed to the fact that star formation in smaller halos is more easily affected \nby the energetic explosions. As expected, the inclusion of the runaway OB stars \nincreases the instantaneous escape fraction on average. The photon production rate-weighted average \nof \\mbox{$f_{\\rm esc}$}\\ (right panel in Figure~\\ref{fig:fesc_runaway}) shows this more clearly. In our fiducial run ($\\textsf{FR}$), 11.4\\% of \nthe ionizing photons produced escaped from the halos of mass $\\mbox{${M}_{\\rm vir}$}\\ge10^8\\,\\mbox{${M}_\\odot$}$ at $z\\ge7$. \nOn the other hand, the $\\textsf{FRU}$\\ run yields higher $\\mbox{$\\left<f_{\\rm esc}\\right>$}$ of 13.8\\%, \nwhich is enhanced by 22\\% compared with that of the $\\textsf{FR}$\\ run.\nAlthough this increase is not as large as claimed in \\citet{conroy12}, \nthe contribution from the runaway OB stars is certainly significant. \nSimilarly as in the $\\textsf{FR}$\\ run, no clear dependence of $\\mbox{$\\left<f_{\\rm esc}\\right>$}$ on halo mass is found.\n\nIt is interesting to discuss possible origins of the significantly different enhancement in the escape fraction\ndue to runaway OB stars found in our simulations compared with the estimate by \\citet{conroy12}. \nFirst, while their model predicts \\mbox{$f_{\\rm esc}$}\\ of non-runaways to be about 2--4\\% in halos of mass \n$10^8 \\le \\mbox{${M}_{\\rm vir}$} \\le 10^9 \\, \\mbox{${M}_\\odot$}$, we find that the self-regulation of star formation via SN explosions \nleads to a high escape of $\\sim$ 10\\% in our fiducial model ($\\textsf{FR}$). \nSecond, while their model finds that runaway OB stars are found to have high $\\mbox{$f_{\\rm esc}$}$ (=30--80\\%), \nour results imply that the mean escape fraction of ionizing photons from runaway OB stars\nis about $20\\%$ ($11.4\\%\\times 70\\% + {\\it 20\\%}\\times 30\\%\\approx13.8\\%$).\nWe also make a more elaborate estimate as follows.\nWe measure the optical depth in the Lyman continuum for the gas inside each halo along 768 sightlines \nper star particle, and combine the attenuated spectral energy distributions. These are used to count \nthe number of hydrogen ionizing photons for runaways and non-runaways separately. \nWe find that the relative contribution from the runaways to the total number of escaping photons is\ncomparable with that of the non-runaways. Considering that the runaway particle is assumed to explain only 30\\% of \nall the OB stars, the net $\\mbox{$f_{\\rm esc}$}$ for the runaways can be estimated to be roughly 23\\% ($=13.8\\%\/2\/0.3$). \nThis is twice higher chance of escaping than the non-runaways, but much smaller than computed in \nthe analytic model. If the escape fraction of non-runaway OB stars were 2\\% in our simulations, \nthe total escape fraction would become $2\\%\\times 70\\% + 23\\%\\times 30\\%=8.3\\%$, \ncorresponding to an increase of a factor of 4.2.\nIt is thus clear that most of the discrepancies arise in a large part due to different escape fraction values \nfor non-runaway OB stars and also due to different escape fraction values for runaway OB stars.\n\n\n\n\\begin{figure}\n   \\centering\n            \\includegraphics[width=8.6cm]{fig11.eps}\n   \\caption{Balance between the ionizing photons escaping from the dark matter halo and the recombination rate \n   in the $\\textsf{FRU}$\\ run.\n   The thick grey line shows the balance condition when the clumping of $C_{\\rm HII}=3$ is used. \n   Enough photons to keep the universe ionized  escape from the halo after $z\\sim8$.\n   }\n   \\label{fig:budget}\n\\end{figure}\n\nAlthough $\\mbox{$\\left<f_{\\rm esc}\\right>$}$ is 22\\% larger in the $\\textsf{FRU}$\\ run than $\\textsf{FR}$, the cumulative number of photons escaped in halos \nwith $\\mbox{${M}_{\\rm vir}$}\\ge10^8\\,\\mbox{${M}_\\odot$}$ by $z=7$ ($N_{\\rm ion}\\approx1.3\\times10^{69}$) is found to be similar to that of the $\\textsf{FR}$\\ run ($N_{\\rm ion}\\approx1.6\\times10^{69}$).\nThis is because star formation is suppressed in relatively massive halos ($\\mbox{${M}_{\\rm vir}$} \\ge 10^9\\,\\mbox{${M}_{\\rm vir}$}$).\n\n\n\n\n\\begin{figure}\n   \\centering\n            \\includegraphics[width=8.6cm]{fig12.eps}\n   \\caption{Rest-frame ultraviolet luminosity function from the $\\textsf{FRU}$\\ run at $z=7$.  Error bars denote the Poissonian error.\n   Observational data from \\citet{bouwens11a} and \\citet{mclure13} are shown as the shaded region \n   and empty squares, respectively. Also included as solid and dashed lines are the Schechter fits to the \n   data provided in these studies.\n   }\n   \\label{fig:uvlf}\n\\end{figure}\n\nOne question is whether or not enough photons escape to keep the \nuniverse at $z\\sim7$ ionized.  The critical photon rate density that can balance the recombination of ionized hydrogen is \n\\begin{equation}\n\\dot{n}_{\\rm ion}^{\\rm crit}  = \\alpha_{\\rm B} \\, n_e \\, n_{\\rm HII} \\simeq 10^{47.2} C_{\\rm HII} (1+z)^3 \\, {\\rm [s^{-1}\\,Mpc^{-3}]},\n\\end{equation}\nwhere $\\alpha_B$ is the case B recombination coefficient, \n$n_e$ is the number density of electron, $n_{\\rm HII}$ is \nthe number density of ionized hydrogen, and $C_{\\rm HII} \\equiv \\left<n_{\\rm HII}^2\\right>\/\\left<n_{\\rm HII}\\right>^2$ is the \nclumping factor of ionized gas. For a  choice of the clumping factor $C_{\\rm HII}\\sim3$ \\citep{pawlik09,raicevic11} and \nthe temperature $T=20000K$, $\\dot{n}_{\\rm ion}^{\\rm crit} = 10^{50.4}\\,[(1+z)\/8]^3  \\, {\\rm s^{-1}\\,Mpc^{-3}}$.\nFigure~\\ref{fig:budget} shows that the escaped photons in $\\textsf{FRU}$\\ can balance the recombination at $z\\le 9$. \nWe find that the photon rate density at $z\\sim7$ is $\\dot{n}_{\\rm ion}=10^{50.7-50.9}   \\, {\\rm s^{-1}\\,Mpc^{-3}}$,\nconsistent with observational findings. \\citet{ouchi09} estimated the ionizing photon density to be \n$\\log \\dot{n}_{\\rm ion} \\simeq 49.8 - 50.3$ by integrating the UV luminosity function (UVLF) down to $M_{\\rm UV}=-18$ (lower) \nor $L=0$ (upper estimate) with a slope of $\\alpha=-1.72$ at $z\\sim7$ with $\\mbox{$f_{\\rm esc}$}=20\\%$. \nIf the slope found in the more recent literature \\citep{mclure13}, $\\alpha=-1.90$, \nis used,  the maximum photon rate density derived would increase to $\\log \\dot{n}_{\\rm ion} \\simeq 50.8$,\nwhich is in agreement with our estimation. Note that the photons escaping from halos of mass \n$\\mbox{${M}_{\\rm vir}$}\\ge10^8\\,\\mbox{${M}_\\odot$}$ account for more than 90\\% of the total escaping photons \nif the baryon-to-star conversion efficiency derived in our simulation \nis extrapolated to smaller halos ($\\mbox{${M}_{\\rm vir}$}<10^{8.5}\\,\\mbox{${M}_\\odot$}$, see below), \nand hence our results should be compared with the maximum photon rate density.\nGiven that their chosen \\mbox{$\\left<f_{\\rm esc}\\right>$}\\  is closed to what our simulation yields (13.8\\%), \nthe agreement implies that SFRs of the galaxies are well reproduced in our simulation. Indeed, \nwe find that our simulated UVLF measured at 1500\\AA\\ (rest-frame)  shows excellent agreement with \nthe LF with the slope of $\\alpha=-1.90$ \\citep{mclure13} down to $M_{\\rm 1500}=-13$ (Figure~\\ref{fig:uvlf}).\nHere we neglect the effect of dust extinction, as the galaxies in our sample are very metal-poor \n($Z_{\\rm star}\\lesssim10^{-3}$).\n\n\n\\begin{table} \n\\caption{Photon number-weighted $f_{\\rm esc}$ at $7\\le z \\lower.5ex\\hbox{$\\; \\buildrel < \\over \\sim \\;$} 15$ from the FRU run}\n\\centering\n\\begin{tabular}{@{}cc}\n\\hline \n$\\log M_{\\rm vir}$ & $\\left<f_{\\rm esc}\\right>$       \\\\\n\\hline \n8.25     & 0.144 $\\pm$ 0.038\\\\\n8.75     & 0.146 $\\pm$ 0.064\\\\\n9.25     & 0.148 $\\pm$ 0.077\\\\\n9.75     & 0.128 $\\pm$ 0.069\\\\\n10.25   & 0.113 $\\pm$ 0.079\\\\\n\\hline \n\\label{table2}\n\\end{tabular} \n\\end{table}\n\nIn Figure~\\ref{fig:cstar}, we plot the product of photon number-weighted escape fraction ($\\mbox{$\\left<f_{\\rm esc}\\right>$}$) and \n baryon-to-star conversion efficiency ($f_{\\star}\\equiv \\Omega_{\\rm m} M_{\\rm star}\/\\Omega_{\\rm b} M_{\\rm vir}$) \n  at $z=7$. Notice that we include all stars within the virial radius of a dark matter halo \n  in this measurement. Since there is little evolution in $\\mbox{$\\left<f_{\\rm esc}\\right>$}$ with redshift (Figure~\\ref{fig:fesc_runaway}, \n  right panel), we combine $\\mbox{$\\left<f_{\\rm esc}\\right>$}$ of the halos in the same mass \n  range at $7\\le z < 20$ to obtain the mean escape fraction as a function of halo mass (Table~\\ref{table2}).\n  We then use a simple fit to the mean, as\n  \\begin{equation}\n  \\log \\mbox{$\\left<f_{\\rm esc}\\right>$} (\\mbox{${M}_{\\rm vir}$}) \\approx -0.510 - 0.039 \\log \\mbox{${M}_{\\rm vir}$}.\n  \\label{fescg_fit}\n  \\end{equation}\nWe limit our fit to the sample with $\\mbox{${M}_{\\rm vir}$}\\ge10^{8.5}\\,\\mbox{${M}_\\odot$}$, where each halo is \n resolved with $\\sim$ 2000 dark matter particles and more.\nThere is a trend that more massive halos contribute more to the total number of ionizing photons per mass,\nwhich essentially reflects the fact that low-mass halos are inefficient in forming stars \n(see also Figure~\\ref{fig:mstar}). The average $\\mbox{$\\left<f_{\\rm esc}\\right>$} f_{\\star}$ of different halo masses can be  \nfitted with \n\\begin{equation}\n\\log \\mbox{$\\left<f_{\\rm esc}\\right>$} f_{\\star} \\approx -7.342 + 0.474\\, \\log \\mbox{${M}_{\\rm vir}$},\n  \\label{fstar_fit}\n\\end{equation}\nshown as the red dashed line in Figure~\\ref{fig:cstar}.\nWe note that $\\mbox{$\\left<f_{\\rm esc}\\right>$} f_{\\star}$  becomes as low as $\\sim 5\\times10^{-4}$ \nin small halos ($\\mbox{${M}_{\\rm vir}$}\\sim10^{8.5}\\,\\mbox{${M}_\\odot$}$), which is roughly 40 times smaller than the results \nfrom \\citet{wise09} ($\\mbox{$\\left<f_{\\rm esc}\\right>$} f_\\star \\approx0.02$). \nThe difference can be attributed to two factors.  \nFirst, our \\mbox{$\\left<f_{\\rm esc}\\right>$}\\ is smaller by a factor of $\\sim3-4$ than that of \\citet{wise09}. \nThis is probably due to the fact that their cosmological runs start from the initial condition \nextracted from adiabatic simulations in which no prior star formation is included. \nSince radiative cooling and star formation are suddenly turned on at some redshift,\nthe gas in the halo rapidly collapses and forms too many stars in their cosmological runs.\nThis is likely to have resulted in stronger starbursts in the galaxies, leading to a higher escape probability. \nSecond, because of the same reason, $f_{\\star}$ is considerably higher in the \\citet{wise09} halos than in our halos. \nFor halos of masses with $\\mbox{${M}_{\\rm vir}$}\\sim10^{8.5}\\,\\mbox{${M}_\\odot$}$, we find that $f_{\\star}\\approx0.003$, \nwhich is smaller by a factor of $\\sim 10$ than those in \\citet{wise09}. \nIndeed, we find fairly good agreement with the latest determination of $\\mbox{$\\left<f_{\\rm esc}\\right>$} f_{\\star}$ in \nhalos of $\\mbox{${M}_{\\rm vir}$}\\sim10^{8.5}$ by \\citet{wise14},\nwho model star formation self-consistently in their cosmological radiation hydrodynamics simulations.\n\n\\begin{figure}\n   \\centering\n            \\includegraphics[width=8.6cm]{fig13.eps}\n   \\caption{Product of the stellar mass fraction within the virial radius of a dark matter halo \n   ($f_\\star=\\Omega_{\\rm m} \\mbox{${M}_{\\rm star}$} \/ \\Omega_{\\rm b} \\mbox{${M}_{\\rm vir}$}$)  at $z=7$ \n   and halo mass-dependent photon production rate-averaged escape fraction from the cosmological simulation with \n   runaway OB stars ($\\textsf{FRU}$).\n   Averages are shown as red empty squares, with the simple regression (dashed line). \n   A smaller number of photons is escaped per unit mass in smaller halos, reflecting the results that\n   star formation is inefficient in the low-mass halos. \n   }\n   \\label{fig:cstar}\n\\end{figure}\n\nIt is worth mentioning that adopting high spatial resolution (or gravitational softening length) \nis important to accurately predict the escape fraction. If the resolution is not high enough to capture \nthe rapid collapse of gas clouds, the resulting star formation histories would become less episodic, \nleading to a longer time delay between the peak of star formation and escape fraction. \nThis in turn would reduce the fraction of escaping photons.  To examine this issue, \nwe run two additional simulations with the identical initial condition and other parameters, \nbut with one less or more level of refinement, corresponding to 8.5 pc or 2.1 pc (physical) resolution, respectively.\nWe find that the run with the lower resolution yields a factor of two smaller mean escape fraction at z=9 \n($\\mbox{$\\left<f_{\\rm esc}\\right>$}=7.6\\%$, see Appendix). On the contrary, higher resolution run exhibits a comparable mean \nescape fraction of $\\mbox{$\\left<f_{\\rm esc}\\right>$}=13.9\\%$ at $z=10$, suggesting that the results are reasonably converged \nfor the parameters used in the $\\textsf{FRU}$\\ run.\n\n\n\\section{Discussion}\n\nRecent studies show that the escape fraction should be larger than 20\\% to re-ionize the \nuniverse by $z=6$ matching the Thomson optical depth inferred from the CMB \n\\citep{kuhlen12,shull12,robertson13}. This can be obtained by numerically solving the simple differential \nequation for the \\mbox{{\\sc H ii}}\\ bubble\n\\begin{equation}\n\\frac{d Q_{\\rm HII} }{dt} = \\frac{\\dot{n}_{\\rm ion}}{\\left<n_{\\rm H}\\right>} - \\frac{Q_{\\rm HII}}{t_{\\rm rec}(C_{\\rm HII})},\n\\end{equation}\nwhere $Q_{\\rm HII}$ is the volume filling fraction of the bubble, $\\left< n_{\\rm H}\\right>$ is the comoving mean density \nof the universe, and \n$t_{\\rm rec} (C_{\\rm HII})= \\left[ C_{\\rm HII}\\,\\alpha_{\\rm B}(T)\\, f_e\\, \\left<n_{\\rm H}\\right> \\,(1+z)^3\\right]^{-1}$ is the \nrecombination timescale for a given clumping factor and temperature. Here $f_e$ is a correction factor that accounts \nfor the additional contribution of singly ($z>4$) or doubly ($z<4$) ionized helium to \nthe number density of electron \\citep[e.g.,][]{kuhlen12}. \nWe adopt a redshift-dependent clumping factor of $C_{\\rm HII} = 1 + \\exp(-0.28\\, z +3.59)$ at $z\\ge10$ or\n$C_{\\rm HII} = 3.2$ at $z<10$ following \\citet{pawlik09}.\nOnce $Q_{\\rm HII}$ is determined, the Thomson optical depth \nas a function of redshift can be calculated as \n\\begin{equation}\n\\tau_e (z)= \\int_0^z c \\left<n_{\\rm H}\\right>\\,\\sigma_T\\,f_e\\,Q_{\\rm HII}(z') \\frac{(1+z')^2 dz'}{H(z')},\n\\end{equation}\nwhere $\\sigma_T$ is the Thomson electron cross section, and $H(z)$ is the Hubble parameter.\nWe follow the exercise by using the ionizing photon density from Figure~\\ref{fig:budget} to examine \nwhether our models provide a reasonable explanation for the reionization history.\nFor $\\dot{n}_{\\rm ion}$ at  $z<7$, we extrapolate based on the simple fit to the results in Figure~\\ref{fig:budget}. \nThis simple experiment indicates that the universe can be re-ionized by $z=7.25$. \nHowever, the evolution of the photon density from the $\\textsf{FRU}$\\ run predicts a smaller volume filling fraction of the \\mbox{{\\sc H ii}}\\ bubble\nat $z=10$ ($Q_{\\rm HII}=12\\%$), compared with other analytic models \\citep[$Q_{\\rm HII}\\gtrsim20\\%$, e.g.][]{shull12} \nthat could reproduce the CMB measurement \\citep[$\\tau_e\\sim0.09$,][]{komatsu11}. Consequently,  the FRU run yields \nthe Thomson optical depth of $\\tau_e=0.065$, which is consistent only within 2$\\sigma$ with the \nCMB measurement. This implies that more ionizing photons are required to escape from halos at high redshift \nto explain the reionization history of the Universe.\n\n\\begin{figure}\n   \\centering\n            \\includegraphics[width=8.6cm]{fig14.eps}\n   \\caption{ Importance of the dwarf galaxy population to the Thomson optical depth measurement \n   in semi-analytic calculations. The top and middle panels show the escape fraction and \n   stellar mass inside the virial radius of a halo as a function of halo mass, respectively, \n   which are used to compute the optical depth (the bottom panel). The measurements from \n   our radiation cosmological simulations with runaway stars ($\\textsf{FRU}$) are shown as \n   blue filled squares with the standard deviations. Empty squares with error \n   bars are the results from \\citet{wise14}.    The optical depth is obtained \n   by taking into account the escaping ionizing photons from halos more massive than $\\mbox{${M}_{\\rm vir}$}$.\n   We neglect the contribution from rare massive halos with $\\mbox{${M}_{\\rm vir}$}>10^{12}\\,\\mbox{${M}_\\odot$}$. \n   Different colors in the bottom panel corresponds to the results with different assumptions on \n   the stellar-to-halo mass relation for minihalos, as indicated in the middle panel.\n   The shaded region denotes the Thomson optical depth inferred from the Planck+WMAP\n   measurements.\n   }\n   \\label{fig:tau_es}\n\\end{figure}\n\n\nThe deficiency of ionizing photons may in part be attributed to the fact that \nour simulations cannot resolve the collapse of small-mass halos ($\\mbox{${M}_{\\rm vir}$}\\lesssim10^8\\,\\mbox{${M}_\\odot$}$)\ndue to finite mass resolution. \\citet{Paardekooper13} argue that reionization is driven by \ndwarf-sized halos of masses  $\\mbox{${M}_{\\rm vir}$}=10^7-10^8\\,\\mbox{${M}_\\odot$}$ with high $\\mbox{$\\left<f_{\\rm esc}\\right>$}$ of $\\approx$0.4--0.9.\nSimilarly, \\citet{wise14} find that the ionizing photons from the minihalos with \n$\\mbox{${M}_{\\rm vir}$}=10^{6.25}-10^{8.25}\\,\\mbox{${M}_\\odot$}$ is crucial at reproducing the Thompson optical \ndepth from the CMB measurements. In order to examine the importance of the minihalos \nin light of our new results, we estimate the optical depth as a function of the minimum halo mass\nthat can contribute to reionization. To do so, we use the theoretical halo mass functions at different \nredshifts \\citep{jenkins01},  convolved with the baryon-to-star conversion efficiency measured at $z=7$ from \nthe $\\textsf{FRU}$\\ run for $\\mbox{${M}_{\\rm vir}$}\\ge10^{7.5}\\,\\mbox{${M}_\\odot$}$ and \\citet{wise14} for $\\mbox{${M}_{\\rm vir}$}<10^{7.5}\\,\\mbox{${M}_\\odot$}$ \n(orange line in the middle panel of Figure~\\ref{fig:tau_es}), to derive the increase in the stellar mass density \nwith redshift. The number of escaping ionizing photons is then calculated by multiplying the number of \nphotons produced with the halo mass-dependent escape fraction based on our results and \\citet{wise14}, \nas (Figure~\\ref{fig:tau_es}, top panel)\n\\begin{equation}\n\\log \\mbox{$\\left<f_{\\rm esc}\\right>$}=\\left\\{\n\\begin{array}{ll}\n -0.51 - 0.039\\,\\log \\mbox{${M}_{\\rm vir}$} &  (\\log \\mbox{${M}_{\\rm vir}$} \\ge 8.5) \\\\\n 2.669 - 0.413\\,\\log \\mbox{${M}_{\\rm vir}$} &  (7 \\le \\log \\mbox{${M}_{\\rm vir}$} < 8.5) \\\\\n -0.222 &  (\\log \\mbox{${M}_{\\rm vir}$} < 7)\\\\\n\\end{array} \n\\right. .\n\\end{equation}\nWe neglect the contribution from rare massive halos with $\\mbox{${M}_{\\rm vir}$}>10^{12}\\mbox{${M}_\\odot$}$.\nFigure~\\ref{fig:tau_es} (orange line, bottom panel) shows that \nthe minihalos of $\\mbox{${M}_{\\rm vir}$}<10^7\\,\\mbox{${M}_\\odot$}$ can indeed provide enough photons \nto match $\\tau_e$ inferred from the CMB measurement. \nWhile the ionizing photons from $\\mbox{${M}_{\\rm vir}$}>10^7\\,\\mbox{${M}_\\odot$}$ only gives $\\tau_e=0.072$,\nthe additional photons arising from the minihalos augment the optical depth to 0.122.\nHowever, we note that this sensitively depends on the assumption\non the baryon-to-star conversion efficiency in the minihalos. For example, when the stellar \nmass-halo mass relation found in the $\\textsf{FRU}$\\ is extrapolated to the minihalos \n(blue line in the bottom panel), the optical depth for the entire halos is only $\\tau_e=0.073$.\nGiven that these minihalos would host a handful of star particles with $m_{\\rm star}\\sim10^2-10^3\\,\\mbox{${M}_\\odot$}$ \nin current numerical simulations, it is unclear how the mass resolution affects the conversion efficiency, \nand further investigations on star formation in the minihalos will be useful to better understand \ntheir relative role to the total ionizing budget.\n\n\n\nIn our simulation, we approximate that massive stars ($M>8\\mbox{${M}_\\odot$}$) evolve off and \nexplode after 10 Myr. We note that this is roughly the timescale \nof the delay between the peak of star formation and escape fraction.\nIn reality, the SN can emerge as early as $\\sim$ 3 Myr for a simple population \\citep{schaller92}. \nStellar winds, photo-ionization, and radiation pressure acting on electron and dust can come into play even earlier. \n\\citet{walch12} claims that a $10^4\\,\\mbox{${M}_\\odot$}$ molecular cloud of the radius 6.4 pc can be dispersed \non a 1-2 Myr timescale by the overpressure of \\mbox{{\\sc H ii}}\\ regions.\nMoreover, it is also plausible that the ionization front instabilities may lead to the higher escape probability \nof ionizing photons \\citep{whalen08b}.\nIf these mechanisms played a role in shaping the evolution of individual molecular clouds, the escape fraction \nmeasured in our simulations would have been higher than 14\\%. \nIn this regard, our photon number-weighted mean is likely to represent the minimum escape of ionizing photons. \nWhen a higher \\mbox{$\\left<f_{\\rm esc}\\right>$}\\ of 30\\% is assumed for the star formation history in the $\\textsf{FRU}$\\ run, \ndark matter halos of $\\mbox{${M}_{\\rm vir}$}>10^8\\,\\mbox{${M}_\\odot$}$ alone can achieve $\\tau_e=0.076$,\nsuggesting that a more precise determination of the escape fraction is as equally important \nas resolving ultra-faint galaxies with $M_{\\rm 1500} > -13$. \nFuture studies focusing on the interplay between the feedback processes will shed more light \non the reionization history of the Universe.\n\n\n\n\n\n\n\\section{Conclusions}\n\nThe escape fraction of hydrogen ionizing photons is a critical ingredient in the theory of reionization. \nDespite its importance, only a handful of studies examined the escape fraction ($\\mbox{$f_{\\rm esc}$}$)\nof high-$z$ galaxies in a cosmological context \\citep{wise09,razoumov10,yajima11,Paardekooper13,wise14}.\nTo better understand the physics behind the escape of ionizing photons and quantify \\mbox{$f_{\\rm esc}$}, \nwe have carried out two zoomed-in cosmological radiation hydrodynamics simulations of \n$3.8\\times4.8\\times9.6$ Mpc$^3$ box (comoving) with \nthe \\mbox{{\\sc \\small Ramses}}\\ code  \\citep{teyssier02,rosdahl13} with high spatial ($\\sim$ 4 pc, physical) and \nstellar mass resolution of 49 $\\mbox{${M}_\\odot$}$.\nBecause energy-based feedback from SN explosions suffers from the artificial\nradiative cooling if the cooling length is under-resolved, we have implemented a new \nmechanical feedback scheme that can approximate all stages of a SN explosion \nfrom the free expansion to snowplow phase. \nWith the physically based feedback model, \nwe have investigated the connection between the regulation of star formation and \ncorresponding evolution of the escape of ionizing photons.  \nWe have also explored the relative importance of runaway OB stars to the escape fraction \nby comparing the twin simulations with ($\\textsf{FRU}$) and without ($\\textsf{FR}$) runaways.\nOur findings can be summarized as follows.\n\n\\begin{enumerate}\n\n\\item When a dense cloud begins to form a cluster of stars, the escape fraction is negligible. \nAs energetic explosions by massive stars follow after $\\sim$ 10 Myr, it blows the star forming gas away,\nincreasing the {\\it instantaneous} escape fraction (\\mbox{$f_{\\rm esc}$}) to \\gtrsim10\\%. Although \\mbox{$f_{\\rm esc}$}\\ is kept high in this phase,\nsubsequent star formation is markedly suppressed, \nand only a small number of photons escapes from their host dark matter halo (Figure~\\ref{fig:ex}). \nThis time delay between the peak of star formation and the escape fraction is crucial in predicting \nthe actual escape probability of ionizing photons. While the instantaneous \\mbox{$f_{\\rm esc}$}\\ can easily \nattain $\\gtrsim30\\%$ in halos of mass $\\mbox{${M}_{\\rm vir}$} \\ge 10^8\\,\\mbox{${M}_\\odot$}$ on average (Figure~\\ref{fig:fesc_stat}), \nthe photon number-weighted mean of the escape fraction (\\mbox{$\\left<f_{\\rm esc}\\right>$}) is found to be 11.4\\% (Figure~\\ref{fig:fesc_wei}).\n\n\\item  \\mbox{$f_{\\rm esc}$}\\ tends to be higher in less massive halos and at lower redshift for a give halo mass (Figure~\\ref{fig:fesc_stat}).\nThis is essentially because less dense and smaller galaxies are more susceptible to SN explosions.\nHowever, the photon production rate-averaged escape fractions show no clear dependence \non halo mass and redshift, again implying that the interplay between star formation and the delay in the onset of \nnegative feedback is more important in determining the actual escape probability. \n\n\\item Absorption of ionizing photons by neutral hydrogen in the ISM is significant (Figure~\\ref{fig:tau}). For galaxies \nwith a low escape fraction ($\\mbox{$f_{\\rm esc}$}<10\\%$), the effective optical depth by the gas within 100 pc \nfrom each young star particles is found to be $\\tau_{\\rm eff,100pc}\\sim 1.9-3.8$ at $z\\sim8$.\nThe nearby neutral gas alone can reduce the number of ionizing photons by 7--45 in this case, \ndemonstrating the importance of properly resolving the ISM to predict a more accurate escape fraction.\n\n\\item Our physically based SN feedback effectively regulates star formation. \nOnly 0.1\\% to 10\\% of the baryons are converted into stars in galaxies at $z=7$ (Figure~\\ref{fig:mstar}).\nThe energetic explosions sometimes completely shut down star formation when galaxies are small.\nThe baryon-to-star conversion ratio is smaller in less massive halos. \nConsequently, halos of different masses contribute comparably to the total number \nof ionizing photons escaped by $z=7$ (Figure~\\ref{fig:fesc_wei}).\n\n\n\\item Inclusion of runaway OB stars increases the escape fraction to $\\mbox{$\\left<f_{\\rm esc}\\right>$}=13.8\\%$ from 11.4\\% \n(Figure~\\ref{fig:fesc_runaway}). Since the runaway OB stars tend to move to lower density regions, \nphotons from them have a higher chance of escaping. Moreover, as the runaway OB stars explode in a less \ndense medium, feedback from SNe becomes more effective, resulting in reduced star formation in halos $\\mbox{${M}_{\\rm vir}$} \\ge 10^9\\,\\mbox{${M}_\\odot$}$, \ncompared with the $\\textsf{FR}$\\ run. Because of the balance between the increase in \\mbox{$\\left<f_{\\rm esc}\\right>$}\\ and the decrease \nin star formation, the total number of ionizing photons escaped by $z=7$ is found to be comparable in\nthe two runs.\n\n\\item \nA sufficient amount of photons escape from the dark \nmatter halos with $\\mbox{${M}_{\\rm vir}$}\\ge10^8\\,\\mbox{${M}_\\odot$}$ to keep the universe ionized at $z\\le9$. \nThe simulated UV luminosity function with a faint end slope of -1.9 is consistent with observations.\n\\end{enumerate}\n\n\n\n\n\n\n\n\\acknowledgements{\nWe thank an anonymous referee for constructive suggestions that improved this paper.\nWe are grateful to Julien Devriendt, Sam Geen, Chang-Goo Kim, Eve Ostriker,  Adrianne Slyz,\nand John Wise for insightful discussions.\nSpecial thanks go to Romain Teyssier and Joakim Rosdahl for sharing their radiation \nhydrodynamics code with us. Computing resources were provided in part by the NASA High-\nEnd Computing (HEC) Program through the NASA Advanced\nSupercomputing (NAS) Division at Ames Research Center and in part by \nHorizon-UK program through DiRAC-2 facilities.  \nThe research is supported by NSF grant AST-1108700 \nand NASA grant NNX12AF91G.\n}\n\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgement}\nWe would like to thank J.Y. Kim for usefull discussions.\nThis work was supported in part by the Basic Science Research Institute \nProgram, Minstry of Education, Project NOs. BSRI-98-2441 and \nBSRI-98-2413. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}