diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzkku" "b/data_all_eng_slimpj/shuffled/split2/finalzkku" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzkku" @@ -0,0 +1,5 @@ +{"text":"\\section{The Mu2e Experiment}\n\\label{sec:intro}\n\nThe Mu2e experiment \\cite{TDR} at Fermilab will search for the\ncharged-lepton flavor violating neutrino-less coherent conversion\nof a negatively charged muon into an electron in the field of an\naluminum nucleus. The process produces a mono-energetic electron\nwith an energy slightly below the muon rest mass (104.967 MeV). If\nno events are observed, Mu2e will set a\nlimit on the ratio between the conversion rate and the muon capture\nrate of $R_{\\mu e}$~$\\leq 8.4\\ \\times\\ 10^{-17}$ (@ 90$\\%$ C.L.).\nThis will improve the current limit \\cite{Sindrum-II} by four\norders of magnitude.\nOn the other hand, an observation of Charge Lepton Flavour Violation\n(CLFV) events will provide a clear indication of New Physics (NP)\nbeyond the Standard Model up to mass scales of nearly $10^4$ TeV,\nfar beyond the direct search reach at colliders, complementing and\nextending other CLFV searches on a wide range of NP scenarios\n\\cite{CLFV-theory}.\n\nThe Mu2e design is based on the MELC concept \\cite{MELC}. An\nintense pulsed muon beam ($\\sim 10^{10} \\mu\/$sec) is produced by\n8 GeV, 8 kW protons hitting a tungsten target and it is stopped\non an aluminum target after travelling inside a very long, curved\nseries of solenoids (Fig.~\\ref{Fig:Mu2e}).\nThe strong negative gradient of the Production Solenoid, from 4 to\n2.5 T, confines soft pions and increases the yield through magnetic\nreflection. The S-shaped Transport Solenoid efficiently transfers\nlow energy, negatively charged particles while allowing a large\nfraction of pions to decay into muons.\nThe Detector Solenoid has a graded field from 2 to 1 Tesla in\nthe upstream region of the stopping target to increase acceptance\nfor Conversion Electron (CE) events. \n\n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width=1.0\\textwidth]{mu2e_layout}\n \\end{center}\n \\caption{The Mu2e experiment. Cosmic Ray Veto and Stopping\n Target Monitor are not shown.}\n \\label{Fig:Mu2e}\n\\end{figure}\n\nThe Mu2e detector, just downstream of the aluminum target inside\na 1T solenoid, is composed of a tracker and an electromagnetic\ncalorimeter. The Mu2e tracker measures the momentum of the conversion\nelectron and separates it from the background. The crystal\ncalorimeter plays an important role in providing particle\nidentification capabilities and a fast online trigger filter, while\nalso aiding the track reconstruction capabilities. The detector\nsolenoid is in vacuum, at $10^{-4}$ Torr, and in a high radiation\nenvironment. The entire detector region and part of the transport\nsolenoid are surrounded by a Cosmic Ray Veto (CRV) that reduces the\ncosmic ray background.\nA High Purity Germanium Detector and a Lanthanum Bromide crystal\nconstitute the Stopping Target Monitor, placed $\\sim 35$ m after\nthe stopping target, which provides normalization to CLFV events by\ndetecting $\\gamma$-rays emitted from muon capture in the aluminum\ntarget.\n\nIn order to reach the required sensitivity, control of the background\nto the level of less than 0.5 expected events is required.\nThe background coming from the\nbeam is reduced by means of a pulsed beam structure with a proton\nextinction lower than 10$^{-10}$: a delay in the start of the live\nwindow of $\\sim$ 700 ns after the bunch arrival time removes the\nprompt background from the acquired data.\nThe extinction level is monitored by detecting scattered protons from\nthe production target to evaluate the fraction of out-of-time beam.\n\n\\section{The tracking system}\n\\label{sec:tracker}\n\nThe Mu2e tracker system \\cite{Tracker} is designed to maximize\nacceptance for conversion electrons while minimizing the contamination\nfrom the muon Decay-In-Orbit (DIO) background, where nuclear\nmodifications push the DIO spectrum towards the CE signal\n(Fig.~\\ref{Fig:tracker} left). Energy loss and detector resolution\nproduce an overlap of the two processes. The selected design is based\non nearly 20,000 low mass straw drift tubes of 5 mm in diameter,\nwith a 15 $\\mu$m Mylar wall and 25 $\\mu$m sense wire.\nStraws of lengths ranging from 430 to 1220 mm are oriented transversely\nto the solenoid axis and arranged in 18 stations (Fig.~\\ref{Fig:tracker}\nright), for a total length of 3.2 metres along the solenoid axis. A\ncentral hole, 38 cm in diameter, makes the device blind to low momentum\nbackground particles ($p<55$ MeV\/c) which are constrained to low radius\nby the solenoidal field.\n\n\\begin{figure}[!th]\n \\centering\n \\begin{tabular}{cc}\n \\includegraphics[width=0.5\\textwidth]{dio_nores} &\n \\includegraphics[width=0.5\\textwidth]{trk-station.png} \\\\\n \\end{tabular}\n \\caption{Left: energy spectrum for electrons produced from\n free muon decays (blue), muon decays in orbit (red) and\n conversion electrons (purple). Right: Sketch of the Mu2e\n straw tracker system. The basic element is the panel, where\n straws are organized in two staggered layers. Six panels\n arranged as shown in in the middle figure above form a plane;\n two planes rotated by $30^\\circ$ constitute a station, right.\n The tracker, containing 18 stations, is 3.2 meters long.}\n \\label{Fig:tracker}\n\\end{figure}\n\nAn eight channel tracker prototype was built and tested with\ncosmics rays to measure performances and tune detector simulations.\nIn Fig.~\\ref{Fig:trk-proto}, the position resolution and straw\nefficiency are compared with Monte Carlo expectations.\nGood reproducibility of data is observed.\n\n\\begin{figure}[!th]\n \\vspace{0.3cm}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{prototypedata3.png}\n \\caption{Longitudinal (left), transverse (center) position\n resolution and efficiency (right) for an eight channel\n prototype of the tracker.\n Data from minimum ionizing particles (blue triangles) are\n compared with Monte Carlo simulation (red crosses). \n Resolution is extracted with Gaussian fits to the spectra.}\n \\label{Fig:trk-proto}\n\\end{figure}\n\nThe tracker performance is studied with Monte Carlo using the\nfull Mu2e simulation. Results are reported in Fig.~\\ref{Fig:trk-reso}.\nThe core momentum resolution of 159 keV\/c is well within physics\nrequirements and stable when increasing accidental hit rate.\nThe total track efficiency of $\\sim 9\\%$ is fully dominated by\ngeometric acceptance.\n\n\\begin{figure}[!th]\n \\vspace{0.3cm}\n \\centering\n \\includegraphics[width=0.5\\textwidth,height=6cm]{trk-reso}\n \\caption{Momentum resolution evaluated with the fully tuned\n Mu2e simulation.}\n \\label{Fig:trk-reso}\n\\end{figure}\n\nAt the moment of writing, twelve pre-production panels are under\nconstruction and testing. In Fig.~\\ref{Fig:trk-plane}, three\npanels are assembled to form a tracking plane.\nA vertical slice test on fully instrumented panels with the entire\nFront-End Electronics chain will be performed.\n\n\\begin{figure}[!th]\n \\centering\n \\includegraphics[width=\\textwidth]{3panels.png}\n \\caption{Tracking plane being assembled with pre-production\n panels.}\n \\label{Fig:trk-plane}\n\\end{figure}\n\n\\section{The calorimeter system}\n\\label{sec:calo}\n\nThe Mu2e calorimeter \\cite{TDR-EMC} has to provide confirmation for\nCE signal events, a powerful $e\/\\mu$ separation - with a muon\nrejection factor of $\\sim 200$, a standalone trigger and seeding for\ntrack reconstruction. An energy resolution of $O(10\\%$) and a time\nresolution of $500$ ps for 100 MeV electrons are sufficient to fulfil\nthese requirements.\nThe calorimeter design consists of two disks made from 674 undoped\nCsI scintillating crystals with ($34\\times 34\\times 200$) mm$^3$\ndimension. Each crystal is read-out by two custom array large area\n($2\\times 3$ of $6\\times 6$ mm$^2$ cells) UV-extended Silicon\nPhoto-Multipliers (SiPMs). Each SiPM is connected to a Front-End\nElectronics (FEE) board providing amplification and shaping of the\nsignal. Groups of 20 signals are sent to a custom digitizer module\n(DIRAC, DIgitizer and ReAdout Controller) where they are sampled at\n200 Mega samples per second and transferred to the Mu2e data acquisition\nsystem. A radioactive source and a laser system allow setting the energy\nscale and monitor the fast changes of response and resolution.\nThe crystals will receive an ionizing dose of 90 krad and a fluence\nof $3\\times10^{12}$ n\/cm$^2$.\nThe photosensors,\nbeing shielded by the crystals, will get a three times smaller dose.\nThe layout of the calorimeter system and pictures of crystals and a\nreadout channel are shown in Fig.~\\ref{Fig:calo}.\n\n\\begin{figure}[!th]\n \\begin{center}\n \\includegraphics[width=1.0\\textwidth]{mu2e_calo.png}\n \\end{center}\n \\caption{Left: sketch of the calorimeter system. The cooling\n pipes and the on board racks containing the DIRAC boards\n are visible. Right: pure CsI calorimeter crystals (top) and\n a readout channel, composed by two UV-extended SiPMs and the\n corresponding analog FEE boards (bottom).}\n\\label{Fig:calo} \n\\end{figure}\n\nA long R\\&D phase with small prototypes demonstrates that the calorimeter\ndesign easily satisfies the requirements\n\\cite{NIM-LYSO1,NIM-LYSO2,NIM-BaF2,Proto-EMC}.\nPre-production components have been used to build a large size calorimeter \nprototype, Module-0 (Fig.~\\ref{Fig:emc-mod0}), with 51 crystals and 102\nSiPMs and front end boards \\cite{Module-0}. It represents a portion of the\nfinal disk and has been used to test the integration and assembly procedures\nand to evaluate the operations of running in vacuum and at low temperatures.\nModule-0 performance was tested with an electron beam of 60-120 MeV at the\nINFN Beam Test Facility in Frascati \\cite{BTF}. The energy distribution for\n100 MeV electrons is well reproduced by the calorimeter simulation,\nFig.~\\ref{Fig:calo-testbeam} left. Energy and time resolution are evaluated\nwith particles impinging on the calorimeter surface both at 0 and 50 degrees.\nThe latter is the expected incidence angle for conversion electrons in Mu2e.\nAn energy resolution of 5\\% (7\\%) and a time resolution of 120 ps (150 ps)\nare obtained for 100 MeV particles impinging at $0^{\\circ}$ ($50^{\\circ}$),\nFig.~\\ref{Fig:calo-testbeam} center and right. Results satisfy\nphysics requirements and are well reproduced by simulation.\n\n\\begin{figure}[!th]\n \\vspace{0.3cm}\n \\centering\n \\begin{tabular}{cc}\n \\includegraphics[width=0.5\\textwidth]{mod0-front} &\n \\includegraphics[width=0.5\\textwidth]{mod0-testbeam} \\\\\n \\end{tabular}\n \\caption{Module-0, a large size prototype of the Mu2e calorimeter.\n Left: front view before mounting the source panel, where the\n staggered crystal structure is visible. Right: rear side, with\n readout channels and cooling circuit.}\n \\label{Fig:emc-mod0}\n\\end{figure}\n\n\\begin{figure}[!t]\n \\centering\n \\begin{tabular}{ccc}\n \\includegraphics[width=0.33\\textwidth]{calo_ene} &\n \\includegraphics[width=0.33\\textwidth]{calo_eres} &\n \\includegraphics[width=0.33\\textwidth]{calo_tres} \\\\\n \\end{tabular}\n \\caption{Calorimeter performance evaluated with a large-scale prototype,\n Module-0, using 60-120 MeV electron beam. Left: data-MC comparison of\n the energy distribution for 100 MeV beam. Energy (center) and time\n (right) resolution for orthogonal and 50$^\\circ$ impinging electrons.\n The energy resolution is compared with results expected from simulated\n data.}\n \\label{Fig:calo-testbeam}\n\\end{figure}\n\nThe complete production components for SiPMs and 85\\% of production\ncrystals have been received and characterized. For all of the 4000\nsensors, the breakdown voltage and the dark current are measured\nat different temperatures. The spread of these quantities over\nthe six cells of each sensor is used as quality control parameter\n(Fig.~\\ref{Fig:calo-qc} bottom). The overall rejection factor is\n1.2\\%, dominated by those sensors whose dark current RMS is too large.\nThe Quality Control of CsI crystals foresees a dimensional control,\nwith 0.1 mm tolerance with respect to nominal values, and a\nmeasurement of the optical properties \\cite{QAcrystals}.\nIn Fig.~\\ref{Fig:calo-qc} (top) the number of photoelectrons and\nthe uniformity response along the crystals are reported for both\nof the CsI producers. About 10\\% of the crystals have been rejected,\nmostly due to problems with mechanical tolerances.\nIrradiation tests have been carried out for small CsI and SiPM\nproduction subsamples. Results show that the calorimeter will be\nable to operate at the end of the Mu2e lifetime at a temperature\nbelow $0^\\circ$ C.\nMean Time To Failure tests on photosensors demonstrate an MTTF\nvalue 10 times larger than the experiment needs.\n\n\\begin{figure}[!t]\n \\centering\n \\includegraphics[width=\\textwidth]{calo-qa}\n \\caption{Summary of Quality Control measurements for production\n CsI crystals and Silicon Photo-Multipliers: CsI light yield\n (top left) and longitudinal response uniformity (top right);\n RMS of the breakdown voltage (bottom left) and of the dark\n current over the six cells of SiPMs (bottom right).\n Vertical lines represent the Quality Control acceptance cuts.}\n \\label{Fig:calo-qc}\n\\end{figure}\n\nThe prototypes of FEE and DIRAC have been exposed to a large ionization\ndose and neutron fluence to qualify rad-hard components. A slice test\nwith the whole calorimeter electronic chain provides results comparable\nto those achieved using a commercial digitizer. A DIRAC prototype is\ncurrently used to read 16 channels of Module-0.\n\n\\section{Cosmic Ray Veto}\n\\label{sec:crv}\n\nIn absence of the vetoing system, cosmic ray muons interacting with\nthe detector materials produce false signal CE candidates at a rate\nof approximately one\/day. In order to maintain the background under\nthe required level, the CRV has to provide a vetoing efficiency of at\nleast 99.99\\% for cosmic ray tracks while withstanding an intense\nradiation environment.\nThe Cosmic Ray Veto system \\cite{CRV} is made by four staggered layers\nof extruded plastic scintillation counters with two embedded 1.4 mm\ndiameter Wavelength Shifting Fibers\/counter, alternated with absorber\nslabs (Fig.~\\ref{Fig:crv-design}).\nEach fiber is readout by means of 2$\\times$2 mm$^2$ SiPMs. To achieve\nthe required coverage, a total of 5,504 counters are needed, organized\nin 86 modules of six different lengths for a total surface coverage of\n327 m$^2$.\n\n\\begin{figure}[!th]\n \\vspace{0.5cm}\n \\centering\n \\begin{tabular}{cc}\n \\includegraphics[width=0.5\\textwidth]{crvnew.png} &\n \\includegraphics[width=0.5\\textwidth]{crv_layout} \\\\\n \\end{tabular}\n \\caption{Left: the Mu2e Cosmic Ray Veto system, covering\n the detector solenoid and part of the transport solenoid.\n Right: layout of a CRV module.}\n \\label{Fig:crv-design}\n\\end{figure}\n\nMeasurements on a full size prototype with 120 GeV protons in the\nFermilab test beam area was carried out (Fig.~\\ref{Fig:crv-testbeam})\ndemonstrating that the needed light yield can be reached: the number\nof photo-eletrons obtained at 1 meter from the readout end provides\na safety factor of $\\sim 40\\%$ with respect to the requirements\n\\cite{CRVnpe}.\nIn Fig.~\\ref{Fig:crv-pe} test beam results are compared with the\nresults obtained from the CRV counter simulation, which includes\nscintillation and Cerenkov photon production\/transport, SiPM and\nelectronics responses. Good agreement is obtained after tuning\nthe Monte Carlo parameters.\nIrradiation of CRV SiPMs with neutrons was also tested to\nunderstand the maximum level of fluence acceptable for operations\n\\cite{CRVradhard}: neutrons could deteriorate the sensors response\nand increase the detector occupancy and dead-time so that shielding\nis mandatory.\n\n\\begin{figure}[!t]\n \\centering\n \\includegraphics[width=\\textwidth]{crvtestbeam.png}\n \\caption{Set-up of the CRV test beam. Protons are tracked with\n multi-wire proportional chambers. Front-End Boards (FEB) are\n visible on the top of the counter.}\n \\label{Fig:crv-testbeam}\n\\end{figure}\n\n\\begin{figure}[!t]\n \\vspace{0.5cm}\n \\centering\n \\begin{tabular}{cc}\n \\includegraphics[width=0.5\\textwidth]{crvtestbeampe} &\n \\includegraphics[width=0.5\\textwidth]{crv_pe_y} \\\\\n \\end{tabular}\n \\caption{Comparison of number of photo-electrons between simulated\n and test beam data for 120 GeV protons normally incident at\n different locations along (left) and across (right) a CRV\n counter.}\n \\label{Fig:crv-pe}\n\\end{figure}\n\nThe assembly of CRV di-counters started in June 2018 and about\nhalf of them have been produced. Production of photosensors and\nelectronics are also underway and 6\\% of the modules have been\nassembled.\nA test stand with cosmic rays is used to control the modules after\nproduction. An example of a cosmic ray event, as recorded by the\ntest stand and by the CRV module under test, is shown in\nFig.~\\ref{Fig:crv-display}.\n\n\n\\begin{figure}[!thb]\n \\begin{center}\n \\includegraphics[width=\\textwidth]{CRVscreen.png} \n \\end{center}\n \\caption{Example of event display at the cosmic ray test stand\n used to qualify CRV modules.} \n \\label{Fig:crv-display} \n \\vspace{1cm}\n\\end{figure}\n\n\\section{Conclusions and perspectives}\n\\label{sec:theend}\n\nThe Mu2e experiment will exploit the world's highest intensity muon\nbeams of the Fermilab Muon Campus to search for CLFV, improving current\nsensitivity by a factor $10^4$ and with a discovery capability over a\nwide range of New Physics models.\nA low mass straw tube tracker, a pure CsI crystal calorimeter with\nSiPM readout and a high efficiency cosmic ray veto have been selected\nto satisfy the demanding requirements.\nTests on prototypes and pre-production modules meet the experimental\nneeds.\nDetector construction is in progress and is expected to be completed\nby the end of 2020.\nInstallation will begin in 2021, followed by commissioning, with data\nbeginning in late 2023.\n\n\n\n\\acknowledgments\n\nWe are grateful for the vital contributions of the Fermilab staff\nand the technical staff of the participating institutions.\nThis work was supported by the US Department of Energy; \nthe Istituto Nazionale di Fisica Nucleare, Italy;\nthe Science and Technology Facilities Council, UK;\nthe Ministry of Education and Science, Russian Federation;\nthe National Science Foundation, USA; \nthe Thousand Talents Plan, China;\nthe Helmholtz Association, Germany;\nand the EU Horizon 2020 Research and Innovation Program under the\nMarie Sklodowska-Curie Grant Agreement No.~690835 and 734303. \nThis document was prepared by members of the Mu2e Collaboration\nusing the resources of the Fermi National Accelerator Laboratory\n(Fermilab), a U.S. Department of Energy, Office of Science, HEP\nUser Facility. Fermilab is managed by Fermi Research Alliance, LLC\n(FRA), acting under Contract No. DE-AC02-07CH11359.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe study of time dependent classical as well as quantum harmonic oscillators \nhas appealed to theoretical physicists since time immemorial. In the literature \nthe work by Lewis {\\it et al.}~\\cite{Lewis} has lead to an upsurge of analysis \nof the Hamiltonian for the time dependent quantum harmonic oscillator using a \nclass of exact invariants designed for such systems~\\cite{Lewis2,Lewis3}. The problem \nbecomes even more fascinating when one has a system of two such oscillators in \ntwo-dimensional space. Now, in order to address practical situations one needs \nto include damping in the system. Although there are several studies on the \none-dimensional damped quantum harmonic oscillator in the past~~\\cite{Sebawe}-\\cite{Pedrosa3}, it's two-dimensional equivalent \nis a less explored system~\\cite{Gouba}. The work by Lawson {\\it et.al.}~\\cite{Gouba} is \none of the very few which analyses a two-dimensional damped quantum harmonic oscillator \nsystem. The solutions obtained by them for the mentioned system provides a \nplatform to explore the construction of various coherent states with \nintriguing properties. \n\nIn the present work we extend the study by Lawson \n{\\it et.al.}~\\cite{Gouba} and consider the two-dimensional damped quantum harmonic \noscillator in noncommutative (NC) space. It has been argued that study of quantum mechanical systems in NC space is essential to ensure the attainment of \ngravitational stability~\\cite{Doplicher} in the present theories of quantum \ngravity, namely, string theory \\cite{amati, sw} and loop quantum gravity \\cite{rov}. \nThe simplest quantum mechanical setting in two dimensional NC space\nconsists of replacing the standard set of commutation relations between the\ncanonical coordinates by NC commutation relations $[X, Y]=i\\theta$, where $\\theta$ is a positive real constant. Quantum mechanical systems in such spaces have been studied extensively in the literature \\cite{suss}-\\cite{fgs}.\nThe study of a two-dimensional quantum harmonic oscillator in NC space with time dependent NC parameters was done in \\cite{Dey}. However, their system was an undamped oscillator. The parametrized form of solutions obtained there offered an interesting possibility for study of generalized version of Heisenberg's \nuncertainty relations. Quantum damped harmonic oscillator \non noncommuting two-dimensional space was studied in \\cite{anto} where the exact propagator of the system was obtained and the thermodynamic properties of the system was investigated using the standard canonical density matrix.\n\nIn this work, a two-dimensional damped quantum harmonic oscillator in NC space\nis considered once again. However, our focus of study is different than the work carried out in \\cite{anto}. We first construct the Hamiltonian and then express it in terms of standard commutative variables. This is done in Section 2. Then we solve the Hamiltonian using the method of invariants~\\cite{Lewis} and obtain the corresponding eigenfunction in \nSection 3. In doing so, although we start with the Hamiltonian and corresponding \ninvariant in Cartesian coordinates, eventually we transform our operators to \npolar coordinates (following closely the procedure suggested in \\cite{Dey}) for ease of solution. The form of the Lewis invariant in Cartesian coordinates with a Zeeman term in the Hamiltonian is an interesting result in itself and it also makes it easier to make a transition to it's polar form. It is to be noted that the eigenfunction \nof the Hamiltonian is a product of the eigenfunction of the invariant and a \nphase factor. Both the eigenfunction and phase factor are expressed in terms \nof time dependent parameters which obey the non-linear differential equation known as Ermakov-Pinney (EP) equation~\\cite{Ermakov,Pinney}. \nNext, in Section 4 we judiciously choose the parameters of the damped system \nsuch that they satisfy all the equations representing the system as well as \nprovide us with an exact closed form solution of the Hamiltonian. The \nsolutions of the NC parameters obtained in our analysis turns out to be such that the phase factor in an integral form given in \\cite{Dey} is exactly integrable for various kinds of dissipation. Then \nin Section 5 we device a procedure to calculate the matrix element of a finite arbitrary power of the position operator with respect to the exact solutions \nfor Hamiltonian eigenstates. Using these expressions we proceed to calculate \nthe expectation value of energy and study the \nevolution of the energy expectation value of the system with time \nfor various types of damping. In Section 6 we summarize our results.\n\n\n\n\\section{Model of the two-dimensional harmonic oscillator}\nThe system we consider is a combination of two non-interacting damped harmonic oscillators in two dimensional NC space. The oscillators have equal time dependent \nfrequencies, time dependent coefficients of friction and equal mass in \nNC space. Such a model of damped harmonic oscillator was \nconsidered in an earlier communication~\\cite{Gouba} in commutative space. In this work, we extend the model by considering the system in NC space\\footnote{We shall be considering NC phase space in our work. However, we shall generically refer this as NC space.}.\n\n\\noindent The Hamiltonian of the system has the following form,\n\\begin{equation}\nH(t)=\\dfrac{f(t)}{2M}({P_1}^2+{P_2}^2)+\\dfrac{M\\omega^2(t)}{2f(t)}({X_1}^2+{X_2}^2)\\label{1}\n\\end{equation}\nwhere the damping factor $f(t)$ is given by,\n\\begin{equation}\nf(t)=e^{-\\int_{0}^t\\eta(s)ds} \n\\label{1x}\n\\end{equation}\nwith $\\eta(s)$ being the coefficient of friction. Here $\\omega(t)$ is the \ntime dependent angular frequency of the oscillators and $M$ is their mass.\nIt should be noted that in commutative space, the model with $f(t)=e^{-\\Gamma t}$ and $\\omega(t)=\\omega_0$, with $\\Gamma$ and $\\omega_0$\nbeing positive constants, is said to be the two-dimensional Caldirola and Kanai Hamiltonian \\cite{caldi, kanai}.\nThe position and momentum \ncoordinates $(X_i,P_i)$ are noncommuting variables in NC space, that is, \ntheir commutators are $[X_1,X_2]~\\neq~0$ and $[P_1,P_2]~\\neq~0$. The \ncorresponding canonical variables $(x_i,p_i)$ in commutative space are such \nthat the commutator $[x_i,p_j]=i\\hbar\\delta_{i,j}$, $[x_i,x_j]=0=[p_i,p_j]$; ($i,j=1,2$).\n\nIn order to express the NC Hamiltonian in terms of the standard commutative variables explicitly, we apply the standard Bopp-shift relations \\cite{mez} ($\\hbar=1$): \n\\begin{eqnarray}\n& X_1=x_1-\\dfrac{\\theta(t)}{2}p_2\\,\\,\\,;\\,\\,\\,X_2=x_2+\\dfrac{\\theta(t)}{2}p_1\\\\\n& P_1=p_1+\\dfrac{\\Omega(t)}{2}x_2\\,\\,\\,;\\,\\,\\,P_2=p_2-\\dfrac{\\Omega(t)}{2}x_1 \\,\\,.\n\\label{eqn1}\n\\end{eqnarray}\nHere $\\theta(t)$ and $\\Omega(t)$ are the NC parameters for space \nand momentum respectively, such that $[X_1,X_2]~=i\\theta(t)$, \n$[P_1,P_2]~=i\\Omega(t)$ and $[X_1,P_1]=i[1+\\frac{\\theta(t)\\Omega(t)}{4}]=[X_2,P_2]$; ($X_1\\equiv X$, $X_2 \\equiv Y$, $P_1\\equiv P_x$, $P_2 \\equiv P_y$). \n\n\\noindent The Hamiltonian in terms of $(x_i,p_i)$ coordinates is therefore given by the following \nrelation,\n\\begin{equation}\nH=\\dfrac{a(t)}{2}({p_1}^2+{p_2}^2)+\\dfrac{b(t)}{2}({x_1}^2+{x_2}^2)+c(t)({p_1}{x_2}-{p_2}{x_1})\\,\\,\\,.\\label{eqn2}\n\\end{equation}\nThe time dependent coefficients in the above Hamiltonian are given as,\n\\begin{eqnarray}\na(t)&=&\\dfrac{f(t)}{M}+\\dfrac{M{\\omega^2(t)}\\theta^2(t)}{4f(t)}\\label{3} \\\\\n b(t)&=&\\dfrac{f(t){\\Omega^2(t)}}{4M}+\\dfrac{M{\\omega^2(t)}}{f(t)}\\label{4} \\\\\n c(t)&=&\\dfrac{1}{2}\\left[\\dfrac{f(t)\\Omega(t)}{M}+\\dfrac{M\\omega^2(t)\\theta(t)}{f(t)} \\right]. \n\\label{eqn3}\n\\end{eqnarray}\nHere it must be noted that although our Hamiltonian given by Eqn.(\\ref{eqn2}) has the same form as that in \n\\cite{Dey} to study a system of a two dimensional harmonic oscillator in NC space, the time dependent \nHamiltonian coefficients (given by Eqn(s).(\\ref{eqn3})) have very different form. This is because our system is that of \na damped harmonic oscillator in two-dimensional NC space. Thus, the damping factor $f(t)$ modulates and alters \nthe Hamiltonian coefficients from the form considered in earlier study \\cite{Dey}.\n\n\n\n\\section{Solution of the model Hamiltonian}\nIn order to find the solutions of the model Hamiltonian $H(t)$ (Eqn.(\\ref{eqn2}))\nrepresenting the two-dimensional damped harmonic oscillator in \nNC{ space, we follow the route suggested by Lewis {\\it et.al.}~\\cite{Lewis} in their work. First we \nconstruct the time-dependent Hermitian invariant operator $I(t)$ corresponding to our Hamiltonian operator $H(t)$ \n(given by Eqn.(\\ref{eqn2})). This is because if one can solve for the eigenfunctions of $I(t)$, $\\phi(x_1,x_2)$, such \nthat,\n\\begin{equation}\nI(t)\\phi(x_1,x_2)=\\epsilon \\phi(x_1,x_2)\n\\label{eqnegn}\n\\end{equation}\nwhere $\\epsilon$ is an eigenvalue of $I(t)$ corresponding to eigenstate $\\phi(x_1,x_2)$, one can obtain the \neigenstates of $H(t)$, $\\psi(x_1,x_2,t)$, using the relation given by Lewis {\\it et. al.}~\\cite{Lewis} which is as \nfollows, \n\\begin{equation}\n\\psi(x_1,x_2,t)=e^{i\\Theta(t)}\\phi(x_1,x_2)\n\\label{eqnpsi}\n\\end{equation}\nwhere the real function $\\Theta(t)$ which acts as the phase factor will be discussed in details later. \n\n\n\n\\subsection{The Time Dependent Invariant}\nNext, following the approach taken by Lewis {\\it et.al.}~\\cite{Lewis}, we need to construct the operator $I(t)$ which \nis an invariant with respect to time, corresponding to the Hamiltonian $H(t)$, as mentioned earlier, such \nthat $I(t)$ satisfies the condition,\n\\begin{equation}\n\\dfrac{dI}{dt}=\\partial_t{I}+\\dfrac{1}{i}[I,H]=0.\n\\label{eqn4}\n\\end{equation}\nThe procedure is to choose the Hermitian invariant $I(t)$ to be of the same homogeneous quadratic form defined by Lewis \n{\\it et. al.}~\\cite{Lewis} for time-dependent harmonic oscillators. However, since we are dealing with a \ntwo-dimensional system in the present study, $I(t)$ takes on the following form,\n\\begin{equation}\nI(t)=\\alpha(t)({p_1}^2+{p_2}^2)+\\beta(t)({x_1}^2+{x_2}^2)+\\gamma(t)(x_1{p_1}+p_2{x_2}).\n\\label{eqn5}\n\\end{equation}\nHere we will consider $\\hbar=1$ since we choose to work in natural units. Now, using the form of $I(t)$ defined by \nEqn.(\\ref{eqn5}) in Eqn.(\\ref{eqn4}) and equating the coefficients of the canonical variables, we get the \nfollowing relations,\n\\begin{eqnarray}\n\\dot{\\alpha}(t)&=&-a(t)\\gamma(t)\\label{eqn6}\\\\\n\\dot{\\beta}(t)&=&b(t)\\gamma(t)\\label{eqn7}\\\\\n\\dot{\\gamma}(t)&=&2\\left[\\,b(t)\\alpha(t)-\\beta(t)a(t)\\,\\right]\n\\label{eqn8}\n\\end{eqnarray}\nwhere dot denotes derivative with respect to time $t$.\n\n\\noindent To express the above three time dependent parameters $\\alpha$,$\\beta$ and $\\gamma$ in terms of a single time \ndependent parameter, we parametrize $\\alpha(t)=\\rho^{2}(t)$. Substituting this in Eqn(s).(\\ref{eqn6}, \\ref{eqn8}), we \nget the other two parameters in terms of $\\rho(t)$ as, \n\\begin{eqnarray}\n\\gamma(t)&=&-\\dfrac{2\\rho\\dot{\\rho}}{a(t)}\\label{eqn9}\\\\\n\\beta(t)&=&\\dfrac{1}{a(t)}\\left[\\dfrac{{\\dot{\\rho}^2}}{a(t)}+{{\\rho}^2}b+\\dfrac{\\rho\\ddot{\\rho}}{a(t)}-\\dfrac{\\rho\\dot{\\rho}\\dot{a}}{a^2} \\right].\\label{eqn10}\n\\end{eqnarray}\nNow, substituting the value of $\\beta$ in Eqn.(\\ref{eqn7}), we get a non-linear equation in \n$\\rho(t)$ which has the form of the non-linear Ermakov-Pinney (EP) equation with a dissipative \nterm~\\cite{Dey, Ermakov, Pinney}. The form of the non-linear equation is as follows, \n\\begin{equation}\n\\ddot{\\rho}-\\dfrac{\\dot{a}}{a}\\dot{\\rho}+ab\\rho={\\xi^2}\\dfrac{a^2}{\\rho^3}~.\n\\label{eqn11}\n\\end{equation} \nwhere ${\\xi^2}$ is a constant of integration. This equation has similar form to the EP equation obtained in \\cite{Dey}, which is expected since \nour $H(t)$ has the same form as theirs. However, once again we should recall the fact that the explicit form of the time-dependent \ncoefficients are different due to the presence of damping. \n\n\n\\noindent Now, using the EP equation we get a simpler form of $\\beta$ as,\n\\begin{eqnarray}\n\\beta(t)&=&\\dfrac{1}{a(t)}\\left[\\dfrac{{\\dot{\\rho}^2}}{a(t)}+\\dfrac{{\\xi^2}{a(t)}}{\\rho^2} \\right].\n\\label{eqnew}\n\\end{eqnarray} \n\\noindent Next, substituting the expressions of $\\alpha$, $\\beta$ and $\\gamma$ in \nEqn.(\\ref{eqn5}), we get the following \nexpression for $I(t)$,\n\\begin{equation}\nI(t)=\\rho^2({p_1}^2+{p_2}^2)+\\left(\\dfrac{\\dot{\\rho}^2}{a^2}+\\dfrac{{\\xi^2}}{\\rho^2}\\right)({x_1}^2+{x_2}^2)-\\dfrac{2\\rho\\dot{\\rho}}{a}(x_1{p_1}+p_2{x_2}).\n\\label{eqn12}\n\\end{equation}\nThe form of the Lewis invariant in Cartesian coordinates will be used later to go over to it's polar coordinate form.\nThe solution of the EP equation under various physically significant conditions shall be discussed later.\n\n\n\n \n\\subsection{Construction of Ladder operators} \nNow that we have the required Hermitian invariant $I(t)$, we proceed to calculate it's eigenstates using the operator approach. For \nthis purpose we need to first construct some ladder operators. To do this, we first need to transform the form of $I(t)$ \n(given by Eqn.(\\ref{eqn12})) to a more manageable form. For this we invoke a unitary transformation using a suitable unitary \noperator $\\hat{U}$ having the following form, \n\\begin{eqnarray}\n\\hat{U}=exp\\left[-\\dfrac{i\\dot{\\rho}}{2a(t)\\rho}({x_1}^2+{x_2}^2)\\right],\\,\\,\\,\n\\hat{U^{\\dagger}}\\hat{U}=\\hat{U}\\hat{U^{\\dagger}}=\\textbf{I}.\n\\label{eqn13}\n\\end{eqnarray}\nDefining, \n\\begin{eqnarray}\n\\phi^{'}(x_1,x_2)=\\hat{U}\\phi(x_1,x_2) \\,\\,\\,,\\,\\,\\,\nI^{'}(t)&=\\hat{U}I\\hat{U^\\dagger}\\,\\,\\;\n\\label{eqn14}\n\\end{eqnarray}\nwhere $\\phi(x_1,x_2)$ is an eigenfunction of $I(t)$ as introduced in Eqn.(\\ref{eqnegn}), \nthen, using Eqn(s).(\\ref{eqnegn},\\ref{eqn14}), we get, \n\\begin{align}\nI^{'}\\phi^{'}=\\hat{U}I\\hat{U^\\dagger}\\hat{U}\\phi=\\hat{U}I\\phi=\\hat{U}\\epsilon\\phi=\\epsilon\\phi^{'}.\n\\end{align}\nThe transformed expression of the invariant, $I^{'}(t)$, using Eqn.(\\ref{eqn14}), has the following form, \n\\begin{align}\nI^{'}(t)=\\rho^2({p_1}^2+{p_2}^2)+\\dfrac{{\\xi^2}}{\\rho^2}({x_1}^2+{x_2}^2)\\,\\,.\n\\label{eqn15} \n\\end{align}\nThis transformed form of the invariant, $I^{'}(t)$, has exactly the same form as that of the Hamiltonian for a time dependent \ntwo-dimensional simple harmonic oscillator. So, we can introduce the corresponding ladder operators for $\\hat{I^{'}}(t)$ to be \ngiven by,\n\\begin{eqnarray}\n{\\hat{a}_j}^{'}=\\dfrac{1}{\\sqrt{2\\xi}}\\left(\\dfrac{\\xi}{\\rho}{\\hat{x}}_j+i\\rho{\\hat{p}}_j\\right)\\,\\,\\,,\\,\\,\\,{{\\hat{a}_j}^{'\\dagger}}=\\dfrac{1}{\\sqrt{ 2\\xi}}\\left(\\dfrac{\\xi}{\\rho}{\\hat{x}}_j-i\\rho{\\hat{p}}_j\\right)\n\\label{eqn16}\n\\end{eqnarray}\nwhere $j=1,2$ and the operators satisfy the commutation relation $[{{\\hat{a}_i}^{'}},{{\\hat{a}_j}^{'\\dagger}}]=\\delta_{ij}$.\n\n\\noindent Now we make the reverse transformation to get the expression of the unprimed ladder operators:\n\\begin{eqnarray}\n\\hat{a_j}(t)&=&{\\hat{U}}^\\dagger{\\hat{a}_j}^{'}\\hat{U}=\\dfrac{1}{\\sqrt{2\\xi}}\\left[\\dfrac{\\xi}{\\rho}x_j+i\\rho{p_j}-\\dfrac{i\\dot{\\rho}}{a(t)}x_j\\right]\\\\\n\\label{eqn17}\n\\hat{{a_j}^{\\dagger}}(t)&=&{\\hat{U}}^\\dagger{{\\hat{a}_j}^{'\\dagger}}\\hat{U}=\\dfrac{1}{\\sqrt{2\\xi}}\\left[\\dfrac{\\xi}{\\rho}x_j-i\\rho{p_j}+\\dfrac{i\\dot{\\rho}}{a(t)}x_j\\right].\n\\label{eqn18} \n\\end{eqnarray}\nIt can be easily checked using the algebra of the primed ladder operators that $[{{\\hat{a}_i}},{{\\hat{a}_j}^{\\dagger}}]=\\delta_{ij}$.\n\n\\noindent We now set $\\xi=1$ and consider two linear combinations of the above two operators such that,\n\\begin{eqnarray}\n\\hat{a}(t)=-\\dfrac{i}{\\sqrt{2}}(\\hat{a}_1-i\\hat{a}_2)\n=\\dfrac{1}{2}\\left[\\rho(\\hat{p_1}-i\\hat{p_2})-\\left(\\dfrac{i}{\\rho}+\\dfrac{\\dot{\\rho}}{a(t)} \\right)(\\hat{x_1}-i\\hat{x_2})\\right]\n\\label{eqn19}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n{\\hat{a}}^\\dagger(t)=\\dfrac{i}{\\sqrt{2}}({\\hat{a}_1}^\\dagger+i{\\hat{a}_2}^\\dagger)\n=\\dfrac{1}{2}\\left[\\rho(\\hat{p_1}+i\\hat{p_2})+\\left(\\dfrac{i}{\\rho}-\\dfrac{\\dot{\\rho}}{a(t)} \\right)(\\hat{x_1}+i\\hat{x_2})\\right].\n\\label{eqn20}\n\\end{eqnarray}\nThese also satisfy the commutation relation $[\\hat{a},\\hat{a}^\\dagger]=1$.\n\n\n\n\n\\subsection{Transformation to polar coordinates}\n\nWith the above results in place, we now transform the invariant $I(t)$ and the corresponding ladder operators to polar \ncoordinates for calculational convenience. For this we invoke the transformation of coordinates of the form,\n\\begin{equation}\nx=rcos\\theta\\,\\,\\,;\\,\\,\\,y=rsin\\theta~.\n\\label{eqn21}\n\\end{equation}\nThe canonical coordinates in polar representation takes the following form, \n\\begin{eqnarray}\np_r&=&\\dfrac{1}{2}\\left(\\dfrac{x_1}{r}{p_1}+{p_1}\\dfrac{x_1}{r}+\\dfrac{x_2}{r}{p_2}+p_2\\dfrac{x_2}{r}\\right)\\nonumber\\\\\n&=&\\dfrac{x_1{p_1}+x_2{p_2}}{r}-\\dfrac{i}{2r}\\nonumber\\\\&=&-i\\left({\\partial}_r+\\dfrac{1}{2r} \\right)\\\\\np_{\\theta}&=&(x_1{p_2}-x_2{p_1})=-i{\\partial_{\\theta}}.\n\\label{eqn22}\n\\end{eqnarray} \nThe commutation relations between ($r$, $p_r$) and \n($\\theta$, $p_\\theta$) have the form\n\\begin{equation}\n[r,p_r]=[\\theta,p_{\\theta}]=[x_1,p_1]=[x_2,p_2]=i.\n\\label{eqn23}\n\\end{equation}\nThe corresponding anticommutation relation can be found to be,\n\\begin{equation}\n[r, p_r]_{+}=[x_1, p_1]_{+}+[x_2, p_2]_{+}\n=2(x_1 p_1+p_2 x_2)\n\\label{eqn24}\n\\end{equation}\nwhere $[A, B]_{+}=AB+BA$ represents anticommutator between operators $A$, $B$.\n\n\\noindent In order to transform the invariant $I(t)$ in polar coordinates, we need to have few other relations which are,\n\\begin{eqnarray}\n({p_1}^2+{p_2}^2)&=&\\left({p_r}^2+\\dfrac{{p_{\\theta}}^2}{r^2}-\\dfrac{1}{4r^2}\\right)\\\\\n(p_1+i{p_2})&=&e^{i\\theta}\\left[p_r+\\dfrac{i}{r}p_{\\theta}+\\dfrac{i}{2r} \\right] \\\\\n(p_1-i{p_2})&=&e^{-i\\theta}\\left[p_r-\\dfrac{i}{r}p_{\\theta}+\\dfrac{i}{2r} \\right].\n\\label{eqn25}\n\\end{eqnarray}\nHence the invariant in polar coordinate system is given by,\n\\begin{eqnarray}\nI(t)=\\dfrac{\\xi^2}{\\rho^2}r^2+\\left(\\rho{p_r}-\\dfrac{\\dot{\\rho}}{a}r\\right)^2+\\left({\\dfrac{\\rho{p_\\theta}}{r}}\\right)^2-\\left({\\dfrac{\\rho\\hbar}{2r}}\\right)^2\n\\label{eqn26}\n\\end{eqnarray}\nand the ladder operators in polar coordinate system have the following form,\n\\begin{eqnarray}\n\\hat{a}(t)&=&\\dfrac{1}{2}\\left[\\left(\\rho{p_r}-\\dfrac{\\dot{\\rho}}{a(t)}r \\right)-i\\left(\\dfrac{r}{\\rho}+\\dfrac{\\rho{p_\\theta}}{r}+\\dfrac{\\rho}{2r} \\right) \\right]e^{-i\\theta}\\nonumber\\\\\n{\\hat{a}}^{\\dagger}(t)&=&\\dfrac{1}{2}e^{i\\theta}\\left[\\left(\\rho{p_r}-\\dfrac{\\dot{\\rho}}{a(t)}r \\right)+i\\left(\\dfrac{r}{\\rho}+\\dfrac{\\rho{p_\\theta}}{r}+\\dfrac{\\rho}{2r} \\right) \\right].\n\\label{eqn27}\n\\end{eqnarray}\nNow we note from Eqn(s).(\\ref{eqn26}, \\ref{eqn27}) that both the invariant $I(t)$ \nand the ladder operators have the same form as those used in \\cite{Dey} to study the undamped harmonic oscillator in \nNC space. The time-dependent coefficients involved in the \npresent study however differ due to the damping present in our system. Thus, we \ncan just borrow the expression of eigenfunction and the phase factors \nfrom \\cite{Dey} for our present system.\n\n\n\n\\subsection{Eigenfunction and phase factor}\nWe depict the set of eigenstates of the invariant operator $I(t)$ as $\\ket{n,l} $, following the convention in \n\\cite{Dey}. Here, $n$ and $l$ are integers such that $n+l\\geqslant0$. So we have the condition $l\\geqslant-n$. \nThus, if $l=-n+m$, then $m$ is a positive integer; and the corresponding eigenfunction in polar coordinate system has the following form (restoring $\\hbar$), \n\\begin{eqnarray}\n\\phi_{n,m-n}(r,\\theta)&=&\\braket{r,\\theta|n,m-n}\\\\\n&=&\\lambda_{n}\\dfrac{{(i\\sqrt{\\hbar}\\rho)}^m}{\\sqrt{m!}}r^{n-m}e^{i\\theta(m-n)-\\dfrac{a(t)-i\\rho\\dot{\\rho}}{2a(t)\n\\hbar{\\rho}^2}r^2}U\\left(-m,1-m+n,\\dfrac{r^2}{\\hbar\\rho^2} \\right)\n\\label{eqn28}\n\\end{eqnarray}\nwhere $\\lambda_n$ is given by\n\\begin{eqnarray}\n\\lambda_n^2=\\dfrac{1}{\\pi{n!}{(\\hbar\\rho^2)}^{1+n}}~. \n\\label{eqn28lam}\n\\end{eqnarray}\nHere, $U\\left(-m,1-m+n,\\dfrac{r^2}{\\hbar\\rho^2} \\right)$ is \nTricomi's confluent hypergeometric function \\cite{Arfken, uva} and the eigenfunction $\\phi_{n,m-n}(r,\\theta)$ satisfies the following \northonormality relation,\n\\begin{equation}\n\\int_0^{2\\pi}d\\theta\\int_0^{\\infty}rdr\\phi^{*}_{n,m-n}(r,\\theta)\\phi_{n^{'},m^{'}-n^{'}}(r,\\theta)=\\delta_{nn^{'}}\\delta_{mm^{'}}.\n\\label{eqn29}\n\\end{equation}\nAgain following \\cite{Dey}, the expression of the phase factor $\\Theta(t)$ is given by, \n\\begin{equation}\n\\Theta_{\\,n\\,,\\,l}(t)\\,=\\,(\\,n\\,+\\,l\\,)\\,\\int_0^t \\left(c(T)-\\dfrac{a(T)}{\\rho^2(T)} \\right)dT~.\n\\label{eqn30}\n\\end{equation}\nFor a given value of $l=-n+m$, it would be given by \\cite{Dey},\n\\begin{equation}\n\\Theta_{\\,n\\,,\\,m\\,-\\,n\\,}(t)=m\\int_0^t \\left(c(T)-\\dfrac{a(T)}{\\rho^2(T)} \\right)dT~.\n\\label{eqn31}\n\\end{equation}\nWe shall use this expression to compute the phase explicitly as a function of time for various physical cases in the subsequent discussion.\n\n\\noindent The eigenfunction of the Hamiltonian therefore reads (using Eqn(s).(\\ref{eqnpsi}, \\ref{eqn28}, \\ref{eqn31}))\n\\begin{eqnarray}\n\\psi_{n,m-n}(r,\\theta,t)&=&e^{i\\Theta_{n, m-n}(t)}\\phi_{n, m-n}(r,\\theta)\\nonumber\\\\\n&=&\\lambda_{n}\\dfrac{{(i\\sqrt{\\hbar}\\rho)}^m}{\\sqrt{m!}}\\exp{\\left[im\\int_0^t \\left(c(T)-\\dfrac{a(T)}{\\rho^2(T)} \\right)dT \\right]}\n\\nonumber\\\\\n&&\\times~r^{n-m}e^{i\\theta(m-n)-\\dfrac{a(t)-i\\rho\\dot{\\rho}}{2a(t)\\hbar{\\rho}^2}r^2}U\\left(-m,1-m+n,\\dfrac{r^2}{\\hbar\\rho^2} \\right).\n\\label{eqn32}\n\\end{eqnarray}\n\n\n\n\n\n\n\n\\section{Solutions for the noncommutative damped oscillator}\nIn this paper we are primarily interested in damped oscillators in \nNC space. For this purpose we want to find the eigenfunctions of \nthe corresponding Hamiltonian under various types of damping. The various \nkinds of damping are represented by various forms of the time dependent \ncoefficients of the Hamiltonian, namely, $a(t)$, $b(t)$ and $c(t)$. \nHowever, the various forms must be constructed in such a way that they satisfy \nthe non-linear EP equation given by Eqn.(\\ref{eqn11}). \nThe procedure of this construction of exact analytical solutions is based on the Chiellini integrability condition \\cite{chill} and this formalism was followed in \\cite{Dey}. We shall do the same in this paper.\nSo, for various forms of $a(t)$ and $b(t)$, we get the corresponding form of $\\rho(t)$ \nusing the EP equation together with the Chiellini integrability condition. In other words, the set of values of $a(t)$, $b(t)$ and $\\rho(t)$ that we use must be a solution set of the EP equation consistent with the Chiellini integrability condition. In the subsequent discussion we shall proceed to obtain solutions of the EP equation for the damped NC oscillator.\n\n\n\\subsection{Solution Set-I for Ermakov-Pinney equation : Exponentially \\\\ decaying solutions } \n\\subsubsection{The Solution Set}\nThe simplest kind of solution set of EP equation under damping is the \nexponentially decaying set used in \\cite{Dey}. The solution set is given by the following\nrelations, \n\\begin{eqnarray}\na(t)=\\sigma e^{-\\vartheta{t}}\\,\\,\\,,\\,\\,\\,b(t)=\\Delta e^{\\vartheta{t}}\\,\\,\\,,\\,\\,\\rho(t)={\\mu}e^{-\\vartheta{t\/2}}\\,\\,\\,\\,\\,\n\\label{EPsoln1}\n\\end{eqnarray}\nwhere $\\sigma,\\Delta$ and $\\mu$ are constants. Here, $\\vartheta$ is any \npositive real number. Substituting the expression of $a(t),b(t) \\,$and$\\, \\rho(t)$ in the EP equation, we can easily verify the relation between these constants to be as follows, \n\\begin{equation}\n\\mu^4=\\dfrac{\\xi^2{\\sigma^2}}{\\sigma\\Delta-\\dfrac{1}{4}\\vartheta^2}~.\n\\label{EPreln1}\n\\end{equation}\n\n\n\n\n\\subsubsection{Study of the corresponding eigenfunctions}\nWe now write down the eigenfunctions of the Hamiltonian for the choosen set of \ntime-dependent coefficients. For this endeavour we need to choose explicit \nforms of the damping factor $f(t)$ and angular frequency of the oscillator \n$\\omega(t)$. The eigenfunction of the invariant $I(t)$ (which is given by \nEqn.(\\ref{eqn28})) takes on the following form for the solution set-I:\n\\begin{eqnarray}\n\\phi_{n,m-n}(r,\\theta)=\\lambda_{n}\\dfrac{{(i{\\mu}e^{-\\vartheta{t\/2}})}^m}{\\sqrt{m!}} r^{n-m}e^{i\\theta(m-n)-\\dfrac{2\\sigma+i\\mu^2\\vartheta}{4\\sigma\\mu^2{e^{-\\vartheta{t}}}}r^2}U\\left(-m,1-m+n,\\dfrac{r^2{e^{\\vartheta{t}}}}{\\mu^2} \\right)\n\\label{eqn33}\n\\end{eqnarray}\nwhere $\\lambda_n$ is given by\n\\begin{eqnarray}\n\\lambda_n^{\\,2}\\,=\\,\\dfrac{1}{\\pi\\,n!\\,[\\mu^2\\,exp\\,(-\\vartheta{t})]^{1+n}}~.\n\\label{eqn33lam}\n\\end{eqnarray}\nIn order to obtain explicit expressions of the phase factors for various cases of the damping factor, we choose both the \nfunctions $\\omega(t)$ and $\\eta(t)$ as follows.\\\\\n\n\n\n\\noindent{\\bf $\\langle A\\rangle$ Solution Set-Ia}\n\n\\noindent Firstly, we choose the damping factor $f(t)=1$. Thus, in this case the damping in the system is due to the exponentially \ndecaying frequency $\\omega(t)$. For this purpose we set, \n\\begin{eqnarray}\n\\eta(t)=0\\,\\,\\Rightarrow\\,\\,f(t)=1\\\\\n\\omega(t)={\\omega_0}\\,exp(-\\Gamma{t}\/2)\\,\\,\\,.\n\\label{eqn34}\n\\end{eqnarray}\nSubstituting the expressions for $a(t)$, $b(t)$, $\\omega(t)$ and $f(t)$ in the \nEqn(s).(\\ref{3}, \\ref{4}), we get the time dependent NC parameters as,\n\\begin{eqnarray}\n\\theta(t)=\\dfrac{2}{M\\omega_0}\\,exp\\,[\\Gamma{t}\/2] \\sqrt{M\\sigma\\,exp(-\\vartheta{t})-1}\\label{eqn35} \\\\\n\\Omega(t)=2\\sqrt{\\,M[\\Delta\\,exp\\,(\\vartheta{t})-M\\omega_0^2\\,exp\\,(-\\Gamma{t})]}. \\label{eqn36}\n\\end{eqnarray}\nIt can be checked that in the limit $\\Gamma\\rightarrow0$, that is, for \nconstant frequency, the expressions for $\\theta(t)$ and $\\Omega(t)$ reduce \nto those in \\cite{Dey}. When $\\vartheta=\\Gamma$, then the solutions take the \nform,\n\\begin{eqnarray}\n\\theta(t)&=&\\dfrac{2}{M\\omega_0}\\, \\sqrt{M\\sigma\\,-e^{\\Gamma t}}\\label{eqn35b} \\\\\n\\Omega(t)&=&2\\sqrt{\\,M[\\Delta\\,exp\\,(\\Gamma{t})-M\\omega_0^2\\,exp\\,(-\\Gamma{t})]}. \\label{eqn36b}\n\\end{eqnarray}\nSubstituting these relations in the expression for $c(t)$ in Eqn.(\\ref{eqn3}), \nwe get an expression for the phase in a closed form as, \n\\begin{equation}\nc(t)=\\sqrt{\\dfrac{ \\Delta\\,exp\\,(\\Gamma{t})-M{\\omega_0}^2\\,exp\\,(-\\Gamma{t}) }{M}} \n\\,+\\,\\omega_0\\,exp\\,(-\\Gamma{t}\/2)\\sqrt{\\,M\\sigma\\,exp\\,(-\\Gamma{t})-1 \\,}.\\label{eqn38}\n\\end{equation}\nSubstituting the expressions of $a(t)$, $\\rho(t)$ and $c(t)$ in Eqn.(\\ref{eqn31}), we get,\n\\begin{eqnarray}\n\\Theta_{\\,n\\,,\\,l}(t)&=&\\,(\\,n\\,+\\,l\\,)\\,\\dfrac{\\omega_0}{2\\sqrt{M\\sigma}\\Gamma}\\ \\left[log_{e}\\dfrac{e^{{\\Gamma}t}-2M\\sigma-2\\sqrt{M\\sigma(M\\sigma-e^{{\\Gamma}t})}}{1-2M\\sigma-2\\sqrt{M\\sigma(M\\sigma-1)}}\\right.\\nonumber\\\\ \n&&\\left. -{\\Gamma}t-2\\sqrt{M\\sigma(M{\\sigma}e^{-{2\\Gamma}t}-e^{-{\\Gamma}t})}+2\\sqrt{M\\sigma(M\\sigma-1)}\\ \\right]\\nonumber\\\\\n&&+\\dfrac{2(n+l)}{\\Gamma}\\,\\left[\\sqrt{\\frac{\\Delta}{M}e^{{\\Gamma}t}-{\\omega_0^2}e^{-{\\Gamma}t}}-\\sqrt{\\dfrac{\\Delta}{M}-{\\omega_0^2}}\\right.\\nonumber\\\\\n&&\\left.+2i{\\omega_0}\\left\\{e^{-{\\Gamma}t\/2} {_{2}F_{1}}\\left(-\\frac{1}{4},\\frac{1}{2},\\frac{3}{4},\\frac{{\\Delta}e^{{2\\Gamma}t}}{M\\omega_0^2}\\right)- {_{2}F_{1}}\\left(-\\frac{1}{4},\\frac{1}{2},\\frac{3}{4},\\frac{\\Delta}{M\\omega_0^2}\\right)\\right\\}\\right]\n-\\frac{\\sigma}{\\mu^2}(n+l)t \\nonumber\\\\\n\\label{eqn39} \n\\end{eqnarray}\nwhere $_{2}F_{1}(a,b,c;t)$ is said to be the Gauss hypergeometric function. It is interesting to note that the solutions of the time dependent NC parameters enable us to get an exact analytic expression for the phase factor.\nIt is further interesting to observe that the phase has a complex part which indicates that the wave function decays with time.\\\\\n\n\n\n\\noindent {\\bf $\\langle B\\rangle$ Solution Set-Ib}\n\n\\noindent Here the oscillator is damped due to the damping factor $f(t)$ and the frequency $\\omega(t)$ \nis a constant. This situation can be depicted by the following relations, \n\\begin{eqnarray}\nf(t)= exp\\,(-\\Gamma{t})~;~\\omega(t)={\\omega_0}.\n\\label{10x}\n\\end{eqnarray}\nSubstituting these relations in Eqn(s).(\\ref{3}, \\ref{4}), we get the time dependent NC parameters \nas,\n\\begin{eqnarray}\n\\theta(t)&=&\\dfrac{2}{M\\omega_0}\\sqrt{M{\\sigma}\\,exp\\,(-\\vartheta{t})-exp\\,(-\\Gamma{t})}\\,\\,e^{-\\Gamma t\/2}\\label{eqn40}\\\\\n\\Omega(t)&=&2e^{\\Gamma{t}}\\sqrt{M\\,[\\Delta\\,exp\\,(\\vartheta-\\Gamma)t-M{\\omega_0}^2]}.\n\\label{eqn41}\n\\end{eqnarray}\nIt can be checked that in the limit $\\Gamma\\rightarrow0$, that is, for \nconstant frequency, the expressions for $\\theta(t)$ and $\\Omega(t)$ reduce \nto those in \\cite{Dey}. When $\\vartheta=\\Gamma$, then the solutions take the \nform,\n\\begin{eqnarray}\n\\theta(t)&=&\\dfrac{2}{M\\omega_0}\\sqrt{M{\\sigma}\\,- 1}\\,\\,e^{-\\Gamma t}\\label{eqn40b}\\\\\n\\Omega(t)&=&2e^{\\Gamma{t}}\\sqrt{M\\,[\\Delta\\,-M{\\omega_0}^2]}.\n\\label{eqn41b}\n\\end{eqnarray}\nSubstituting these relations in the expression for $c(t)$ in Eqn.(\\ref{eqn3}), we \nget,\n\\begin{equation}\nc(t)= \\sqrt{\\dfrac{ \\Delta\\,-M{\\omega_0}^2\\, }{M}} \\,+\\,\\omega_0\\sqrt{M\\sigma-1}\\,=\\,constant ~. \\label{eqn43}\n\\end{equation}\nSubstituting the expressions of $a(t)\\,,\\rho(t)$\\, and $c(t)$\\, in Eqn.(\\ref{eqn31})\\,,\\,we get an expression for the phase in a closed form as,\n\\begin{align}\n\\Theta_{\\,n\\,,\\,l}(t)\\,=&(\\,n\\,+\\,l\\,)\\left[-\\frac{\\sigma}{\\mu^2}\\,+\\,\\sqrt{\\frac{ \\Delta\\,-M{\\omega_0}^2\\,}{M} } \\,+\\omega_0\\sqrt{M\\sigma-1} \\right]\\,t ~.\\label{eqn44}\n\\end{align}\nOnce again we are able to obtain an exact expression for the phase, in this case varying linearly with time.\nIt is important to note that the reality of the phase in this case depends crucially on the parameters $\\Delta$, $M$, $\\sigma$, $\\omega_0$. The phase $\\Theta_{n,l}$ is real if $\\Delta-M\\omega_{0}^2 \\geq 0$ and $M\\sigma \\geq 1$, else it is complex.\\\\\n\n\n\n\\noindent {\\bf $\\langle C\\rangle$ Solution Set-Ic}\n\n\n\\noindent Here the oscillator is damped due to the damping factor $f(t)$ and the time-dependent \nfrequency $\\omega(t)$; both of which are exponentially decaying. Thus, we set, \n\\begin{eqnarray}\nf(t)=exp\\,(-\\Gamma{t})~;~\\omega(t)={\\omega_0}\\,exp\\,(-\\Gamma{t}\/2).\n\\label{eqn45}\n\\end{eqnarray}\nSubstituting these relations in Eqn.(s)(\\ref{3}, \\ref{4}), we get the time dependent NC \nparameters to be, \n\\begin{eqnarray}\n\\theta(t)&=&\\dfrac{2}{M\\omega_0}\\sqrt{(M{\\sigma}e^{-(\\vartheta-\\Gamma)t}\\,- 1)}e^{-\\Gamma{t}\/2}\\label{eqn46}\\\\\n\\Omega(t)&=&2\\sqrt{M\\,[\\Delta\\,exp(\\vartheta{t})-M{\\omega_0}^2\\, \\,]}\\,\\,e^{\\Gamma t\/2}.\\label{eqn47}\n\\end{eqnarray}\nIt can be checked that in the limit $\\Gamma\\rightarrow0$, that is, for \nconstant frequency, the expressions for $\\theta(t)$ and $\\Omega(t)$ reduce \nto those in \\cite{Dey}. When $\\vartheta=\\Gamma$, then the solutions take the \nform,\n\\begin{eqnarray}\n\\theta(t)&=&\\dfrac{2}{M\\omega_0}\\sqrt{(M{\\sigma}\\,- 1)}e^{-\\Gamma{t}\/2}\\,\\,\\label{eqn46}\\\\\n\\Omega(t)&=&2\\sqrt{M\\,[\\Delta\\,exp(\\Gamma{t})-M{\\omega_0}^2\\, \\,]}\\,\\,e^{\\Gamma t\/2}.\\label{eqn47}\n\\end{eqnarray}\nSubstituting these relations in the expression for $c(t)$ in Eqn.(\\ref{eqn3}), we \nget,\n\\begin{align}\nc(t)= \\sqrt{\\dfrac{ \\Delta\\,-M{\\omega_0}^2\\,\\exp\\left[-\\Gamma{t}\\right] }{M}} \n + {\\omega_0}\\,e^{-\\Gamma t\/2}\\sqrt{M{\\sigma}\\,-1}~.\\label{eqn49}\n\\end{align}\nSubstituting the expressions of $a(t)$, $\\rho(t)$ and $c(t)$\\, in Eqn.(\\ref{eqn31}), we obtain an expression for the phase in a closed form as,\n\\begin{eqnarray}\n\\Theta_{\\,n\\,,\\,l}(t)\\,&=&\\,\\dfrac{(\\,n+l\\,)}{\\Gamma\\,\\sqrt{M}}\\,\\left[\\,\\sqrt{\\Delta}\\,\\Gamma\\,t\\,+\\,2\\sqrt{\\Delta-M\\omega_0^2}\\,-2\\sqrt{\\Delta-M\\omega_0^2{\\exp\\,(-\\Gamma{t})}}\\right.\\nonumber\\\\\n&&\\left.+\\,2\\sqrt{\\Delta}\\,log\\,\\left(\\frac{\\Delta+\\sqrt{\\Delta[\\Delta-M\\omega_0^2\\,\\exp\\,(-\\Gamma{t}) ]}}{\\Delta+\\sqrt{\\Delta[\\Delta-M\\omega_0^2\\,]}}\\right) \\right]\\nonumber \\\\\n&&-\\,(\\,n+l\\,)\\left[\\dfrac{\\sigma\\,t}{\\mu^2}\\,+\\,\\dfrac{2}{\\Gamma}\\,\\omega_0\\,\\left(e^{-\\Gamma t\/2}-1\\right)\\sqrt{\\,M\\sigma-1} \\right].\\label{eqn50}\n\\end{eqnarray}\n\n\n\n\n\n\n\\subsection{Solution Set-II for Ermakov-Pinney equation: Rationally decaying solutions}\n\\subsubsection{The Solution Set}\nWe now consider rationally decaying solutions of the EP equation similar to that used in~\\cite{Dey} which is of the form,\n\\begin{eqnarray}\n&a(t)=\\dfrac{\\sigma\\,\\left(1+\\dfrac{2}{k}\\right)^{\\,(k+2)\/k}}{(\\Gamma{t}+\\chi)^{\\,(k+2)\/k}}\\nonumber \\\\ \\nonumber\\\\\n&b(t)=\\dfrac{\\Delta\\,\\left(\\dfrac{k}{k+2} \\right)^{(2-k)\/k} }{(\\Gamma{t}+\\chi)^{\\,(k-2)\/k}} \\,\\,\\,\\Rightarrow\\, \\,\\,\\,\\dfrac{\\Delta\\,\\left(1+\\dfrac{2}{k}\\right)^{\\,(k-2)\/k} }{(\\Gamma{t}+\\chi)^{\\,(k-2)\/k}}\\nonumber \\\\ \\nonumber\\\\\n&\\rho(t)=\\dfrac{\\mu\\left(1+\\dfrac{2}{k}\\right)^{1\/k} }{(\\Gamma{t}+\\chi)^{1\/k}}\n\\label{EPsoln2}\n\\end{eqnarray}\nwhere $\\sigma$, $\\Delta$, $\\mu$, $\\Gamma$ and $\\chi$ are constants such that $(\\Gamma{t}+\\chi)~\\neq~0$, and $k$ is an integer. Substituting the expressions of $a(t)$, $b(t)$, and $\\rho(t)$ in the EP equation, we can easily verify the relation between these constants to be as follows, \n\\begin{equation}\n\\Gamma^2\\mu=(k+2)^2\\,(\\sigma\\Delta\\mu-\\frac{\\xi^2\\sigma^2}{\\mu^3}).\n\\label{EPreln2}\n\\end{equation}\n\n\n\n\\subsubsection{Study of the corresponding eigenfunctions}\nThe eigenfunction of the invariant operator $I(t)$ (given by \nEqn.(\\ref{eqn28})) for this solution Set-II is given by,\n\\begin{eqnarray}\n\\phi_{n\\,,\\,m-n}(r,\\theta)=\\lambda_{n}\\,\\dfrac{{(i\\mu)}^{\\,m}}{\\sqrt{m!}}\\left[\\dfrac{k+2}{k(\\Gamma{t}+\\chi)}\\right]^{m\/k} r^{n-m}e^{i\\theta(m-n)-\\dfrac{[\\sigma\\,(k+2)\\,+\\,i\\mu^2\\Gamma]\\,\\,(\\Gamma{t}+\\chi)^{2\/k}\\,\\,\\,k^{2\/k} }{2\\sigma\\,(k+2)^{\\,(k+2)\/k}\\mu^2}r^2}\\nonumber \\\\\n\\times\\,\\,\\,U\\left(-m,1-m+n,\\,\\dfrac{r^2[k(\\Gamma{t}+\\chi)]^{2\/k}}{\\mu^2\\left(k+2\\right)^{2\/k}}\\,\\right)\n\\label{eqn51}\n\\end{eqnarray}\nwhere $\\lambda_n$ is given by \n\\begin{eqnarray}\n\\lambda_n^{\\,2}=\\dfrac{1}{\\pi\\,n!\\mu^{2n+2}}\\left[\\dfrac{k(\\Gamma{t}+\\chi)}{k+2}\\right]^{2(1+n)\/k}.\n\\label{eqn51lam}\n\\end{eqnarray}\nIn order to get the eigenfunction of the Hamiltonian $H(t)$, we need to calculate the associated phase factor. Once again for this we need to fix up the forms of the damping factor $f(t)$ and angular frequency $\\omega(t)$ of the oscillator. In order to \nexplore the solution of $H(t)$ for rationally decaying coefficients, we choose a rationally decaying form for $\\omega(t)$ \nand set $f(t)=1$. Thus, we have the following relations, \n\\begin{eqnarray}\n\\eta(t)=0\\,\\,\\Rightarrow\\,\\,f(t)=1\\\\\n\\omega(t)=\\dfrac{\\omega_0}{(\\Gamma\\,t+\\chi)}~.\n\\end{eqnarray}\nSubstituting these relations in Eqns.(\\ref{3}, \\ref{4}), we get the time dependent NC parameters as, \n\\begin{eqnarray}\n\\theta(t)&=& \\dfrac{2\\,(\\Gamma\\,t+\\chi)}{M\\,\\omega_0}\\,\\sqrt{M\\sigma\\,\\left[\\dfrac{(k+2)}{k\\,(\\Gamma{t}\\,+\\,\\chi)}\\right]^{(k+2)\/k}\\,-\\,1} \\label{eqn52} \\\\ \\nonumber \\\\\n\\Omega(t)&=& \\,2\\,\\sqrt{M\\Delta\\,\\left[\\dfrac{k+2}{k(\\Gamma{t}+\\chi)}\\right]^{\\,(k-2)\/k}\\,-\\,\\dfrac{M^{\\,2}\\omega_0^{\\,2}}{(\\Gamma\\,t+\\chi)^2}}~.\\label{eqn53}\n\\end{eqnarray}\nWe now consider $k=2$. This enables us to integrate the expression for the phase factor (given by Eqn.(\\ref{eqn31})).\nThe simplified forms of $a(t)$, $b(t)$ and $\\rho(t)$ for $k=2$ read,\n\\begin{eqnarray}\na(t)=\\dfrac{4\\sigma}{(\\Gamma{t}+\\chi)^{\\,2}}\\,\\,,\\,\\,b(t)\\,=\\,\\Delta\\,\\,,\\,\\,\\rho(t)=\\left[\\dfrac{2\\mu^{\\,2}}{\\Gamma{t}+\\chi}\\right]^{1\/2}.\n\\end{eqnarray}\nSubstituting these relations in the expression for $c(t)$ in Eqn.(\\ref{eqn3}) gives, \n\\begin{equation}\nc(t)\\,=\\,\\dfrac{\\omega_0}{(\\Gamma\\,t+\\chi)}\\,\\sqrt{\\dfrac{4\\sigma\\,M}{(\\Gamma{t}+\\chi)^2}\\,-\\,1}\\,+\\,\\sqrt{\\dfrac{\\Delta}{M}\\,-\\,\\dfrac{\\omega_0^{\\,2}}{(\\Gamma\\,t+\\chi)^{\\,2}}}\\,\\,\\,\\,.\n\\end{equation} \nSubstituting these expressions for $a(t)$, $\\rho(t)$ and $c(t)$ \nfor $k=2$ in Eqn.(\\ref{eqn31}), we get the following expression \nfor the phase factor in a closed form as,\n\\begin{eqnarray}\n\\Theta_{\\,n, l\\,}(t)&=&\\dfrac{(n+l)}{\\Gamma}\\,\\left[\\omega_0\\,\\,tan^{\\,-1}\\left(\\dfrac{\\omega_0}{\\sqrt{\\frac{\\Delta}{M}{(\\Gamma\\,t+\\chi)^2}-\\omega_0^2}}\\right)+\\sqrt{\\dfrac{\\Delta\\,{(\\Gamma\\,t+\\chi)^2}}{M}-\\omega_0^2}\\,-\\,\\frac{2\\sigma}{\\mu^2}\\,log_{e}\\,\\frac{(\\chi+\\Gamma\\,t)}{\\chi}\\right.\\nonumber\\\\\n&&\\left.-{\\sqrt{\\frac{\\Delta}{M}{\\chi^2}-\\omega_0^2}}-\\omega_0\\,\\,tan^{\\,-1}\\left(\\dfrac{\\omega_0}{\\sqrt{\\frac{\\Delta}{M}{\\chi^2}-\\omega_0^2}}\\right)\\right] \\nonumber\\\\\n&& +\\dfrac{\\omega_0(n+l)}{\\Gamma}\\left[\\dfrac{\\sqrt{4\\,\\sigma\\,M-\\chi^2}}{\\chi}-\\dfrac{\\sqrt{4\\,\\sigma\\,M-(\\chi+\\Gamma\\,t)^2}}{(\\chi+\\Gamma\\,t)}\n\\right.\\nonumber\\\\\n&&\\left. +ilog_{e}\\dfrac{(\\chi+\\Gamma\\,t)+{\\sqrt{(\\chi+\\Gamma\\,t)^2-4\\,\\sigma\\,M}}}{\\chi+\\sqrt{\\chi^2-4\\,\\sigma\\,M}} \\right].\n\\label{eqn55}\n\\end{eqnarray}\nWe can now get the eigenfunction of this rationally decaying damped system using \nEqn.(\\ref{eqnpsi}).\n\n\n\n\n\n\n\n\\subsection{Solution Set-III for Ermakov-Pinney equation: Elementary Solution}\n\\subsubsection{The Solution Set}\nWe now propose a simple method of obtaining a solution of the EP equation.\nThe method is as follows. Choosing $\\rho(t)$ to be any arbitrary time dependent function and taking it's time derivative as proportional to $a(t)$, that is,\n$a(t)=constant \\times\\dot{\\rho}$ and setting \n$b(t)=constant \\times \\dfrac{a}{\\rho^4}$, we observe that these would always satisfy the EP equation along with a certain constraint relation among the constants.\n\n\\noindent Here we consider a simple solution which is a special case of the above solution for the EP equation. We call this the elementary solution which reads,\n\\begin{eqnarray}\na(t)={\\sigma}\\,\\,\\,\\,,\\,\\,\\,b(t)=\\dfrac{{\\Delta}}{{(\\Gamma\\,t\\,+\\,\\chi)^4}}\\,\\,\\,,\\,\\,\\rho(t)=\\mu(\\Gamma{t}\\,+\\,\\chi)\n\\label{eqn57}\n\\end{eqnarray}\nwhere $\\Gamma$, $\\chi$, $\\mu$, $\\sigma$ and $\\Delta$ are constants. The above solution set satisfy the EP equation with the following constraint relation,\n\\begin{equation}\n\\Delta\\mu^4=\\xi^2\\sigma\\,\\,.\n\\end{equation}\n\n\n\n\\subsubsection{Study of the corresponding eigenfunctions}\nThe eigenfunctions of the invariant operator $I(t)$ for this solution set is given by,\n\\begin{eqnarray}\n\\phi_{n,m-n}(r,\\theta)&=&\\lambda_{n}\\dfrac{{[i\\mu(\\Gamma\\,t+\\chi)]}^m}{\\sqrt{m!}}r^{n-m}e^{i\\theta(m-n)-\\dfrac{\\sigma-i\\mu^2\\Gamma(\\Gamma\\,t+\\chi)}{2\\sigma\\mu^2(\\Gamma\\,t+\\chi)^2}r^2}\\nonumber \\\\\n&&\\times\\,\\,U\\left(-m,1-m+n,\\dfrac{r^2}{\\mu^2(\\Gamma\\,t+\\chi)^2} \\right)\n\\label{eqn58}\n\\end{eqnarray}\nwhere $\\lambda_n$ is given by\n\\begin{eqnarray}\n\\lambda_n^2=\\dfrac{1}{\\pi{n!}{\\left[\\mu(\\Gamma\\,t+\\chi) \\right]}^{2+2n}}~. \n\\label{eqn58lam}\n\\end{eqnarray}\nIn order to get an eigenfunction of the Hamiltonian, we calculate the phase factor for a particular case of the damped harmonic oscillator where the angular frequency $\\omega(t)$ is rationally decaying and the damping factor $f(t)$=1. Thus, we set, \n\\begin{eqnarray}\n\\eta(t)=0\\,\\,\\Rightarrow\\,\\,f(t)=1\\\\\n\\omega(t)=\\dfrac{\\omega_0}{(\\Gamma\\,t+\\chi)}\n\\end{eqnarray}\nwhere $\\Gamma$ and $\\chi$ are real constants.\nSubstituting these relations in Eqns.(\\ref{3}, \\ref{4}), we get the time dependent NC parameters as, \n\\begin{eqnarray}\n\\theta(t)=\\dfrac{2\\,(\\Gamma\\,t+\\chi)}{\\omega_0\\,M}\\,\\sqrt{M\\,\\sigma-1}\\label{eqn59}\\\\\n\\Omega(t)=2\\sqrt{\\dfrac{M\\Delta}{(\\Gamma\\,t+\\chi)^4}-\\dfrac{M^2\\,\\omega_0^2}{(\\Gamma\\,t+\\chi)^{2}}}~.\\label{eqn60}\n\\end{eqnarray}\nSubstituting these relations \nin the expression for $c(t)$ in Eqn.(\\ref{eqn3}), we get, \n\\begin{align}\nc(t)=\\sqrt{\\dfrac{\\Delta}{M(\\Gamma\\,t+\\chi)^4}-\\dfrac{\\omega_0^2}{(\\Gamma\\,t+\\chi)^2}}\\,+\\,\\dfrac{\\omega_0}{(\\Gamma\\,t+\\chi)}\\sqrt{M\\,\\sigma-1}.\\label{eqn61}\n\\end{align}\nSubstituting these expressions of $a(t)$, $\\rho(t)$ and $c(t)$ in Eqn.(\\ref{eqn31}), we obtain an expression for the phase factor in a closed form as, \n\\begin{align}\n\\Theta_{\\,n\\,,\\,l}(t)&=\\,(n+l\\,)\\left[\\omega_0\\dfrac{\\sqrt{M\\,\\sigma-1}}{\\Gamma}\\log\\,\\dfrac{(\\Gamma\\,t+\\chi)}{\\chi}-\\frac{{\\sigma}t}{\\mu^2\\chi(\\Gamma\\,t+\\chi)} \\right] +\\,\\dfrac{(\\,n+l\\,)}{\\Gamma}\\left[\\sqrt{\\dfrac{\\Delta}{M\\chi^2}-\\omega_0^2}\\right.\\nonumber\\\\\n&\\left.-\\sqrt{\\dfrac{\\Delta}{M(\\Gamma\\,t+\\chi)^2}-\\omega_0^2}\n+\\omega_0\\left\\{\\tan^{-1}\\left(\\dfrac{\\omega_0\\chi}{\\sqrt{\\dfrac{\\Delta}{M}-\\chi^2\\omega_0^2}}\\right)-\\tan^{-1}\\left(\\dfrac{\\omega_0(\\Gamma\\,t+\\chi)}{\\sqrt{\\dfrac{\\Delta}{M}-\\omega_0^2(\\Gamma\\,t+\\chi)^2}}\\right)\\right\\}\\,\\, \\right].\n\\label{eqn62}\n\\end{align} \nWe can now get the eigenfunction of this system by using Eqn.(\\ref{eqnpsi}).\n\n\n\n\\section{Expectation Values}\nIn this section, we intend to calculate the expectation value of energy. For \nthis we need to calculate the expectation value of the Hamiltonian $H(t)$ in \nit's own eigenstates. \nThe expectation value $\\langle H \\rangle$ is given by (using Eqn.(\\ref{eqn2})), \n\\begin{equation}\n\\langle H\\rangle = \\dfrac{a(t)}{2}(\\langle{p_1}^2\\rangle+\\langle{p_2}^2\\rangle)+\\dfrac{b(t)}{2}(\\langle{x_1}^2\\rangle+\\langle{x_2}^2\\rangle)+c(t)(\\langle{p_1}{x_2}\\rangle-\\langle{p_2}{x_1}\\rangle)\\,\\,\\,.\\label{eqn63}\n\\end{equation}\nTo calculate this we need to get the expectation value of the individual canonical \noperators. To set up our notation we denote the eigenstates of the Hamiltonian $H(t)$ \nby $|n,l\\rangle_H$. \n\n\n\n\n\\subsection{Matrix elements of the coordinate operators raised to arbitrary finite powers}\n\\noindent We start by calculating the matrix element of an arbitrary power of $x$, $_{H}\\langle n,l|x^k|n,l\\rangle_{H}$, which is given by\n\\begin{eqnarray} \n_{H}\\langle n,m-n|x^k|n,m'-n\\rangle_{H}&=&\\int r dr d\\theta\n~_{H}\\langle n,m-n |r,\\theta\\rangle\\langle r,\\theta|r^k cos^k\\theta|n,m'-n\\rangle_{H}\\nonumber\\\\\n&=&\\dfrac{1}{2^k}e^{i(\\Theta_{n,m'-n}-\\Theta_{n,m-n})}\\int r^{k+1} dr d\\theta~(e^{i\\theta}+e^{-i\\theta})^k \\nonumber\\\\\n&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times\\phi^*_{n,m-n}(r,\\theta)\\phi_{n,m'-n}(r,\\theta)\\nonumber\\\\ \n\\label{eqn64}\n\\end{eqnarray} \nwhere we have used the relations, \n$|n,l\\rangle_{H}~=~e^{i\\Theta_{n,l}}|n,l\\rangle$ where $|n,l\\rangle_H$ and \n$|n,l\\rangle$ are eigenstates of the Hamiltonian $H(t)$ and Lewis invariant $I(t)$ \nrespectively. We have also used the relation $\\langle r,\\theta|n,m'-n\\rangle$~=~$\\phi_{n,m'-n}(r,\\theta)$, with $\\phi$ being the eigenfunction of $I(t)$.\nNow, Eqn.(\\ref{eqn64}) can be rewritten as,\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^k|n,m'-n\\rangle_{H}&=&\\dfrac{\\pi}{2^{k-1}} \\sum_{r=0}^{k} {^{k}C_r}\\delta_{m',m+2r-k} A(n,m,m+2r-k)\\nonumber\\\\\n&\\times& \\int_0^\\infty r~dr~r^{2(n-m-r+k)}\ne^{\\dfrac{-r^2}{\\hbar \\rho^2}} \\nonumber\\\\\n&\\times& U\\left(-m,1-m+n,\\dfrac{r^2}{\\hbar\\rho^2} \\right)\\nonumber\\\\\n&\\times& U\\left(-m-2r+k,1-m-2r+k+n,\\dfrac{r^2}{\\hbar\\rho^2} \\right)\n\\label{eqn67}\n\\end{eqnarray}\nwhere $A(n,m,m+2r-k)=e^{i(\\Theta_{n,m-n+2r-k}-\\Theta_{n,m-n})}\\lambda_n^2\\dfrac{(-i\\hbar^{1\/2}\\rho)^{m}(i\\hbar^{1\/2}\\rho)^{m+2r-k}}{\\sqrt{m!(m+2r-k)!}}$~. \n\n\\noindent Now defining $w=-\\dfrac{r^2}{\\hbar \\rho^2}$, we have, \n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^k|n,m'-n\\rangle_{H}&=&\\sum_{r=0}^{k}\\frac{\\pi}{2^k}e^{i(\\Theta_{n,m-n+2r-k}-\\Theta_{n,m-n})}(-1)^{k+r}i^{-k}~{^{k}C_r}\\delta_{m',m+2r-k}\\nonumber\\\\\n&\\times& \\lambda_n^2(\\hbar^{1\/2} \\rho)^{2n+k+2}\\sqrt{m!(m+2r-k)!}\\nonumber\\\\\n&\\times& \\int_0^\\infty dw w^{n-m-r+k}e^{-w} L_m^{(n-m)}(w) L_{m+2r-k}^{(n-m-2r+k)}(w)\n\\label{eqn68}\n\\end{eqnarray}\nwhere we have used the following result on special functions \\cite{Arfken, uva},\n\\begin{eqnarray}\nL_n^{(\\zeta)}(w)=\\frac{(-1)^n}{n!}U(-n,\\zeta+1,w) \\label{eqn69}\n\\label{eqn70}\n\\end{eqnarray}\nwhere $L_n^{(\\zeta)}(w)$ are associated Laguerre polynomials.\n\n\\noindent Now, we get using the relation for phase given in \\cite{Dey},\n\\begin{eqnarray}\n\\Theta_{n,l}=(n+l)\\int^t \\left[c(\\tau)-\\frac{a(\\tau)}{\\rho^2 (\\tau)}\\right]d\\tau\n\\label{eqn73}\n\\end{eqnarray} \nthe following relation,\n\\begin{eqnarray}\ne^{i(\\Theta_{n,m-n+2r-k}-\\Theta_{n,m-n})}&=&e^{i[\\{(n+m-n+2r-k)-(n+m-n)\\}\\int^t (c(\\tau)-\\frac{a(\\tau)}{\\rho^2 (\\tau)})d\\tau]} \\nonumber\\\\\n&=& e^{i(0+2r-k)\\int^t (c(\\tau)-\\frac{a(\\tau)}{\\rho^2 (\\tau)})d\\tau}\\nonumber\\\\\n&=&e^{i\\Theta_{0,2r-k}}.\n\\label{eqn74}\n\\end{eqnarray}\nSo, we finally get the following relation for the matrix element of $x^k$,\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^k|n,m'-n\\rangle_{H}&=&\\sum_{r=0}^{k}\\frac{\\pi}{2^k}e^{i\\Theta_{0,2r-k}}(-1)^{k+r}i^{-k}~{^{k}C_r}\\delta_{m',m+2r-k}\\nonumber\\\\\n&\\times& \\lambda_n^2(\\hbar^{1\/2} \\rho)^{2n+k+2}\\sqrt{m!(m+2r-k)!}\\nonumber\\\\\n&\\times& \\int_0^\\infty dw~w^{n-m-r+k}e^{-w} L_m^{(n-m)}(w) L_{m+2r-k}^{(n-m-2r+k)}(w).\n\\label{eqn75}\n\\end{eqnarray}\nThis is a new result in this paper and can be used to obtain the matrix element or expectation value of any power of $x$.\nFor the sake of completeness, we also write down the matrix element of $x^k$ in the eigenstates of the Lewis invariant $I(t)$, which reads\n\\begin{eqnarray}\n\\langle n,m-n|x^k|n,m'-n\\rangle&=&\\sum_{r=0}^{k}\\frac{\\pi}{2^k}(-1)^{k+r}i^{-k}~{^{k}C_r}\\delta_{m',m+2r-k}\\nonumber\\\\\n&\\times& \\lambda_n^2(\\hbar^{1\/2} \\rho)^{2n+k+2}\\sqrt{m!(m+2r-k)!}\\nonumber\\\\\n&\\times& \\int_0^\\infty dw~w^{n-m-r+k}e^{-w} L_m^{(n-m)}(w) L_{m+2r-k}^{(n-m-2r+k)}(w).\n\\label{eqn75I}\n\\end{eqnarray}\nNote that the phase factor does not appear in the above result.\n\n\\noindent Now, we proceed to evaluate the matrix element \n$_{H}\\langle n,m-n|x|n,m'-n\\rangle_{H}$ using the expression \nobtained in Eqn.(\\ref{eqn75}). This reads\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x|n,m'-n\\rangle_{H}=\n_{H}\\langle n,m-n|x^k\\mid_{\\,k=1;\\,r=0}|n,m'-n\\rangle_{H}\\nonumber\\\\\n+ _{H}\\langle n,m-n|x^k\\mid_{\\,k=1;\\,r=1}|n,m'-n\\rangle_{H}.\n\\label{eqn76}\n\\end{eqnarray}\nEvaluating the above matrix elements give,\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^k\\mid_{\\,k=1;\\,r=0}|n,m'-n\\rangle_{H}= -\\frac{i}{2}\n{(\\rho\\hbar^{1\/2})}\\sqrt{m}e^{-i\\Theta_0,1}\\delta_{m,m'+1}\n\\label{eqn77}\n\\end{eqnarray}\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^k\\mid_{\\,k=1;\\,r=1}|n,m'-n\\rangle_{H}=\\frac{i}{2}\n{(\\rho\\hbar^{1\/2})}\\sqrt{m'}e^{i\\Theta_0,1}\\delta_{m',m+1}.\n\\label{eqn78}\n\\end{eqnarray}\nIn order to obtain Eqn(s).(\\ref{eqn77}, \\ref{eqn78}), we \nused the following relations involving the associated Laguerre polynomials,\n\\begin{eqnarray}\nL_n^{(\\zeta)}(w)=L_n^{(\\zeta+1)}(w)-L_{n-1}^{(\\zeta+1)}(w) \\nonumber\\\\\n\\int_0^\\infty dw~w^{\\zeta} e^{-w}L_n^{(\\zeta)} (w)L_m^\\zeta (w) =\\frac{(n+\\zeta)!}{n!}\\delta_{n,m}~.\n\\label{eqn80}\n\\end{eqnarray}\nCombining Eqn(s).(\\ref{eqn77}, \\ref{eqn78}), we get the following expression,\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x|n,m'-n\\rangle_{H}~=~\\frac{i}{2}\n{(\\rho\\hbar^{1\/2})}[\\sqrt{m'}e^{i\\Theta_{0,1}}\\delta_{m',m+1}\n-\\sqrt{m}e^{-i\\Theta_{0,1}}\\delta_{m,m'+1}].\n\\label{eqn79}\n\\end{eqnarray}\nNext, we evaluate,\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^2|n,m'-n\\rangle_{H}= _{H}\\langle n,m-n|x^k\\mid_{\\,k=2;\\,r=0}|n,m'-n\\rangle_{H}\\nonumber\\\\\n+_{H}\\langle n,m-n|x^k\\mid_{\\,k=2;\\,r=1}|n,m'-n\\rangle_{H}+ _{H}\\langle n,m-n|x^k\\mid_{\\,k=2;\\,r=2}|n,m'-n\\rangle_{H}. \n\\label{eqn81}\n\\end{eqnarray}\nEvaluation of the above matrix elements yield, \n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^k\\mid_{\\,k=2;\\,r=0}|n,m'-n\\rangle_{H}&=& \n-\\frac{1}{4}{(\\hbar\\rho^2)}e^{-i\\Theta_{0,2}}\\delta_{m',m-2}\\sqrt{m(m-1)}\\nonumber\\\\\n_{H}\\langle n,m-n|x^k\\mid_{\\,k=2;\\,r=1}|n,m'-n\\rangle_{H}&=&~\\frac{1}{2}{(\\hbar\\rho^2)}e^{-i\\Theta_{0,0}}\\delta_{m,m'}(m+n+1)\\nonumber\\\\\n_{H}\\langle n,m-n|x^k\\mid_{\\,k=2;\\,r=2}|n,m'-n\\rangle_{H}&=& \n-\\frac{1}{4}{(\\hbar\\rho^2)}e^{i\\Theta_{0,2}}\\delta_{m',m+2}\\sqrt{(m+2)(m+1)}.\n\\label{eqn82}\n\\end{eqnarray}\nIn order to calculate the above expressions, apart from the relations between \nspecial functions given by Eqn.(\\ref{eqn80}), we need the following relation,\n\\begin{eqnarray}\n\\int_{0}^{\\infty} dw~w^{k+p}e^{-w}L_n^k(w)L_n^k(w)~=~\\frac{(n+k)!}{n!}\\times(2n+k+1)^p ~.\n\\label{eqn83}\n\\end{eqnarray}\nSo we have,\n\\begin{eqnarray}\n_{H}\\langle n,m-n|x^2|n,m'-n\\rangle_{H}&=&\\frac{(\\hbar\\rho^2)}{2}\\delta_{m,m'}(m+n+1)\\nonumber\\\\\n&&-\\frac{(\\hbar\\rho^2)}{4}\\left[e^{-i\\Theta_{0,2}}\\delta_{m',m-2}\\sqrt{m(m-1)}\\right.\\nonumber\\\\\n&& \\left. +e^{i\\Theta_{0,2}}\\delta_{m',m+2}\\sqrt{(m+2)(m+1)}\\right].\n\\label{eqn84}\n\\end{eqnarray}\nIt is to be noted that the matrix elements for $x$ and $x^2$ in the \neigenstates of the Hamiltonian [given by Eqn(s).(\\ref{eqn79}, \\ref{eqn84}) \nrespectively], matches exactly with the corresponding expression \ngiven in \\cite{Dey}, although the result \nquoted in \\cite{Dey} is in the eigenstate of the invariant $I(t)$.\n\n\\noindent The matrix element of $y^k$ in the \neigenstates of the Hamiltonian can be obtained similarly, and reads,\n\\begin{eqnarray}\n_{H}\\langle n,m-n|y^k|n,m'-n\\rangle_{H}&=&\\sum_{r=0}^{k}\\frac{\\pi}{2^k}e^{i\\Theta_{0,2r-k}}~{^{k}C_r}\\delta_{m',m+2r-k}\\nonumber\\\\\n&\\times& \\lambda_n^2(\\hbar^{1\/2} \\rho)^{2n+k+2}\\sqrt{m!(m+2r-k)!}\\nonumber\\\\\n&\\times& \\int_0^\\infty dw~w^{n-m-r+k}e^{-w} L_m^{(n-m)}(w) L_{m+2r-k}^{(n-m-2r+k)}(w).\n\\label{eqn85}\n\\end{eqnarray}\nOnce again we write down the matrix element of $y^k$ in the eigenstates of the Lewis invariant $I(t)$. This reads\n\\begin{eqnarray}\n\\langle n,m-n|y^k|n,m'-n\\rangle&=&\\sum_{r=0}^{k}\\frac{\\pi}{2^k}~{^{k}C_r}\\delta_{m',m+2r-k}\\nonumber\\\\\n&\\times& \\lambda_n^2(\\hbar^{1\/2} \\rho)^{2n+k+2}\\sqrt{m!(m+2r-k)!}\\nonumber\\\\\n&\\times& \\int_0^\\infty dw~w^{n-m-r+k}e^{-w} L_m^{(n-m)}(w) L_{m+2r-k}^{(n-m-2r+k)}(w).\n\\label{eqn85I}\n\\end{eqnarray}\nUsing Eqn.(\\ref{eqn85}), we may evaluate the matrix element of $y$ and $y^2$ \nin the eigenstate of the Hamiltonian. We find, \n\\begin{eqnarray}\n_{H}\\langle n,m-n|y|n,m'-n\\rangle_{H}&=&\n_{H}\\langle n,m-n|y^k\\mid_{\\,k=1;\\,r=0}|n,m'-n\\rangle_{H}\\nonumber\\\\&&+ _{H}\\langle n,m-n|y^k\\mid_{\\,k=1;\\,r=1}|n,m'-n\\rangle_{H}\n\\nonumber\\\\\n&=&-\\frac{1}{2}\n{(\\rho\\hbar^{1\/2})}[\\sqrt{m}e^{-i\\Theta_{0,1}}\\delta_{m',m-1}\n+\\sqrt{m+1}e^{i\\Theta_{0,1}}\\delta_{m',m+1}].\n\\label{eqn86}\n\\end{eqnarray}\n\\begin{eqnarray}\n_{H}\\langle n,m-n|y^2|n,m'-n\\rangle_{H}&=&_{H}\\langle n,m-n|y^k\\mid_{\\,k=2;\\,r=0}|n,m'-n\\rangle_{H}\\nonumber\\\\\n&&+_{H}\\langle n,m-n|y^k\\mid_{\\,k=2;\\,r=1}|n,m'-n\\rangle_{H}\\nonumber\\\\\n&&+_{H}\\langle n,m-n|y^k\\mid_{\\,k=2;\\,r=2}|n,m'-n\\rangle_{H} \\nonumber\\\\\n&=&\\frac{\\hbar\\rho^2}{4}\\delta_{m',m-2}\\sqrt{m(m-1)}e^{-i\\Theta_{0,2}}\\nonumber\\\\\n&&+\\frac{1}{2}\\delta_{m,m'}{(\\hbar\\rho^2)}(m+n+1)+\\frac{\\hbar\\rho^2}{4}\\delta_{m',m+2}\\sqrt{(m+2)(m+1)}e^{i\\Theta_{0,2}}.\\nonumber\\\\\n\\label{eqn87}\n\\end{eqnarray}\nFrom the above analysis, we find that even the expression for the matrix element of the operator $y^k$ in the eigenstate of $H(t)$ \nmatches with that found in \\cite{Dey} for $k=1, 2$, \nthough again they had inappropriately quoted the results in the \neigenstate of the Lewis invariant. \n\n\n\n\\subsection{Analysis of the expectation value of energy}\nAs we have already seen from Eqn.(\\ref{eqn2}), in order to calculate the expectation value of energy one needs the expectation values $\\braket{{p_1}^2}$, $\\braket{{p_2}^2}$, $\\braket{{x_1}^2}$, $\\braket{{x_2}^2}$, $\\braket{{p_1}{x_2}}$ and \n$\\braket{{p_2}{x_1}}$. As we have seen in the previous subsection, our calculated generalized expressions for matrix elements $_{H}\\langle n,m-n|x^k\\mid_{\\,k=1;\\,r=0}|n,m'-n\\rangle_{H}$ and $_{H}\\langle n,m-n|y^k\\mid_{\\,k=1;\\,r=0}|n,m'-n\\rangle_{H}$ matched exactly with the calculations in \\cite{Dey} for $k=1, 2$. Hence, we use the matrix elements quoted in the said work to calculate \nthe following expectation values,\n\\begin{eqnarray}\n\\braket{x_j^2}=\\dfrac{\\rho^2}{2}(n+m+1)\\,\\,;\\,\\,\\braket{p_j^2}=\\dfrac{1}{2}\\left(\\dfrac{1}{\\rho^2}+\\dfrac{\\dot{\\rho}^2}{a^2} \\right)\\,(n+m+1)\\,\\,;\\,\\,\\braket{x_j\\,p_k}=\\dfrac{1}{2}\\,\\epsilon_{jk}(m-n)\\,\\,;\\label{eqn88}\n\\end{eqnarray}\nwhere $j,k=1,2$ and $\\epsilon_{jk}=-\\epsilon_{kj}$ with $\\epsilon_{12}=1$. So, the expectation value of energy $\\braket{E_{n,m-n}(t)}$ with \nrespect to energy eigenstate $\\psi_{n,m-n}(r,\\theta,t)$ can be expressed as,\n\\begin{align}\n&\\braket{E_{n,m-n}(t)}=\\dfrac{1}{2}\\,(n+m+1)\\left[b(t)\\rho^2(t)+\\dfrac{a(t)}{\\rho^2(t)}+\\dfrac{\\dot{\\rho}^2(t)}{a(t)} \\right]+c(t)\\,(n-m)\\,\\,.\\nonumber\\\\\n&=\\dfrac{1}{2}\\left[\\,(n+m+1)\\left(b(t)\\rho^2(t)+\\dfrac{a(t)}{\\rho^2(t)}+\\dfrac{\\dot{\\rho}^2(t)}{a(t)} \\right)+(n-m)\\left(\\dfrac{f(t)\\Omega(t)}{M}+\\dfrac{M\\omega^2(t)\\theta(t)}{f(t)}\\right) \\right].\n\\label{eqn89}\n\\end{align}\n\n\n\n\n\\noindent It is interesting to note that even when the frequency of oscillation $\\omega{\\rightarrow}0$, the \nexpectation value of energy is non-zero. This is because all the three parameters of the Hamiltonian $a(t)$, $b(t)$ and $c(t)$ are finite even as $\\omega{\\rightarrow}0$, as is clear from the Eqn(s).(\\ref{3},\\ref{4},\\ref{eqn3}). Now we will proceed to study the time-dependent behaviour of $\\braket{E_{n,m-n}(t)}$ for various types of damping.\n\n\n\n\n\\subsubsection{Exponentially decaying solution}\nFor the exponentially decaying solution given by Eqn.(\\ref{EPsoln1}), the energy expectation value takes the following form,\n\\begin{equation}\n\\braket{E_{n,m-n}(t)}=(n+m+1)\\mu^2\\Delta+c(t)\\,(n-m)\n\\label{eqn90}\n\\end{equation}\nwhere we have set the constant $\\xi^2$ to unity and used the constraint relation given by Eqn.(\\ref{EPreln1}).\n\n\\vskip 0.1cm\n\n\n\n\\noindent{\\bf $\\langle A\\rangle$ Solution Set-Ia}\n\\vskip 0.15cm\n\n\\noindent For this case we consider $f(t)=1$ and $\\omega(t)=\\omega_0\\,e^{-\\Gamma\\,t\/2}$. The expectation value of \nenergy for the ground state has the following expression,\n\\begin{eqnarray}\n\\braket{E_{n,-n}(t)}&=&(n+1)\\mu^2\\Delta+\\,n\\,\\left[\\sqrt{\\dfrac{ \\Delta\\,exp\\,(\\Gamma{t})-M{\\omega_0}^2\\,exp\\,(-\\Gamma{t}) }{M}} \n\\,\\right.\\nonumber\\\\\n&&~~~~~~~~~~~~~~~~~~~~~~~~~~\\left. +\\,\\omega_0\\,exp\\,(-\\Gamma{t\/2})\\sqrt{\\,M\\sigma\\,exp\\,(-\\Gamma{t})-1 \\,}\\right].\n\\label{eqn91}\n\\end{eqnarray}\nFrom Eqn.(\\ref{eqn91}), we see that the expectation value of the energy becomes complex beyond a certain time limit. The \ncondition for getting the expectation value of energy to be real is as follows,\n\\begin{eqnarray}\nM\\,\\sigma\\,e^{-\\Gamma\\,t}>1\\,\\,\\Rightarrow\\,t\\leq\\,\\dfrac{ln(M\\,\\sigma)}{\\Gamma}~.\\label{eqn92}\n\\end{eqnarray}\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.4]{expdecay.eps}\n\\caption{\\textit{A study of the variation of expectation value of energy, scaled by \n$\\frac{1}{\\omega_0}$ ($\\frac{\\langle E \\rangle}{\\omega_0}$) in order to make \nit dimensionless, as we vary $\\Gamma$t (again a dimensionless quantity). Here \nwe consider mass M=1, $\\mu$=1,$\\Delta$=$10^7$, $\\sigma$=$10^7$, \n$\\omega_0$=$10^3$ and $\\Gamma$=1 in natural units. The expectation value of \nenergy $\\langle E \\rangle$ is calculated for exponentially decaying \nHamiltonian parameters when $\\langle A\\rangle$ Set-IA $f(t)=1$ and \n$\\omega(t)=\\omega_0 e^{-{\\Gamma}t\/2}$; $\\langle B\\rangle$ Set-IB \n$f(t)=e^{-{\\Gamma}t}$ and $\\omega(t)=\\omega_0$ and $\\langle C\\rangle$ \nSet-IC $f(t)=e^{-{\\Gamma}t}$ and $\\omega(t)=\\omega_0 e^{-{\\Gamma}t\/2}$. While \nfor $\\langle A\\rangle$ the energy first decreases, then increases with time, \nfor $\\langle B\\rangle$ the energy remains constant as we vary time. \nFor $\\langle C\\rangle$ the energy decays off with time.}} \n\\label{fig1}\n\\end{figure}\n\n\n\n\\noindent We see from Fig.(\\ref{fig1}), that the energy initially decays but then increases with time. This is \nbecause for large time at which $exp\\,(-\\Gamma{t\/2})\\approx\\,0 $, the \napproximated expression of energy reads\n\\begin{equation}\nE_{n,-n}(t)\\approx\\,(n+1)\\mu^2\\Delta+\\,n\\,\\sqrt{\\dfrac{ \\Delta\\,exp\\,(\\Gamma{t}) }{M}}\\,\\,\n\\label{e7}\n\\end{equation}\nwhich is still increasing with time. The reason for the increase of energy with time is the form of the \ncoefficient $b(t)$ in the Hamiltonian. Although the coefficient $a(t)$ is exponentially decaying with \ntime, the coefficient $b(t)$ exponentially increases with time in order to satisfy EP equation. However, since there is an upper limit of time within which the energy remains real, so the energy remains finite within the allowed time interval.\n\n\\vskip 0.1cm\n\n\n\n\n\\noindent{{\\bf{$\\langle B\\rangle$ Solution Set-Ib}}}\n\n\\noindent Here we set $f(t)=e^{-\\Gamma\\,t}$ and $\\omega(t)=\\omega_0$. With this the energy expression for the ground state takes the form,\n\\begin{equation}\n\\braket{E_{n,-n}(t)}=(n+1)\\mu^2\\Delta+\\,n\\,\\left[\\sqrt{\\dfrac{ \\Delta\\,-M{\\omega_0}^2\\, }{M}} \\,+\\,\\omega_0\\sqrt{M\\sigma-1}\\right].\\label{e8}\n\\end{equation}\n\n\n\\noindent We note from Fig.(\\ref{fig1}), that the expectation value of the energy remarkably remains constant as we vary time, as is observed from Eqn.(\\ref{e8}). This must be because the effect of the exponentially decaying Hamiltonian \ncoefficient $a(t)$ and damping term $f(t)$ gets balanced out by the exponentially increasing Hamiltonian \ncoefficient $b(t)$.\n\\vskip 0.1cm\n\n\n\n\\noindent{{\\bf{$\\langle C\\rangle$ Solution Set-Ic}}}\n\n\\noindent Here we set $f(t)=e^{-\\Gamma\\,t}$ and $\\omega(t)=\\omega_0\\,e^{-\\Gamma\\,t\/2}$. \nWith this the expectation value of the energy expression takes the form,\n\\begin{equation}\n\\braket{E_{n,-n}(t)}=(n+1)\\mu^2\\Delta+\\,n\\,\\left[\\sqrt{\\dfrac{ \\Delta\\,-M{\\omega_0}^2\\,exp\\left[-\\Gamma{t}\\right] }{M}} \n + {\\omega_0}\\,exp\\,(-\\Gamma{t\/2})\\sqrt{M{\\sigma}\\,-1}\\right].\\label{e9}\n\\end{equation}\nThe above expression gives a very nice decaying expression for the expectation value of energy with respect to time, and finally approaching a constant value in the limit \n$t\\rightarrow\\infty$. This behaviour is also exhibited in the nature of the plot of variation of the expectation value of energy with time seen in Fig.(\\ref{fig1}).\n\n\\vskip 0.1cm\n\n\n\n\\subsubsection{Rationally decaying solution}\n\n\\noindent In this case the expectation value of energy for $k=2$ reads\n\\begin{eqnarray}\nE_{n,-n}(t)&=&\\dfrac{(n+1)}{2(\\Gamma\\,t+\\chi)}\\left[2\\left(\\dfrac{\\sigma}{\\mu^2}+\\Delta\\mu^2\\right)+\\dfrac{\\mu^2\\Gamma^2}{8\\sigma} \\right]\\nonumber\\\\\n&&~~~~~~~~~~~~~~~~~~~~~~~ +n\\left[\\dfrac{\\omega_0}{\\Gamma\\,t+\\chi}\\,\\sqrt{\\dfrac{4\\sigma\\,M}{(\\Gamma{t}+\\chi)^2}\\,-\\,1}\\,+\\,\\sqrt{\\dfrac{\\Delta}{M}-\\dfrac{\\omega_0^{\\,2}}{(\\Gamma\\,t+\\chi)^{\\,2}}} \\right].\\label{e10}\n\\end{eqnarray}\n\\noindent Note that although it has a nice decaying property like the damping case on commutative plane, there is an upper bound of time above which the energy ceases to be real. The upper bound on time reads,\n\\begin{eqnarray}\n4\\sigma\\,M\\,\\geq\\,(\\Gamma\\,t+\\chi)^2\\,\\,\\Rightarrow\\,\\,t\\,\\leq\\,\\dfrac{1}{\\Gamma}(2\\sqrt{M\\,\\sigma}-\\chi).\n\\label{e11}\n\\end{eqnarray}\n\n\n\\vskip 0.1cm\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.4]{ratdecay.eps}\n\\caption{\\textit{A study of the variation of expectation value of energy, scaled by \n$\\frac{1}{\\omega_0}$ ($\\frac{\\langle E \\rangle}{\\omega_0}$) in order to make \nit dimensionless, as we vary $\\Gamma$t (again a dimensionless quantity). Here \nwe consider mass M=1, $\\mu$=1,$\\Delta$=$10^7$, $\\sigma$=$10^7$, \n$\\omega_0$=$10^3$, $\\chi=1$ and $\\Gamma$=1 in natural units. The expectation value of \nenergy $\\langle E \\rangle$ is calculated for rationally decaying \nHamiltonian parameters. We consider $f(t)=1$ and $\\omega(t)=\\dfrac{\\omega_0}{(\\Gamma\\,t+\\chi)}$.}} \n\\label{fig2}\n\\end{figure}\n\n\\noindent From Fig.(\\ref{fig2}), we see indeed the expectation value of energy $\\langle E \\rangle$ decays with time \nfollowing power law as expected for the rationally decaying solutions. \n\n\n\\subsubsection{Elementary solution}\nFor the elementary solution set, the expectation value of the energy reads,\n\\begin{eqnarray}\n\\braket{E_{n,-n}(t)}&=&\\dfrac{1}{2}(n+1)\\left[\\left(\\Delta\\mu^2+\\dfrac{\\sigma}{\\mu^2}\\right)\\dfrac{1}{(\\Gamma\\,t+\\chi)^2}+\\dfrac{\\mu^2\\Gamma^2}{\\sigma}\\right]\n\\nonumber\\\\&&+n\\left[\\dfrac{\\omega_0\\sqrt{M\\sigma-1}}{(\\Gamma\\,t+\\chi)}+\\dfrac{1}{(\\Gamma\\,t+\\chi)}\\sqrt{\\dfrac{\\Delta}{M\\,(\\Gamma\\,t+\\chi)^2}-\\omega_0^2 } \\right].\\label{e12}\n\\end{eqnarray}\nFurther, the constraint relation $\\Delta\\mu^4=\\xi^2\\sigma$ results in the following form for the expectation value of energy (setting $\\xi^2=1$),\n\\begin{align}\n\\braket{E_{n,-n}(t)}\n&=\\dfrac{1}{2}(n+1)\\left[\\dfrac{2\\sigma}{\\mu^2(\\Gamma\\,t+\\chi)^2}+\\dfrac{\\mu^2\\Gamma^2}{\\sigma}\\right]+n\\,\\left[\\dfrac{\\omega_0\\sqrt{M\\sigma-1}}{(\\Gamma\\,t+\\chi)}+\\dfrac{1}{(\\Gamma\\,t+\\chi)}\\sqrt{\\dfrac{\\Delta}{M\\,(\\Gamma\\,t+\\chi)^2}-\\omega_0^2}\\right].\\label{e130}\n\\end{align}\nThis expression also provides an upper bound of the time limit above which the expectation value of energy would become complex. This upper bound reads,\n\\begin{eqnarray}\n\\dfrac{\\Delta}{M\\,(\\Gamma\\,t+\\chi)^2}\\,\\geq\\,\\omega_0^2\\,\\Rightarrow\\,t\\,\\leq\\,\\dfrac{1}{\\Gamma}\\left[ \\dfrac{1}{\\omega_0}\\sqrt{\\dfrac{\\Delta}{M}}-\\chi \\right].\\label{e14}\n\\end{eqnarray}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.4]{elmdecay.eps}\n\\caption{\\textit{A study of the variation of expectation value of energy, scaled by \n$\\frac{1}{\\omega_0}$ ($\\frac{\\langle E \\rangle}{\\omega_0}$) in order to make \nit dimensionless, as we vary $\\Gamma$t (again a dimensionless quantity). Here \nwe consider mass M=1, $\\mu$=1,$\\Delta$=$10^7$, $\\sigma$=$10^7$, \n$\\omega_0$=$10^3$, $\\chi=1$ and $\\Gamma$=1 in natural units. The expectation value of \nenergy $\\langle E \\rangle$ is calculated for elementarily decaying \nHamiltonian parameters. We consider $f(t)=1$ and $\\omega(t)=\\dfrac{\\omega_0}{(\\Gamma\\,t+\\chi)}$.}} \n\\label{fig3}\n\\end{figure}\n\n\\noindent In Fig.(\\ref{fig3}), we observe that the expectation value of energy again undergoes a power law decay with time for the elementary solution.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nWe now summarize our results. In this paper we have considered a two-dimensional damped harmonic oscillator in noncommutative space\nwith time dependent noncommutative parameters.\nWe map this system in terms of commutative variables by using a shift of variables connecting the noncommutative and commutative\nspace, known in the literature as Bopp-shift. We have then obtained the exact solution of this time dependent system by using the well known Lewis invariant which in turn leads to a non-linear differential equation known as the Ermakov-Pinney equation. We first obtain the Lewis invariant in Cartesian coordinates. \nWe then make a transformation to polar coordinates and write down our results in these coordinates. Doing \nso, we use the operator approach to obtain the eigenstates of the invariant. With this background in place, we make various choices of the parameters in the problem which in turn leads to solutions for the time dependent noncommutative parameters. We have considered three different sets of choices for which solutions have been obtained, namely, exponentially decaying solutions, rationally decaying solutions and elementary solutions. Interestingly, the solutions obtained make it possible to integrate the phase factor exactly thereby giving an exact solution for the eigenstates of the Hamiltonian. We have then computed the matrix elements of operators raised to a finite integer power in both the eigenstates of the Hamiltonian as well as the Lewis invariant. From these results, we are able to compute the expectation value of the Hamiltonian. Expectedly, the expectation value of the energy varies with time. For the exponentially decaying solutions, we get three kinds of behaviour corresponding to the choices of the damping factor and the frequency of the oscillator. For the case where the damping factor is set to unity and the frequency of the oscillator decays with time, the expectation value of the energy first decreases with time and then increases.\nThe reason for this behaviour is due to the particular form of the solutions of the Ermakov-Pinney equation which fixes the forms of the noncommutative parameters. It is these time dependent forms of the noncommutative parameters that results in the above mentioned behaviour of the expectation value of the energy with time. \nIn this case, we also observe that there is an upper bound of time above which the energy expectation value ceases to be real. \nFor the case where the damping factor has a decaying part and the frequency of the oscillator is a constant, we observe that the expectation value of the energy remarkably remains constant with time. This must be the case because the effect of the exponentially decaying coefficient in the Hamiltonian \nand the damping term gets balanced out by the exponentially increasing coefficient in the Hamiltonian. For the case where both the damping term as well as the frequency of the oscillator decays with time, we find an exponentially decaying behaviour of the expectation value of the energy. For the rationally decaying and the elementary solution, we observe a power law decay of the energy expectation value with time together with an upper bound of time above which the energy expectation value ceases to be real. Investigating these cases of damped oscillators, we conclude that the behaviour corresponding to the exponentially decaying solution, where both the frequency and damping term are decaying exponentially with time, is similar to a damped oscillator in commutative space. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgement}\nMD would like to thank Ms. Riddhi Chatterjee and Ms.Rituparna Mandal for their helpful assistance to operate the software Mathematica.\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n The Bloch--Kato conjecture, which relates the dimension of the Selmer group of a $p$-adic geometric Galois representation to the order of vanishing of its $L$-function, is one of the most important open problems in number theory. In a recent paper \\cite{LZ20}, we proved this conjecture for the 4-dimensional Galois representations arising from automorphic representations of $\\GSp_4$, under various technical hypotheses, using the ``method of Euler systems''; this relied crucially on the construction of an Euler system for $\\GSp_4$ in our earlier work \\cite{LSZ17} with Skinner, and the construction of a $p$-adic spin $L$-function for $\\GSp_4$ in the paper \\cite{LPSZ1} with Pilloni and Skinner.\n\n In this paper, we prove new cases of the Bloch--Kato conjecture, for the 8-dimensional tensor product Galois representation $V(\\pi) \\otimes V(\\sigma)$ associated to a $\\GSp_4$ automorphic representation $\\pi$ and a $\\GL_2$ automorphic representation $\\sigma$. The principal ingredients are the the Euler system for these Galois representations constructed in \\cite{HJS20}, and the formula proved in \\cite{LZ20b-regulator} relating these Euler system classes to periods of $p$-adic modular forms for $\\GSp_4 \\times \\GL_2$ (obtained by integrating a class in $H^2$ arising from $\\pi$, restricted to $\\GL_2 \\times_{\\GL_1} \\GL_2 \\subset \\GSp_4$, against the product of a cusp form in $\\sigma$ and a non-classical $p$-adic Eisenstein series). The main result of this paper, \\cref{thm:BKconj}, proves the Bloch--Kato conjecture for a certain twist of $V(\\pi)^* \\otimes V(\\sigma)^*$, corresponding to a critical value of the $L$-function $L(\\pi \\times \\sigma, s)$.\n\n The main new technical input needed in order to prove this theorem is to interpolate the $p$-adic automorphic periods arising from \\cite{LZ20b-regulator} in $p$-adic families, with both of the weights $(r_1, r_2)$ of the $\\GSp_4$ automorphic representation allowed to vary. This is not accessible by the methods of our earlier work \\cite{LPSZ1}, since the version of higher Hida theory used in that paper (based on the earlier work \\cite{pilloni20}) is only applicable to 1-parameter families in which $r_1$ varies for a fixed $r_2$. A similar issue arises in our earlier work \\cite{LZ20}, but in that setting, we were able to bypass the problem by applying the functorial lift from $\\GSp_4$ to $\\GL_4$, and applying the results of \\cite{DJR18, barreradimitrovwilliams} on $p$-adic $L$-functions for $\\GL_{2n}$. However, this does not work for $\\GSp_4 \\times \\GL_2$, since there appears to be no known construction of $p$-adic $L$-functions for $\\GL_4 \\times \\GL_2$.\n\n We therefore develop a direct approach to interpolating these $p$-adic periods in 2-parameter families for $\\GSp_4$, with both $r_1$ and $r_2$ varying, using the new ``higher Coleman theory'' introduced in \\cite{boxerpilloni20}. Our main result in this direction is \\cref{thm:bspairing}, whose proof occupies the majority of the present paper. This result shows that there is a well-defined pairing between the higher Coleman theory spaces for $\\GSp_4$ and spaces of overconvergent modular forms for $\\GL_2 \\times \\GL_2$; and in the final sections of the paper, we use this to define $p$-adic $L$-functions in families for $\\GSp_4 \\times \\GL_2$ by pairing a family of $H^2$ eigenclasses for $\\GSp_4$ with the product of a $\\GL_2$ cusp-form family and an auxiliary Eisenstein series. The existence of this $p$-adic $L$-function then allows us to prove a reciprocity law relating the Euler system of \\cite{HJS20} to critical complex $L$-values, and thus to prove the Bloch--Kato conjecture.\n\n These new methods can also be used to strengthen the results of \\cite{LZ20} for the degree 4 motive of $\\GSp_4$; for reasons of space, we shall pursue this in a forthcoming paper. Our methods also give, as a by-product, the construction of a ``$p$-adic Eichler--Shimura isomorphism in families'' for $\\GSp_4$, interpolating the comparison isomorphisms between de Rham and \\'etale cohomology for all (or almost all) specialisations of a $\\GSp_4$ Hida family. Our results give an interpolation of the comparison isomorphism after projecting to a specific filtration step of de Rham cohomology, corresponding to $H^2$ of automorphic vector bundles. Our results are thus complementary to the recent work of Diao et al \\cite{diao-rosso-wu21} which interpolates the filtration step corresponding to $H^0$.\n\n\\begin{remark}\n The switch from ``Hida'' to ``Coleman'' theory allows us to define $p$-adic $L$-functions for finite-slope families, rather than just for ordinary (i.e.~slope 0) families. However, this comes at a price: the use of Coleman theory requires an overconvergence condition on the Eisenstein series, which does not hold for the 2-parameter family of Eisenstein series used in \\cite{LPSZ1}. So the price we pay for including the second weight variable $r_2$ is that we lose sight of the cyclotomic variable -- for each automorphic representation $\\pi \\times \\sigma$ of $\\GSp_4 \\times \\GL_2$, there is an interval of integers $n$ such that $V(\\pi)^* \\otimes V(\\sigma)^*(-n)$ is critical, but in the present paper we can only prove the Bloch--Kato conjecture for a specific $n$, corresponding to the lower endpoint of this interval. Even in the ordinary case, to prove the Bloch--Kato conjecture for all of the critical twists, we would need a version of higher Hida (rather than Coleman) theory for $\\GSp_4$ with both $r_1$ and $r_2$ varying. Such a theory is not available at present, although analogous results for Hilbert modular groups have been announced by Giada Grossi \\cite{grossi21}.\n\\end{remark}\n\n\\emph{Acknowledgements.} We would like to thank George Boxer and Vincent Pilloni for answering our questions about their beautiful theory. We are very grateful for their patience.\n\n\n\n\n\\section{Preliminaries}\n\n Throughout this paper $p$ is a prime.\n\n \\subsection{The group $G$}\n\n Let $G=\\GSp(4)$, with respect to the anti-diagonal Hermitian form with matrix $J = \\begin{smatrix} & &&1\\\\&&1\\\\&-1\\\\-1\\end{smatrix}$. Write $B_G$ for the Borel subgroup consisting of upper-triangular matrices, and write $P_{\\Kl}$ and $P_{\\Sieg}$ for the Klingen and Siegel parabolic subgroups containing $B_G$. We then have the Levi decompositions\n \\[ B_G = T N_{B}, \\qquad P_{\\Sieg}= M_{\\Sieg} N_{\\Sieg},\\qquad P_{\\Kl}= M_{\\Kl} N_{\\Kl},\\]\n where $T$ is the diagonal torus.\n\n The Siegel parabolic $P_{\\Sieg}$ and its Levi $M_{\\Sieg}$ plays a distinguished role in our constructions, since it is conjugate to the centraliser of the cocharacter defining the Shimura datum; the Klingen parabolic is less important here (in contrast with our previous paper \\cite{LPSZ1}). Hence we shall often write simply $P_G$, $M_G$ for $P_{\\Sieg}, M_{\\Sieg}$. We identify $M_{G}$ with $\\GL_2 \\times \\GL_1$ via $\\stbt{A}{0}{0}{\\star} \\mapsto (A, \\nu)$, where $\\nu$ is the symplectic multiplier.\n\n Let $W_G=N_G(T)\/T$ denote the Weyl group of $(G,T)$. The group $W_G$ is generated by the $T$-cosets of the elements\n $s_1=\\begin{smatrix} 1\\\\ & & 1 \\\\ & -1 \\\\ &&&1 \\end{smatrix}$\n and\n $s_2=\\begin{smatrix} & 1\\\\ -1 \\\\ &&& -1 \\\\ &&1\\end{smatrix}$.\n \n\n Let $W_{M_G} = \\langle s_2 \\rangle $ denote the Weyl group of $(M_G, T)$, and let ${}^{M} W_G=W_{M_G}\\backslash W_G$. This has a distinguished set of coset representatives (the \\emph{Kostant representatives}) given by\n \\[ {}^MW_G=\\{ \\mathrm{id}, w_1, w_2, w_3 \\}\\]\n where $w_1 = s_1$, $w_2 = s_1 s_2$, $w_3 = s_1 s_2 s_1$. These have lengths $\\ell(w_i) = i$. We use $w_G^{\\max}$ for the long Weyl element of $G$, and $w_{M_G}^{\\max} = s_2$ the long Weyl element of $M_G$.\n\n \\begin{remark}\n Note that:\n \\begin{enumerate}[(i)]\n \\item Since $W_G$ permutes the coordinates of the diagonal torus, we can realize it as subgroup of $S_4$. Then ${}^{M}W_G$ identifies with the permutations $w \\in W_G$ such that $w(1) m} \\mathcal{B}_{m'}, \\qquad \\overline{\\mathcal{B}}^\\circ_m = \\{ |.| : |z| < |p|^m\\}.\n \\]\n Thus $\\mathcal{B}^{\\circ}_m \\subset \\overline{\\mathcal{B}}^\\circ_{m} \\subset \\mathcal{B}_m \\subset\\overline{\\mathcal{B}}_m$, and (as the notation suggests) $\\overline{\\mathcal{B}}_m$ is the closure of $\\mathcal{B}_m$, and similarly $\\overline{\\mathcal{B}}_m^\\circ$ of $\\mathcal{B}^{\\circ}_m$. Moreover, the sets $\\overline{\\mathcal{B}}_m - \\mathcal{B}_m$ and $\\overline{\\mathcal{B}}^\\circ_m -\\mathcal{B}^{\\circ}_m$ consist entirely of rank $> 1$ points.\n\n \\begin{remark}\n Compare the four flavours of root subgroups in \\bp{\\S 3.3.2}. The space $\\overline{\\mathcal{B}}_m$ corresponds to the ``dagger affinoid disc'' in Grosse-Kl\\\"onne's theory of dagger spaces.\n \\end{remark}\n\n More generally, if $A$ is a subset of $\\overline{\\mathbf{Q}}_p$, we write $A + \\mathcal{B}_m = \\bigcup_{a \\in A}(a + \\mathcal{B}_m)$ etc; we shall only use this if $A$ is compact, in which case the union is finite.\n\n \\subsection{Level groups at $p$}\n\n \\begin{notation} Let $t \\in \\mathbf{Z}_{\\ge 1}$.\n \\begin{itemize}\n \\item Let $K_{\\Iw}^G(p^t) = \\{ g \\in G(\\ZZ_p): g\\bmod p \\in B_G(\\mathbf{Z}\/p^t)\\}$ be the depth $t$ upper-triangular Iwahori of $G$, and similarly for $H$.\n \\item Let $K^H_{\\diamondsuit}(p^t)$ denote the group $H(\\QQ_p) \\cap \\hat\\gamma K^G_{\\Iw}(p^t) \\hat\\gamma^{-1}$, which is concretely given by\n \\[ K^H_{\\diamondsuit} =\n \\left \\{ h \\in H(\\ZZ_p): h = \\left(\\stbt{x}{y}{0}{z},\\stbt{x}{-y}{0}{z}\\right) \\bmod p^t \\text{ \\textup{for some} $x,y,z$}\\right\\}.\\]\n \\end{itemize}\n \\end{notation}\n\n Note that $K^H_{\\diamondsuit}(p^t)$ is, fortuitiously, a subgroup of $K^H_{\\Iw}(p^t)$.\n\n \\subsection{Tubes of ``radius one''}\n\n We note that if $\\mathcal{X}$ is the analytic adic space associated to a finite-type $\\ZZ_p$-scheme $X$, then there is a specialisation map $\\sp: \\mathcal{X} \\to X_{\\mathbf{F}_p}$ which is a continuous map of topological spaces. If $Z \\subset X_{\\mathbf{F}_p}$ is a locally closed subset, we let $]Z[$ be the \\emph{interior} of $\\sp^{-1}(Z)$; this is the adic space corresponding to the tube in the sense of classical rigid geometry, while $\\sp^{-1}(Z)$ is not a classical rigid space in general. Of course, if $Z$ is open, then $\\sp^{-1}(Z) =\\ ]Z[$; on the other hand, if $Z$ is closed, then $\\sp^{-1}(Z) =\\ \\overline{]Z[}$.\n\n \\begin{definition}\n Let $\\scalebox{1.15}{$\\mathtt{U}$}^G_0 =\\ ]Y^G_{w_1, \\mathbf{F}_p}[$, $\\scalebox{1.15}{$\\mathtt{Z}$}^G_0 = \\ \\overline{]X^G_{w_1, \\mathbf{F}_p}[}$, and $\\scalebox{1.15}{$\\mathtt{I}$}^G_{0,0} = \\scalebox{1.15}{$\\mathtt{Z}$}^G_0 \\cap \\scalebox{1.15}{$\\mathtt{U}$}^G_0$.\n \\end{definition}\n\n Note that $\\scalebox{1.15}{$\\mathtt{U}$}^G_0$ is open and $\\scalebox{1.15}{$\\mathtt{Z}$}^G_0$ closed, and both are invariant under the Iwahori $K^G_{\\Iw}(p)$ (since the Borel subgroup of $G_{\\mathbf{F}_p}$ fixes the mod $p$ Bruhat cells). Thus $\\scalebox{1.15}{$\\mathtt{I}$}^G_{0, 0}$ is a ``partial closure'' of the Bruhat cell $]C_{w_1, \\mathbf{F}_p}^G[$.\n\n We also write $\\scalebox{1.15}{$\\mathtt{Z}$}^H_0 = \\overline{]X^H_{\\mathrm{id}}[}$ (the preimage of the point $\\{\\mathrm{id}_H\\} \\in \\FL^H_{\\mathbf{F}_p}$) which is stable under $K^H_{\\Iw}(p)$, and we formally set $\\scalebox{1.15}{$\\mathtt{U}$}^H_0 = \\scalebox{1.15}{$\\mathtt{FL}$}^H$.\n\n \\begin{proposition}\n We have $\\scalebox{1.15}{$\\mathtt{I}$}^G_{0,0} \\subset U^G_{w_1}$, and in the coordinates on $U^G_{w_1}$ described in \\cref{sect:explicitparam}, we have\n \\[ \\scalebox{1.15}{$\\mathtt{I}$}^G_{0,0} = \\{ \\stbt x y z x : x, z \\in \\overline{\\mathcal{B}}_{0}^\\circ, y \\in \\mathcal{B}_0 \\}.\\]\n Similarly\n \\[ \\scalebox{1.15}{$\\mathtt{I}$}^H_{0,0} = \\scalebox{1.15}{$\\mathtt{Z}$}^H_0 = \\{ (z_1, z_2) : z_i \\in \\overline{\\mathcal{B}}_{0}^\\circ \\}.\\]\n \\end{proposition}\n\n \\begin{proof}\n This is an instance of \\bp{Lemma 3.21 (5)}.\n \\end{proof}\n\n \\begin{proposition}\n \\label{prop:Cartadic}\n We have a Cartesian diagram of adic spaces\n \\[\n \\begin{tikzcd}\n \\scalebox{1.15}{$\\mathtt{Z}$}^H_0 = \\scalebox{1.15}{$\\mathtt{I}$}^H_{0, 0} \\dar[\"\\hat\\iota\"] \\rar[hook] &\\scalebox{1.15}{$\\mathtt{U}$}^H_0 \\dar[\"\\hat\\iota\"]\\\\\n \\scalebox{1.15}{$\\mathtt{I}$}^G_{0,0} \\rar[hook] & \\scalebox{1.15}{$\\mathtt{U}$}^G_0\n \\end{tikzcd}\n \\]\n in which all the morphisms are closed embeddings.\n \\end{proposition}\n\n \\begin{proof}\n This follows readily from \\cref{prop:intersectcells} and the definition of the $\\scalebox{1.15}{$\\mathtt{U}$}$'s, $\\scalebox{1.15}{$\\mathtt{I}$}$'s and $\\scalebox{1.15}{$\\mathtt{Z}$}$'s.\n \\end{proof}\n\n \\subsection{Tubes of smaller radius}\n\n Let $m, n, t$ be integers with\n \\begin{equation}\\label{eq:mnt}\n 0 \\le n \\le m < t, \\qquad \\text{$m > n$ if $n \\ne 0$}.\n \\end{equation}\n\n \\begin{definition}\n We define subsets $\\scalebox{1.15}{$\\mathtt{I}$}_{m, n}^G \\subset \\scalebox{1.15}{$\\mathtt{U}$}_n^G$ in $\\scalebox{1.15}{$\\mathtt{FL}$}^G$ as follows: we let\n \\[ \\scalebox{1.15}{$\\mathtt{I}$}_{m, n}^G = \\{ \\stbt x y z x : x, z \\in \\overline{\\mathcal{B}}_{m}^\\circ, y \\in \\mathcal{B}_n + \\ZZ_p \\}. \\]\n (consistently with the $(m, n) = (0, 0)$ case described above). For $n \\ge 1$ we set\n \\[ \\scalebox{1.15}{$\\mathtt{U}$}_n^G =\\{ \\stbt x y z x : x, z \\in \\mathcal{B}_n^\\circ, y \\in \\mathcal{B}_n + \\ZZ_p \\}.\\]\n and for $n = 0$ we use the definition above.\n \\end{definition}\n\n \\begin{proposition} \\\n \\begin{enumerate}[(i)]\n \\item The sets $\\scalebox{1.15}{$\\mathtt{U}$}_n^G$ and $\\scalebox{1.15}{$\\mathtt{U}$}_n^G - \\scalebox{1.15}{$\\mathtt{I}$}_{m, n}^G$ are open in $\\scalebox{1.15}{$\\mathtt{FL}$}^G$.\n \\item If $n > 0$ then $\\scalebox{1.15}{$\\mathtt{I}$}_{m, n}^G = \\scalebox{1.15}{$\\mathtt{I}$}_{m, 0}^G \\cap \\scalebox{1.15}{$\\mathtt{U}$}_n^G$.\n \\item The sets $\\scalebox{1.15}{$\\mathtt{U}$}_n^G$ and $\\scalebox{1.15}{$\\mathtt{I}$}_{m, n}^G$ are stable under $K_{G, \\Iw}(p^t)$, and in the notation of \\bp{\\S 3.3.3}, we have\n \\[ \\scalebox{1.15}{$\\mathtt{U}$}_n^G =\\ ]C_{w_1, \\mathbf{F}_p}[_{(n, n)}K_{G, \\Iw}(p^t),\\qquad \\scalebox{1.15}{$\\mathtt{I}$}_{m, n} =\\ ]C_{w_1, \\mathbf{F}_p}[_{(\\overline{m}, n)}K_{\\Iw}^G(p^t).\\]\n \\end{enumerate}\n \\end{proposition}\n\n \\begin{proof}\n The first two statements are obvious. For the stability under $K_{G, \\Iw}(p^t)$, we treat $n = 0$ and $n > 0$ separately: in the $n = 0$ case, the stability of $\\scalebox{1.15}{$\\mathtt{U}$}_0^G$ is already established, and the stability of $\\scalebox{1.15}{$\\mathtt{I}$}_{m, n}$ follows from the identification $\\scalebox{1.15}{$\\mathtt{I}$}_{m, 0} = \\, ]C_{w_1, \\mathbf{F}_p}\\![\\,_{\\overline{m}, 0} = \\mathcal{P}\\backslash \\mathcal{P} w \\mathcal{G}_{\\overline{m}, 0}$ of \\bp{\\S 3.3.3}.\n\n For $n > 0$, we argue as in Lemma 3.18 of \\emph{op.cit.} to describe the spaces $]C_{w_1, \\mathbf{F}_p}[_{(n, n)} K_{\\Iw}^G(p^t)$ and $]C_{w_1, \\mathbf{F}_p}[_{(\\overline{m}, n)}K_{\\Iw}^G(p^t)$ as finite unions of translates of $]C_{w_1, \\mathbf{F}_p}[_{(n, n)}$, indexed by coset representatives for $N_{B_G}(\\mathbf{Z}\/p^n)$ modulo its intersection with $w^{-1}P_G w$. We can take these coset representatives to be of the form\n \\[ \\begin{smatrix} 1 \\\\ &1 & \\alpha \\\\ &&1 \\\\ &&&1\\end{smatrix}, \\qquad \\alpha \\in \\mathbf{Z} \/ p^n, \\]\n which act as $\\stbt x y z x \\mapsto \\stbt x {y + \\alpha} z x$.\n \\end{proof}\n\n \\begin{remark}\n One can choose a closed subset $\\scalebox{1.15}{$\\mathtt{Z}$}_m^G$ such that $\\scalebox{1.15}{$\\mathtt{I}$}_{m, n}^G = \\scalebox{1.15}{$\\mathtt{Z}$}_m^G \\cap \\scalebox{1.15}{$\\mathtt{U}$}_n^G$; however, it is a little awkward to choose such a subset which is invariant under $K^G_{\\Iw}(p^t)$, and in any case the choice of such a subset will not matter for our theory, so we shall not privilege any particular choice.\n \\end{remark}\n\n\n \\begin{remark}\n Formally setting $n = 0$ in the formula defining $\\scalebox{1.15}{$\\mathtt{U}$}_n^G$ for $n > 0$ gives a well-defined and $K^G_{\\Iw}(p)$-invariant set (in fact it is exactly $]C_{w_1}[\\,$), but this set does not contain $\\scalebox{1.15}{$\\mathtt{I}$}_{0, 0}$. Hence we use the formula of the previous section in which $x, z$ are allowed to ``go off to infinity''.\n \\end{remark}\n\n\n\n \\subsection{Iwahori-level tubes for $H$}\n\n For $n < t$ we shall define\n \\[ \\scalebox{1.15}{$\\mathtt{U}$}_n^H = \\hat\\iota^{-1}(\\scalebox{1.15}{$\\mathtt{U}$}_n^G) \\subseteq \\scalebox{1.15}{$\\mathtt{FL}$}^H.\\]\n This is an open $K_{H, \\Iw}(p^t)$-invariant set, containing $\\{\\mathrm{id}\\}$. For $n = 0$ it is the whole of $\\scalebox{1.15}{$\\mathtt{FL}$}^H$.\n\n \\begin{proposition}\n The set $\\hat\\iota^{-1}(\\scalebox{1.15}{$\\mathtt{I}$}_{m, n}^G)$ is closed in $\\scalebox{1.15}{$\\mathtt{FL}$}_H$ and invariant under $K_{H, \\Iw}(p^t)$, and does not depend on $n$; explicitly it is given by\n \\[ \\scalebox{1.15}{$\\mathtt{Z}$}_m^H = \\{ (z_1, z_2): z_i \\in \\overline{\\mathcal{B}}^\\circ_m\\}.\\]\n \\end{proposition}\n\n \\begin{proof}\n Clear from \\eqref{eq:gamma-param}.\n \\end{proof}\n\n We therefore have a Cartesian diagram of closed embeddings generalising \\cref{prop:Cartadic} above, for any $m, n, t$ as in \\eqref{eq:mnt}:\n \\begin{equation}\\label{eq:Cartadic2}\n \\begin{tikzcd}\n \\scalebox{1.15}{$\\mathtt{Z}$}^H_m \\dar[\"\\hat\\iota\"] \\rar[hook] &\\scalebox{1.15}{$\\mathtt{U}$}^H_n \\dar[\"\\hat\\iota\"]\\\\\n \\scalebox{1.15}{$\\mathtt{I}$}^G_{m,n} \\rar[hook] & \\scalebox{1.15}{$\\mathtt{U}$}^G_n.\n \\end{tikzcd}\n \\end{equation}\n\n It is convenient to extend the definition of $\\scalebox{1.15}{$\\mathtt{Z}$}_m$ to remove the requirement that $m < t$. For any integers $t \\ge 1$ and $m \\ge 0$, let us define\n \\[ \\scalebox{1.15}{$\\mathtt{Z}$}_m^H \\coloneqq \\{ (z_1, z_2): z_i \\in \\overline{\\mathcal{B}}^\\circ_m + p^t \\ZZ_p\\}.\\]\n This is invariant under $K_{H, \\Iw}(p^t)$, and hence also under $K_{H, \\diamondsuit}(p^t)$. If $m < t$ then this agrees with the definitions above. We define an open set $\\scalebox{1.15}{$\\mathtt{U}$}_n^H$ similarly. (However, in the $m \\ge t$ case we do not know if these sets $\\scalebox{1.15}{$\\mathtt{U}$}_n^H$ and $\\scalebox{1.15}{$\\mathtt{Z}$}_m^H$ can be fitted into a diagram like \\eqref{eq:Cartadic2}.)\n\n\n\n\n\\section{Pullbacks in overconvergent cohomology}\n \\subsection{Adic spaces and period maps}\n\n We consider the analytifications $\\mathcal{S}_{G,K}^{\\an}=(S_K\\times \\Spec(\\QQ_p))^{\\an}$, $\\mathcal{S}_{G,K}^{\\mathrm{tor}}=(S_{G,K}^{\\mathrm{tor}}\\times\\Spec(\\QQ_p))^{\\an}$ and $\\scalebox{1.15}{$\\mathtt{FL}$}_G=(\\FL_{G}\\times\\Spec(\\QQ_p))^{\\an}$, as well as the groups $\\mathcal{G}^{\\an}=(G\\times \\Spec(\\QQ_p))^{\\an}$, $\\mathcal{P}^{\\an}_{G}=(P_{G}\\times \\Spec(\\QQ_p))^{\\an}$ and $\\mathcal{M}_G^{\\an}=(M_G\\times \\Spec(\\QQ_p))^{\\an}$.\n\n Since we have fixed an integral model of $G$, we have quasi-compact, affinoid groups $\\mathcal{G}\\hookrightarrow \\mathcal{G}^{\\an}$, $\\mathcal{P}_{G,\\Sieg}\\hookrightarrow \\mathcal{P}_{G,\\Sieg}^{\\an}$ and $\\mathcal{M}_G\\hookrightarrow \\mathcal{M}_G^{\\an}$.\n\n Write $\\mathcal{S}_{G,K^p}^{\\mathrm{tor}}$ for the perfectoid space $\\varprojlim_{K_p} \\mathcal{S}_{G,K^pK_p}^{\\mathrm{tor}}$. We can then consider the Hodge--Tate period map\n \\[ \\pi_{\\HT,G}^{\\mathrm{tor}}: \\mathcal{S}_{G,K^p}^{\\mathrm{tor}} \\longrightarrow \\scalebox{1.15}{$\\mathtt{FL}$}_G\\]\n which for every open compact $K_p\\subset G(\\QQ_p)$ descends to a map of topological spaces (c.f. \\cite[\\S 4.5]{boxerpilloni20})\n \\[ \\pi_{\\HT,G,K_p}^{\\mathrm{tor}}: \\mathcal{S}_{G,K^pK_p}^{\\mathrm{tor}}\\longrightarrow \\scalebox{1.15}{$\\mathtt{FL}$}_G\/K_p.\\]\n\n There is an analogous Hodge--Tate period map for $H$ also. To lighten the notation, we shall frequently omit many of the subscripts from $\\pi_{\\HT,G,K_p}^{\\mathrm{tor}}$ when they are clear from context (in particular, we shall almost always omit the ``$\\mathrm{tor}$'', since the non-compactified Shimura variety plays no role here).\n\n We shall attempt to consistently maintain the convention that subsets of flag varieties are denoted by typewriter letters $\\scalebox{1.15}{$\\mathtt{U}$}$ etc, and the preimages of these spaces under the Hodge--Tate period maps are denoted by calligraphic letters $\\mathcal{U}$ etc.\n\n\n \\subsection{Period maps}\n\n \\begin{theorem}\n \\label{thm:HTmaps}\n There is a commutative diagram of Hodge--Tate period maps (where we have omitted some unimportant subscripts and superscripts for clarity)\n \\[\n \\begin{tikzcd}\n \\mathcal{S}^{\\mathrm{tor}}_{H,\\Iw}(p^t) \\rar[\"\\pi^H_{\\Iw}\"] & \\scalebox{1.15}{$\\mathtt{FL}$}_H \/ K_{H, \\Iw}(p^t)\\\\\n \\mathcal{S}^{\\mathrm{tor}}_{H,\\diamondsuit}(p^t) \\rar[\"\\pi^H_{\\diamondsuit}\"] \\dar[\"\\hat\\iota\" left] \\uar[\"\\pr_\\diamondsuit\" left] & \\scalebox{1.15}{$\\mathtt{FL}$}_H \/ K_{H, \\diamondsuit}(p^t) \\dar[\"\\hat\\iota\" right] \\uar[\"\\pr_{\\diamondsuit}\" right]\\\\\n \\mathcal{S}^{\\mathrm{tor}}_{G,\\Iw}(p^t) \\rar[\"\\pi^G_{\\Iw}\"] & \\scalebox{1.15}{$\\mathtt{FL}$}_G \/ K_{G, \\Iw}(p^t)\n \\end{tikzcd}\n \\]\n in which the maps $\\pr_{\\diamondsuit}$ are the natural quotients, and the downward ones are given by the composite of the natural embedding $H \\hookrightarrow G$ and right-translation by $\\hat\\gamma = \\gamma w_1$.\n \\end{theorem}\n\n \\begin{proof}\n It suffices to check that the Hodge--Tate period maps for $H$ and for $G$ at perfectoid infinite level are compatible; but this is a direct consequence of the construction, since the Hodge--Tate period map for Hodge-type Shimura varieties is defined using an embedding into a Siegel Shimura variety. See \\cite[\\S 4.4.7]{boxerpilloni20}).\n \\end{proof}\n\n For $(m, n, t)$ as in \\eqref{eq:mnt}, we define subspaces\n \\[ \\mathcal{I}^G_{m, n} \\subset \\mathcal{U}^G_n \\subset \\mathcal{S}_{G,\\Iw}(p^t), \\qquad \\mathcal{Z}^H_{m} \\subset \\mathcal{U}^H_n \\subset \\mathcal{S}_{H,\\diamondsuit}(p^t)\n \\]\n as the preimages of the subsets $\\scalebox{1.15}{$\\mathtt{I}$}^G_{m, n} \\subset \\scalebox{1.15}{$\\mathtt{U}$}^G_n \\subset \\scalebox{1.15}{$\\mathtt{FL}$}^G$ under $\\pi^G_{\\Iw}$, respectively $\\mathcal{Z}^H_{m} \\subset \\mathcal{U}^H_n \\subset \\scalebox{1.15}{$\\mathtt{FL}$}^H$ under $\\pi^H_{\\diamondsuit}$. Combining \\cref{thm:HTmaps} and \\cref{eq:Cartadic2}, we obtain a Cartesian diagram\n \\begin{equation}\\label{eq:adicSh}\n \\begin{tikzcd}\n \\mathcal{Z}^{H}_m \\dar \\rar[hook] &\\mathcal{U}^H_n \\dar\\\\\n \\mathcal{I}^G_{m, n} \\rar[hook] & \\mathcal{U}^G_n\n \\end{tikzcd}\n \\end{equation}\n in which the horizontal arrows are closed embeddings; and $\\mathcal{Z}^{H}_m$ is closed in $\\mathcal{S}_{H,\\diamondsuit}(p^t)$.\n\n\n \\subsection{Overconvergent pullback}\n We can now define the pullback map on overconvergent cohomology. We give the definitions for the non-cuspidal cohomology, using the coefficient sheaf $\\mathcal{V}= \\mathcal{V}_\\kappa$ for some $M_G$-dominant integral weight $\\kappa$; the definitions are the same for cuspidal cohomology using $\\mathcal{V}= \\mathcal{V}_\\kappa(-D)$ instead.\n\n Using the diagram \\eqref{eq:adicSh} and the functoriality of cohomology with support, we get a map\n \\begin{equation}\n \\label{eq:iota1}\n \\hat\\iota^* : R\\Gamma_{\\mathcal{I}^G_{mn}}(\\mathcal{U}_n^G, \\mathcal{V}) \\longrightarrow R\\Gamma_{\\mathcal{Z}_m^{H}}\\left(\\mathcal{U}_n^H, \\hat\\iota^* \\mathcal{V}_{\\kappa}\\right) \\cong R\\Gamma_{\\mathcal{Z}_m^{H}}\\left(\\mathcal{S}_{H,\\diamondsuit}(p^t), \\hat\\iota^* \\mathcal{V}\\right),\n \\end{equation}\n where the final isomorphism comes from excision, using the fact that $\\mathcal{Z}_m^{H}$ is closed in $\\mathcal{S}_{H,\\diamondsuit}(p^t)$.\n\n As in \\bp{\\S 5.4.1}, for any $t \\ge 1$, we can define the finite-slope overconvergent cohomology for $G$ as\n \\begin{align*}\n R\\Gamma^G_{w_1}(\\kappa)^{-, \\fs} &\\coloneqq R\\Gamma_{\\mathcal{I}_{00}}(\\mathcal{U}^G_0, \\mathcal{V}_\\kappa)^{-, \\fs}.\n \\end{align*}\n So \\eqref{eq:iota1} for $(m, n) = (0,0)$ gives our first definition of the pullback map on overconvergent cohomology, taking values in $R\\Gamma_{\\mathcal{Z}_0^{H}}\\left(\\mathcal{S}_{H,\\diamondsuit}(p^t), \\hat\\iota^* \\mathcal{V}\\right)$.\n\n \\begin{proposition}[Comparison with classical pullback]\n \\label{prop:classicalcomp}\n We have the following commutative diagram:\n \\[\n \\begin{tikzcd}\n R\\Gamma_{\\mathcal{I}_{00}^G}(\\mathcal{U}_0^G,\\mathcal{V}) \\rar[\"\\quad \\hat\\iota^*\\quad\" above, \"\\eqref{eq:iota1}\" below] \\dar[\"\\cores\"] & R\\Gamma_{\\mathcal{Z}_0^{H}}\\left(\\mathcal{S}^{\\mathrm{tor}}_{H, \\diamondsuit}(p^t), \\hat\\iota^* \\mathcal{V}\\right) \\dar[\"\\cores\"]\\\\\n \n R\\Gamma(\\mathcal{U}_0^G,\\mathcal{V}) \\rar{\\quad \\hat\\iota^*\\quad } & R\\Gamma\\left(\\mathcal{S}^{\\mathrm{tor}}_{H,\\diamondsuit}(p^t),\\hat\\iota^*\\mathcal{V}\\right)\\\\\n \n R\\Gamma\\left(\\mathcal{S}^{\\mathrm{tor}}_{G,\\Iw}(p^t),\\mathcal{V}\\right)\\uar[\"\\mathrm{res}\"] \\arrow[\"\\hat\\iota^*\" below]{ru}\n \\end{tikzcd}\n \\]\n in which the bottom horizontal map corresponds to the classical pushforward via the rigid-analytic GAGA theorem. Moreover, the spaces in the left column of the diagram have actions of the prime-to-$p$ Hecke algebra and the operators $\\mathcal{U}'_{\\Sieg}$, $\\mathcal{U}'_{\\Kl}$ at $p$, and the maps $\\mathrm{res}$ and $\\cores$ are compatible with these actions.\n \\end{proposition}\n\n \\begin{proof}\n The only non-obvious step of the diagram is the existence of the middle horizontal map, which follows from \\cref{prop:Cartadic}. The compatibility with Hecke actions is an easy check, cf.~\\bp{Lemma 5.17}.\n \\end{proof}\n\n \\begin{proposition}[Change of support condition]\n \\label{prop:changesupport}\n The maps \\eqref{eq:iota1} for $(m, n)$, $(m, 0)$, and $(0, 0)$ fit into a diagram\n \\[\n \\begin{tikzcd}\n R\\Gamma_{\\mathcal{I}_{mn}^G}(\\mathcal{U}_n^G,\\mathcal{V}) \\arrow{r}{\\hat\\iota^*} &\n R\\Gamma_{\\mathcal{Z}_m^H}\\left(\\mathcal{S}^{\\mathrm{tor}}_{H, \\diamondsuit}(p^t), \\hat\\iota^*(\\mathcal{V})\\right)\\\\\n R\\Gamma_{\\mathcal{I}_{m0}^G}(\\mathcal{U}_0^G,\\mathcal{V}) \\arrow{u}{\\mathrm{res}} \\arrow{r}{\\hat\\iota^*} \\dar[\"\\cores\"]& R\\Gamma_{\\mathcal{Z}_m^H}\\left(\\mathcal{S}^{\\mathrm{tor}}_{H, \\diamondsuit}(p^t), \\hat\\iota^*(\\mathcal{V})\\right) \\uar[equals] \\arrow{d}{\\cores} \\\\\n R\\Gamma_{\\mathcal{I}_{00}^G}(\\mathcal{U}_0^G,\\mathcal{V}) \\arrow{r}{\\hat\\iota^*} &\n R\\Gamma_{\\mathcal{Z}_0^H}\\left(\\mathcal{S}^{\\mathrm{tor}}_{H, \\diamondsuit}(p^t), \\hat\\iota^*(\\mathcal{V})\\right).\n \\end{tikzcd}\n \\]\n The complexes in the left column have compatible actions of the Hecke operators away from $p$, and of $\\mathcal{U}'_{\\Kl}$ and $\\mathcal{U}'_{\\Sieg}$ at $p$, and the maps $\\mathrm{res}$ and $\\cores$ are compatible with these.\n \\end{proposition}\n\n \\begin{proof}\n Immediate from the fact that $\\mathcal{I}^G_{mn} = \\mathcal{I}_{m0}^G \\cap \\mathcal{U}_n^G$ and standard functoriality properties of cohomology with support.\n \\end{proof}\n\n \\begin{proposition}\n The spaces in the left columns of the diagrams in \\cref{prop:classicalcomp,prop:changesupport} all have actions of the prime-to-$p$ Hecke operators, and of the Hecke operators $\\mathcal{U}'_{\\Sieg}$, $\\mathcal{U}'_{\\Kl}$, $\\mathcal{U}'_B$ at $p$. Moreover, the maps in the left column of \\cref{prop:changesupport} become isomorphisms on the finite-slope part for $\\mathcal{U}'_B$.\n \\end{proposition}\n\n \\begin{proof}\n The compatibility with Hecke operators away from $p$ is clear, since the Hodge--Tate period map is invariant under the action of the prime-to-$p$ Hecke algebra. The fact that the maps in the left column of \\cref{prop:classicalcomp} are maps of Hecke modules is an instance of \\bp{Lemma 5.17}.\n\n The assertions regarding the finite-slope part follow from \\bp{Theorem 5.66}, since one can check that $(\\mathcal{I}_{m, n}, \\mathcal{U}_n)$ defines an ``allowed support condition'' in the sense of \\bp{\\S 5.4.3}.\n \\end{proof}\n\n This shows that we have well-defined maps\n \\begin{equation}\n \\label{eq:iota2}\n R\\Gamma^G_{w_1}(\\kappa)^{-, \\fs} \\longrightarrow R\\Gamma_{\\mathcal{Z}_m^H}\\left(\\mathcal{S}^{\\mathrm{tor}}_{H, \\diamondsuit}(p^t), \\hat\\iota^*(\\mathcal{V})\\right)\n \\end{equation}\n for any $0 \\le m < t$, compatible under corestriction, extending \\eqref{eq:iota1} for in the $m = 0$ case.\n\n\n\n\n \\subsection{Functoriality of coefficients}\n \\label{sect:branchcoeffs}\n \\begin{proposition}\n Let $\\kappa_1 = (r_1, -r_2-2; r_1 + r_2)$ with $r_1 \\ge r_2 \\ge -1$, and let $\\tau = (t_1, t_2; r_1 + r_2)$ where $t_i \\ge -1$ and $t_1 + t_2 = r_1 - r_2 - 2$. Then there is a nonzero homomorphism of $(\\gamma^{-1} M_H \\gamma)$-representations\n \\[ V^G_{\\kappa_1} |_{\\gamma^{-1} M_H\\gamma } \\to V^H_{\\tau}, \\]\n uniquely determined up to scaling.\n \\end{proposition}\n\n It will be helpful to fix a normalisation for this map, by choosing a vector $f \\in (V^G_{\\kappa_1})^\\vee$ which transforms by $\\tau^{-1}$ under $\\gamma^{-1} M_H \\gamma$. We have an explicit presentation of $(V^G_{\\kappa_1})^\\vee = V^G_{(-w_{0, M} \\kappa_1)}$ as the space of polynomial functions $f \\in \\mathcal{O}(M_G)$ which satisfy $f(mb) = \\kappa_1(b) f(m)$ for all $b \\in B_{M_G}$ and $m \\in M_G$, with $M_G$ acting by left-translation. Since $\\gamma^{-1} M_H \\gamma \\cdot B_{M_G}$ is open in $M_G$, we can choose a unique $f$ which satisfies $f(\\mathrm{id}) = 1$ and transforms via $\\tau^{-1}$ under the action of $M_H$.\n\n This map gives a homomorphism of sheaves on $\\mathcal{S}^{\\mathrm{tor}}_{H, \\diamondsuit}(p^t)$,\n \\[ \\hat\\iota^*\\left(\\mathcal{V}_\\kappa^G\\right) \\longrightarrow \\mathcal{V}_\\tau^H, \\]\n and combining this with \\cref{eq:iota2}, we obtain maps of complexes\n \\[ R\\Gamma^G_{w_1}(\\kappa_1)^{-, \\fs} \\longrightarrow R\\Gamma_{\\mathcal{Z}_m^H}\\left(\\mathcal{S}^{\\mathrm{tor}}_{H, \\diamondsuit}(p^t), \\mathcal{V}^H_\\tau\\right)\n \\]\n for any $\\tau$ in the appropriate range, and similarly for cuspidal cohomology.\n\n \\begin{remark}\n The map is formally well-defined for a rather wider range of values of the parameters; but we have restricted to the case when $r_1 \\ge r_2 \\ge -1$ and $t_1, t_1 \\ge -1$, in order that there are interesting cuspidal automorphic representations contributing to $H^2$ for both $\\mathcal{V}_{\\kappa_1}^G$ and $\\mathcal{V}^H_{\\tau}$.\n\n If we set $k_i = r_i + 3$ and $c_i = t_i + 2$, then $(k_1, k_2, c_1, c_2)$ will define a point lying on the top edge of the region labelled $(f)$ in \\cite[Diagram 2]{LZvista}. Unfortunately, it seems to be difficult to extend our present analysis to points in the interior of this region; this would require some sort of ``nearly version'' of higher Coleman theory, analogous to the theory of nearly-overconvergent families in $H^0$ of modular curves recently introduced by Andreatta--Iovita \\cite{andreattaiovita21}. See \\cite[\\S 6]{LPSZ1} for an analogous theory in the ordinary case (with $r_2$ fixed, rather than varying as here).\n \\end{remark}\n\n\n \\subsection{Change of level}\n \\label{sect:changelevel}\n Finally, we note that for any $t \\ge 1$, we have $[K_{H, \\diamondsuit}(p^t): K_{H, \\diamondsuit}(p^{t+1})] = p^4 = [K_{G, \\Iw}(p^t) : K_{G, \\Iw}(p^{t+1})]$, and hence the natural map\n \\[ \\mathcal{S}_{H, \\diamondsuit}(p^{t+1}) \\longrightarrow \\mathcal{S}_{H, \\diamondsuit}(p^{t}) \\times_{\\mathcal{S}_{G, \\Iw}(p^{t})} \\mathcal{S}_{G, \\Iw}(p^{t+1})\n \\]\n is an isomorphism. So the pushforward (trace) maps arising from changing $t$ on the two spaces are compatible with the pullback $\\hat\\iota^*$, and similarly for the cohomology with supports, for any support condition invariant under $K_{G, \\Iw}(p^t)$.\n\n Hence, if we temporarily write $\\mathcal{I}_{mn}^G(p^t)$ etc to distinguish our various locally closed subspaces of Shimura varieties at the different levels, then we have trace maps\n \\[ R\\Gamma_{\\mathcal{I}^G_{mn}(p^{t+1})}(\\mathcal{U}^G_n(p^{t+1}), \\mathcal{V}^G_\\kappa) \\longrightarrow R\\Gamma_{\\mathcal{I}^G_{mn}(p^t)}(\\mathcal{U}^G_n(p^t), \\mathcal{V}^G_\\kappa) \\]\n and\n \\[ R\\Gamma_{\\mathcal{Z}^H_{m}(p^{t+1})}(\\mathcal{S}_{H, \\diamondsuit}(p^{t+1}), \\mathcal{V}^H_\\tau) \\longrightarrow R\\Gamma_{\\mathcal{Z}^H_{m}(p^t)}(\\mathcal{S}_{H, \\diamondsuit}(p^{t}), \\mathcal{V}^H_\\tau).\\]\n and these are compatible with the pullback maps $\\hat\\iota^*$, and the restriction\/corestriction maps for varying $m, n$. Moreover, the trace maps for $G$ are isomorphisms on the finite-slope part by \\bp{Theorem 5.14}.\n\n We can thus define a map\n \\[\n R\\Gamma_{w_1}(\\kappa)^{-, \\fs} \\longrightarrow R\\Gamma_{\\mathcal{Z}^H_m(p^t)}(\\mathcal{S}_{H, \\diamondsuit}(p^t), \\mathcal{V}^H_\\tau).\n \\]\n for any $t \\ge 1$, $m \\ge 0$ (not necessarily with $t > m$) by composing with the trace map from level $t'$ for some auxiliary $t' > m$; this allows us to define $\\hat\\iota^*$ as a map\n \\[\n R\\Gamma^G_{w_1}(\\kappa)^{-, \\fs} \\to \\varprojlim_m R\\Gamma_{\\mathcal{Z}^H_m(p^t)}(\\mathcal{S}_{H, \\diamondsuit}(p^t), \\mathcal{V}^H_\\tau).\n \\]\n Since the spaces $\\mathcal{Z}_m^H$ are actually invariant under the Iwahori of $H$, we can trace down further to land in the space\n \\[ \\varprojlim_m R\\Gamma_{\\mathcal{Z}^H_m(p^t)}(\\mathcal{S}_{H, \\Iw}(p^t), \\mathcal{V}^H_\\tau),\\]\n where we have abused notation a little by using $\\mathcal{Z}^H_m(p^t)$ for the preimages of $\\scalebox{1.15}{$\\mathtt{Z}$}_m^H$ at either Iwahori or $\\diamondsuit$ level.\n \\begin{definition}\n We define\n \\[ R\\Gamma_{\\mathrm{id}}\\left(\\mathcal{S}_{H, \\Iw}(p^t), \\tau\\right)^{(-, \\dag)} = \\varprojlim_m R\\Gamma_{\\mathcal{Z}^H_{m}(p^t)}\\left( \\mathcal{S}_{H, \\Iw}(p^t), \\mathcal{V}^H_\\tau\\right) .\\]\n \\end{definition}\n\n This space can be interpreted as the compactly-supported cohomology of the intersection $\\bigcap_m \\mathcal{Z}^H_m(p^t) = \\pi_{H}^{-1}\\left( \\{\\mathrm{id}_H\\} \\right)$; we shall recall this in a little more detail in the next section, where we shall allow more general coefficients.\n\n\n\n\\section{Torsors}\n\n We now begin constructing the ``locally analytic'' version of the pullback map on higher Coleman theory.\n \\subsection{Torsors on flag varieties}\n\n The map $x \\mapsto x^{-1}: G\\rightarrow \\FL_G$ (recall that $\\FL_G=P_G\\backslash G$) allows us to regard $G$ as a right $P_G$-torsor over $\\FL_G$, and similarly to regard $G\/ N_G\\rightarrow \\FL_G$ as a right $M_G$-torsor. We consider their analytifications\n \\[ \\scalebox{1.15}{$\\mathtt{P}$}^G: \\mathcal{G}\\rightarrow \\scalebox{1.15}{$\\mathtt{FL}$}_G \\qquad \\text{and} \\qquad \\scalebox{1.15}{$\\mathtt{M}$}^G:\\mathcal{G}\/ \\mathcal{N}_G\\rightarrow \\scalebox{1.15}{$\\mathtt{FL}$}_G.\\]\n which are torsors over $\\scalebox{1.15}{$\\mathtt{FL}$}_G$ under the (affinoid) analytic groups $\\mathcal{P}_G$ and $\\mathcal{M}_G$ respectively. We similarly define torsors over the flag varieties of $H$ and $H_i$ for $i=1,2$.\n\n \\begin{definition} Define $\\mathcal{P}^G_{\\HT}$ and $\\mathcal{M}^G_{\\HT}$ to be the pullbacks via $\\pi^G_{\\HT}$ of the torsors $\\scalebox{1.15}{$\\mathtt{P}$}^G$ and $\\scalebox{1.15}{$\\mathtt{M}$}^G$; these are (right) torsors over $\\mathcal{S}_{G,\\Iw}(p^t)$ for the groups $\\mathcal{P}_G$ and $\\mathcal{M}_G$. We similarly define $\\mathcal{P}^H_{\\HT}$ and $\\mathcal{M}^H_{\\HT}$, $\\mathcal{P}^{H_i}_{\\HT}$ and $\\mathcal{M}^{H_i}_{\\HT}$ for $i=1,2$.\n \\end{definition}\n\n \\begin{note}\n It is easy to check that $\\mathcal{M}^H_{\\HT}=\\mathcal{M}^{H_1}_{\\HT}\\times_{\\GL_1}\\mathcal{M}^{H_2}_{\\HT}$, where we take the fibre product with respect to the action of $\\nu$ in the parametrisation of $T$.\n \\end{note}\n\n\n \\subsection{Reduction of structure}\n\n \\begin{definition}\n For $n > 0$, let $\\mathcal{M}^1_{G,n} \\triangleleft \\mathcal{M}_G$ be the (affinoid analytic) group of elements which reduce to the identity $\\pmod{p^n}$. Define\n \\[ \\mathcal{M}^\\square_{G,n} = \\mathcal{M}^1_{G,n} \\cdot B_{M_G}(\\ZZ_p), \\]\n which is an affinoid analytic subgroup containing $\\Iw_{M_G}(p^n)$. A similar definition applies to $M_H = T$; we write the group as $\\mathcal{T}^{\\square}_{n} = T(\\ZZ_p) \\mathcal{T}^1_{n}$.\n \\end{definition}\n\n \\begin{note}\n We follow \\cite{boxerpilloni20} here in using affinoid subgroups and affinoid subspaces of flag varieties to develop the locally-analytic theory, rather than the ``mixed'' spaces (products of some copies of $\\mathcal{B}_n$ and some of $\\mathcal{B}^\\circ_n$) used in the previous sections.\n \\end{note}\n\n \\begin{note}\\label{note:identifyMGsq}\n Identifying $M_G$ with $\\GL_2 \\times \\GL_1$ as in the introduction, we have\n \\[\\mathcal{M}^\\square_{G, n} =\n \\left\\{\n (\\stbt x y z w, \\lambda) :\n \\begin{array}{c}\n x,w,\\lambda \\in \\ZZ_p^\\times \\cdot (1 + \\mathcal{B}_n),\\\\ z \\in \\mathcal{B}_n, y \\in \\ZZ_p + \\mathcal{B}_n.\n \\end{array}\n \\right\\}\n \\qedhere\n \\]\n \n \\end{note}\n\n \\begin{notation}\n Define\n \\[ \\mathcal{T}^\\diamondsuit_n = \\{\\diag(t_1, t_2, \\nu t_2^{-1}, \\nu t_1^{-1}) \\in \\mathcal{T}^\\square_n: t_1-t_2 \\in \\mathcal{B}_n\\}. \\]\n \\end{notation}\n\n Thus $\\mathcal{T}^\\diamondsuit_n$ and $\\mathcal{T}^\\square_n$ are both disjoint unions of copies of $\\mathcal{T}^1_n$, but $\\mathcal{T}^\\diamondsuit_n$ has fewer of these components than $\\mathcal{T}^\\square_n$.\n\n \\begin{proposition} \\label{prop:redofstr} Let $t > n > 0$.\n \\begin{enumerate}\n \\item Over $\\mathcal{U}^G_n$, the torsor $\\mathcal{M}_{\\HT}^G$ has a reduction of structure to an \\'etale torsor $\\mathcal{M}^G_{\\HT,n}$ under the group $\\mathcal{M}^\\square_{G,n}$.\n\n \\item Over $\\mathcal{U}^H_{\\Iw,n}$, the torsor $\\mathcal{M}_{\\HT}^H$ has a reduction of structure to an \\'etale torsor $\\mathcal{M}^H_{\\HT,n,\\Iw}$ under the group $\\mathcal{T}^{\\square}_{n}$.\n\n \\item Over $\\mathcal{U}^H_n$, the torsor $\\mathcal{M}_{\\HT}^H$ has a reduction of structure to an \\'etale torsor $\\mathcal{M}^H_{\\HT,n,\\diamondsuit}$ under the group $\\mathcal{T}^\\diamondsuit_{n}$ (and this refines the pullback of $\\mathcal{M}^H_{\\HT,n,\\Iw}$ to level $K^H_{\\diamondsuit}(p^t)$).\n\n \n \\end{enumerate}\n \\end{proposition}\n\n \\begin{proof}\n Part (1) is essentially the result of \\bp{\\S 6.2.1}. The proofs of (2) and (3) are similar.\n \\end{proof}\n\n \\begin{lemma}\n We have the following inclusions of subgroups.\n \\begin{itemize}\n \\item As subgroups of $\\mathcal{M}_G$, we have\n \\[ \\mathcal{T}^\\diamondsuit_{n} = \\mathcal{T} \\cap \\gamma \\mathcal{M}^\\square_{G,n}\\gamma^{-1}. \\]\n \\item As subgroups of $\\mathcal{G}$,\n \\[ \\hat\\gamma^{-1} \\cdot K^H_{\\diamondsuit}(p^n)\\mathcal{H}^1_{n} \\cdot \\hat\\gamma \\subset K^G_{\\Iw}(p^n)\\mathcal{G}^1_{n}, \\]\n where $\\hat\\gamma = \\gamma w_1$ as usual.\n \\end{itemize}\n \\end{lemma}\n\n \\begin{proof}\n \n If $\\tau = (\\stbt{t_1}{}{}{t_2}, \\nu)$ is an element of $\\mathcal{T}$ then $\\gamma^{-1} \\tau \\gamma = ( \\stbt{t_1}{}{t_2-t_1}{t_2}, \\nu)$. It is now clear that $\\gamma^{-1}T_{\\diamondsuit}(p^t) \\gamma \\subset \\Iw_{M_G}(p^t)$ and $\\gamma^{-1}\\mathcal{T}^1_{n} \\gamma \\subset \\mathcal{M}^1_{G,n}$, so the required inclusion follows.\n\n The second statement can be verified similarly; the inclusion on $\\ZZ_p$-points is the definition of $K^H_{\\diamondsuit}$, and the inclusion on $\\mathcal{H}^1_{n}$ follows from the fact that $\\mathcal{H}^1_{n} \\subset \\mathcal{G}^1_n$ and $\\mathcal{G}^1_n$ is normal in $\\mathcal{G}$.\n \\end{proof}\n\n\n \\begin{proposition}\\label{prop:pullbackcomp}\n We have an equality of $\\mathcal{M}^\\square_{G, n}$-torsors over $\\mathcal{U}^H_{n,\\diamondsuit}$:\n \\[ \\hat\\iota^*\\left( \\mathcal{M}^G_{\\HT,n,\\Iw}\\right) = \\mathcal{M}^H_{\\HT,n,\\diamondsuit}\\times^{\\left[\\mathcal{T}^\\diamondsuit_{n}, \\gamma\\right]} \\mathcal{M}^\\square_{G,n}, \\]\n where we regard $\\mathcal{T}^{\\diamondsuit}_{n}$ as a subgroup of $\\Iw_{M_G}(p^t)\\mathcal{M}^1_{G,n}$ via conjugation by $\\gamma$.\n \\end{proposition}\n\n \\begin{proof}\n We check the analogous statement on the flag varities. We first observe that we have a commutative diagram of adic spaces\n \\[\n \\begin{tikzcd}\n K^H_{\\diamondsuit}(p^t)\\mathcal{H}^1_{n} \\dar\\rar & K^G_{\\Iw}(p^t)\\mathcal{G}^1_{n}\\dar\\\\\n \\mathcal{B}^H\\backslash \\mathcal{B}^H K^H_{\\diamondsuit}(p^t)\\mathcal{H}^1_{n}\\rar[hook, \"\\hat\\iota\"]\n & \\mathcal{P}^G\\backslash \\mathcal{P}^G w_1K^G_{\\Iw}(p^t)\\mathcal{G}^1_{n}.\n \\end{tikzcd}\n \\]\n Here, the vertical maps are given by $h\\mapsto \\mathcal{B}^H\\backslash \\mathcal{B}^H h^{-1}$ on the left, and $g\\mapsto \\mathcal{P}^G\\backslash \\mathcal{P}^G w_1g^{-1}$ on the right; the lower horizontal map $\\hat\\iota$ is $\\mathcal{B}^H h\\mapsto \\mathcal{P}^G h \\gamma w_1$, and the map along the top making the diagram commute is $h \\mapsto \\hat\\gamma^{-1} h\\hat\\gamma$, which is well-defined by the preceding lemma. (Note that the commutativity of the diagram relies on the fact that $\\gamma \\in P_G$.)\n\n The right-translation action of $\\mathcal{B}^H$ on $\\mathcal{G}$ makes the left-hand column into a torsor for the group $\\mathcal{B}^H \\cap K^H_{\\diamondsuit}(p^t)\\mathcal{H}^1_{n}$. Similarly, via right-translation conjugated by $w_1$, $K^G_{\\Iw}(p^t)\\mathcal{G}^1_{n}$ becomes a torsor for the group\n \\( \\mathcal{P}\\cap w \\mathcal{G}^1_n K^G_{\\Iw}(p^t)w^{-1}; \\)\n and these structures are compatible if we consider $\\mathcal{B}^H \\cap K^H_{\\diamondsuit}(p^t)\\mathcal{H}^1_{n}$ as a subgroup of $\\mathcal{P} \\cap w \\mathcal{G}^1_n K^G_{\\Iw}(p^t)w^{-1} $ via conjugation by $\\gamma$.\n\n Passing to the $\\mathcal{N}_H$-coinvariants on the left, we obtain a torsor for $T_\\diamondsuit(p^t) \\mathcal{T}^1_{n} = \\mathcal{T}^\\diamondsuit_n$; and passing to $\\mathcal{N}_G$-coinvariants on the right, we obtain a torsor for the projection of $\\mathcal{P} \\cap w \\mathcal{G}^1_n K^G_{\\Iw}(p^t)w^{-1}$ to the Levi $\\mathcal{M}_G$, which is the group $\\mathcal{M}^\\square_{G, n}$. Moreover, these structures are compatible via the $\\gamma$-conjugation inclusion $\\mathcal{T}^\\diamondsuit_n \\hookrightarrow \\mathcal{M}^\\square_{G,n}$ established in the above lemma.\n\n We now note that $\\scalebox{1.15}{$\\mathtt{U}$}^G_n \\subset \\scalebox{1.15}{$\\mathtt{FL}$}^G$ is contained in the subset $\\mathcal{P}^G w_1K^G_{\\Iw}(p^t)\\mathcal{G}^1_{n}$, since $\\scalebox{1.15}{$\\mathtt{U}$}^G_n$ is the orbit of $w_1$ under $K^G_{\\Iw}(p^t) \\mathcal{G}^1_{n, n}$ in the notation of \\bp{\\S 3.3.3}, and $\\mathcal{G}^1_{n, n} \\subset \\mathcal{G}^1_n$. So pulling back to $\\mathcal{U}^G_n$ via the Hodge--Tate period map gives the result.\n \\end{proof}\n\n\\section{Spaces of distributions and branching laws}\n\n \\subsection{Analytic characters}\n\n \\begin{definition}\n Let $n \\in \\mathbf{Q}_{> 0}$. We say a continuous character $\\kappa: \\ZZ_p^\\times \\to A^\\times$, for $(A, A^+)$ a complete Tate algebra, is \\textbf{$n$-analytic} if it extends to an analytic $A$-valued function on the affinoid adic space\n \\[ \\ZZ_p^\\times \\cdot \\mathcal{B}_n \\subset \\mathbf{G}_m^{\\mathrm{ad}}.\\]\n This definition extends naturally to characters $T(\\ZZ_p) \\to A^\\times$: the $n$-analytic characters are exactly those which extend to $\\mathcal{T}^\\square_n$.\n \\end{definition}\n\n \\begin{remark}\n For compatibility with our notations for algebraic weights, we shall denote a $p$-adic character $\\kappa$ of $T(\\ZZ_p)$ by a triple $(\\rho_1, \\rho_2; \\omega)$ of characters of $\\ZZ_p^\\times$, via\n \\[ \\kappa(\\diag(st_1, st_2, st_2^{-1}, st_1^{-1})) = \\rho_1(t_1) \\rho_2(t_2) \\omega(s), \\]\n so that formally\n \\[ \\kappa(\\diag(t_1, t_2, \\nu t_2^{-1}, \\nu t_1^{-1})) = \\rho_1(t_1) \\rho_2(t_2) \\left(\\frac{\\omega}{\\rho_1\\rho_2}\\right)^{\\tfrac{1}{2}}(\\nu).\\]\n This is of course not well-defined as written, since $p$-adic characters do not have a unique square root, so we should understand the triple $(\\rho_1, \\rho_2; \\omega)$ as coming with an implicit choice of square root of $\\omega \/\\rho_1 \\rho_2$ which is being suppressed from the notation.\n \\end{remark}\n\n \\subsection{Analytic inductions}\n\n We recall some definitions from \\bp{\\S 6.1.2}. Let $(A,A^+)$ be a complete Tate algebra over $(\\QQ_p,\\ZZ_p)$. Let $n_0 > 0$, and assume that $\\kappa_A: T(\\ZZ_p)\\rightarrow A^\\times$ is an $n_0$-analytic character. For $?\\in \\{G,H\\}$ and $n \\geq n_0$, let $\\mathcal{M}^1_{?,n}$ be the affinoid subgroup of $\\mathcal{M}_?$ defined above, and let $B_{M_G}$ be the Borel of $M_?$.\n\n \\begin{definition}\n For $n\\geq n_0$, define\n \\begin{align*}\n V^{n-\\an}_{G,\\kappa_A}=&\\, \\an\\Ind^{\\left(\\mathcal{M}^\\square_n\\right) }_{\\left(\\mathcal{M}^\\square_n \\cap \\mathcal{B}_{G}\\right)}(w_{0,M_?}\\kappa_A)\\\\\n =&\\, \\Big\\{ f \\in \\mathcal{O}(\\mathcal{M}^\\square_{G, n}) \\mathop{\\hat\\otimes} A : f(mb)=(w_{0,M}\\kappa_A)(b^{-1})f(m),\\\\\n &\\quad \\forall \\, m\\in \\mathcal{M}^\\square_{G, n},\\, \\forall b\\in \\mathcal{M}^\\square_{G, n} \\cap \\mathcal{B}_G \\Big\\}.\n \\end{align*}\n We define a left action of $\\mathcal{M}^\\square_{G, n}$ on $V^{n-\\an}_{G,\\kappa_A}$ by $(h \\cdot f)(m) = f(h^{-1} m)$.\n\n Write $D^{n-\\an}_{G,\\kappa_A}$ for the dual space, and $\\langle -, - \\rangle$ for the pairing between these; we equip $D^{n-\\an}_{G,\\kappa_A}$ with a left action of the same group $\\mathcal{M}^\\square_{G, n}$, in such a way that $\\langle h\\mu , hf \\rangle = \\langle \\mu ,f \\rangle$.\n \\end{definition}\n\n Let us describe $V_{G,\\kappa_A}^{n-\\an}$ explicitly. We use the description of $\\mathcal{M}_{G, n}^{\\square}$ given in \\cref{note:identifyMGsq}. Since $\\mathcal{M}_{G, n}^{\\square}$ has an Iwahori decomposition, restriction to elements of the form $(\\stbt{1}{}{\\star}{1}, 1)$ identifies $V_{G,\\kappa_A}^{n-\\mathrm{an}}$ with the space of analytic functions of $z \\in \\mathcal{B}_n$; this space is independent of $\\kappa_A$, but the action of $\\mathcal{M}^{\\square}_{G,n}$ does depend on $\\kappa_A$, as follows.\n\n \\begin{propqed}\n Suppose $\\kappa_A$ is the character $(\\rho_1, \\rho_2; \\omega)$. Then the action of $(\\stbt a b c d, \\nu)$ on $f \\in \\mathcal{O}(\\mathcal{B}_n) \\mathop{\\hat\\otimes} A$ is given by\n \\[ \\left((\\stbt a b c d, \\nu) f\\right)(z) = f\\left( \\frac{az-c}{-bz + d}\\right) (-bz + d)^{\\rho_1 - \\rho_2} (ad-bc)^{\\rho_2} \\nu^{(\\omega-\\rho_1-\\rho_2)\/2}.\\qedhere\\]\n \\end{propqed}\n\n \\begin{note}\n For $H$ in place of $G$, we can make the same definitions; but the resulting spaces are much simpler, since $\\mathcal{M}_H = \\mathcal{T}$ is commutative and contained in $\\mathcal{B}_H$. Hence any function $f \\in V^{n-\\an}_{H,\\kappa_A}$ is uniquely determined by its value at 1. So $V^{n-\\an}_{H,\\kappa_A}$ is canonically $A$, with $\\mathcal{T}^\\square_n$ acting via $\\kappa_A$; and dually $D^{n-\\an}_{H,\\kappa_A}$ is $A$ with $\\mathcal{T}^\\square_n$ acting via $\\kappa_A^{-1}$.\n \\end{note}\n\n\n \\begin{note}\n If $\\kappa_A$ is an algebraic character $(k_1, k_2; c)$, then $V_{G,\\kappa_A}^{n-\\an}$ naturally contains the algebraic $M_H$-representation of highest weight $\\kappa_A$ (identified with polynomials in $z$ of degree $\\le k_1 - k_2$); and dually, $D^{n-\\an}_{?,\\kappa_A}$ surjects onto the algebraic representation of highest weight $\\kappa_A^\\vee$ (the dual of the weight $\\kappa_A$ representation).\n \\end{note}\n\n\n\n \\subsection{Branching laws in families}\\label{ss:kraken}\n\n \\begin{definition}\n Let $A$ be a Tate algebra endowed with an $n_0$-analytic character $\\kappa_A: T(\\ZZ_p) \\to A^\\times$ as above, and additionally with a character $\\lambda: (1 + \\mathcal{B}_n)^\\times \\to A^\\times$. Define the \\emph{kraken} to be the function\n \\[ \\mathscr{K}^{\\lambda}(z)=\\lambda(1+z),\\]\n viewed as an element of $V_{G,\\kappa_{A}}^{n-\\an}$.\n \\end{definition}\n\n \\begin{lemma}\n The function $\\mathscr{K}^{\\lambda}$ is an eigenvector for $\\gamma^{-1} \\mathcal{T}^\\diamondsuit_n \\gamma \\subset \\mathcal{M}^\\square_{G, n}$, with eigencharacter $w_{0, M} \\kappa_A + (\\lambda, -\\lambda; 0)$.\n \\end{lemma}\n\n \\begin{proof}\n We have $\\gamma^{-1}\\left( \\stbt x {} {} y, \\nu\\right) \\gamma=(\\stbt x {} {-x+y} y, \\nu)$.\n If this condition is satisfied, then (writing $\\kappa = (\\rho_1, \\rho_2; \\omega)$ as before) we have\n \\[ (\\stbt x {} {-x+y} y, \\nu) \\mathscr{K}^{\\lambda}(z)= x^{\\rho_2}y^{\\rho_1}\\nu^{(\\omega-\\rho_1-\\rho_2)\/2} \\mathscr{K}^{\\lambda}\\left( \\frac{x}{y}(z+1) - 1 \\right)=x^{\\rho_2+\\lambda}y^{\\rho_1-\\lambda} \\nu^{(\\omega-\\rho_1-\\rho_2)\/2}\\mathscr{K}^{\\lambda}(z).\\qedhere \\]\n \\end{proof}\n\n As an immediate consquence, we obtain the following result:\n\n \\begin{propqed}\\label{prop:krakenpower}\n Pairing with the element $\\mathscr{K}^{\\lambda}$ defines a homomorphism of $\\mathcal{T}^\\diamondsuit_n$-representations\n \\[ \\hat\\iota^*( D^{n-\\an}_{G,\\kappa_A}) \\longrightarrow D^{n-\\an}_{H,w_{0, M} \\kappa_A + (\\lambda, -\\lambda;0)}.\\qedhere\\]\n \\end{propqed}\n\n \\begin{note}\n Note that $D^{n-\\an}_{H,w_{0, M} \\kappa_A + (\\lambda, -\\lambda;0)}$ is one-dimensional (and independent of $n$).\n \\end{note}\n\n We now consider a special case. Let $A = \\QQ_p$ and take $\\kappa_A$ to be the algebraic weight $(r_2 + 2, -r_1; -r_1-r_2)$, for some integers $r_1 \\ge r_2\\ge 0$, so that $\\kappa_A^\\vee$ is the weight $\\kappa_1$ of \\eqref{eq:ourweights}. If we choose $\\lambda$ to be an integer in the range $[0, r_1 + r_2 + 2]$, then $\\mathscr{K}^\\lambda$ lies in the polynomial subspace $V_{G, \\kappa_A} \\subset V_{G, \\kappa_A}^{n-\\an}$. Its value at the identity element of $\\mathcal{M}_{G, n}^{\\square}$ is 1, by definition.\n\n So, if $t_i \\ge -1$ are integers with $t_1 + t_2 = r_1 - r_2 - 2$, and we we take $\\lambda$ such that $(r_1 - \\lambda, \\lambda - 2 - r_2) = (t_1, t_2)$, then we obtain a commutative diagram of $\\mathcal{T}^\\diamondsuit_n$-representations\n \\[\\begin{tikzcd}\n \\iota^*(D^{n-\\an}_{G,\\kappa_A}) \\rar \\dar &D^{n-\\an}_{H,-\\tau_A} \\dar[\"\\cong\"] \\\\\n \\iota^*\\left(V_{G,\\kappa_A^\\vee}\\right) \\rar &V_{H, \\tau_A}\n \\end{tikzcd}\\]\n where $\\tau_A^\\vee = (t_1, t_2; r_1 + r_2)$. Hence the homomorphism of Proposition \\ref{prop:krakenpower} is compatible with the classical branching law described in \\cref{sect:branchcoeffs}\n\n\n\\section{Sheaves of distributions}\n\n We use the above function spaces and morphisms as ``models'' for sheaves on the Shimura variety.\n\n \\subsection{Labelling of weights}\n \\label{ss:analyticweights}\n\n We recall some definitions from \\bp{\\S 6.2} (this theory is a bit messy owing to the need to reconcile various different conventions).\n\n As above, we let $(A,A^+)$ be a Tate algebra over $(\\QQ_p,\\ZZ_p)$. Given a weight $\\nu_A : T(\\ZZ_p) \\to A^\\times$ for some coefficient ring $A$, following \\bp{\\S 6}, we define $\\kappa_A: T(\\ZZ_p) \\to A^\\times$ by\n \\[ \\kappa_A = -w_{0, M} w_1 (\\nu + \\rho) - \\rho. \\]\n Explicitly, if $\\nu_A$ is $(\\nu_1, \\nu_2; \\omega)$ for some $\\nu_i, \\omega: \\ZZ_p^\\times \\to A^\\times$, then\n \\[ \\kappa_A = (\\nu_2 - 1, -3-\\nu_1; -\\omega).\\]\n\n We are not so much interested in the linear dual $\\kappa_A^\\vee$ as the ``Serre dual'' $\\kappa_A' = (\\kappa_A + 2\\rho_{nc})^\\vee$. Explicitly this is $(\\nu_1, -2-\\nu_2; c) = w_1(\\nu_A+ \\rho) - \\rho$. So when $A = \\QQ_p$ and $\\nu = (r_1, r_2; r_1 + r_2)$ is an integral algebraic weight, we have $\\kappa_A' = \\kappa_1$ in the notation of \\eqref{eq:ourweights}.\n\n\n \\subsection{Sheaves on $G$}\n\n Let $1 \\le n < t$ be integers.\n\n \\begin{definition}\n We now define two sheaves $\\mathcal{V}^{n-\\an}_{G,\\nu_A}$ and $\\mathcal{D}^{n-\\an}_{G,\\nu_A}$ over $\\mathcal{U}^G_n$. The former can be defined as a subsheaf of $\\pi_* (\\mathcal{M}^G_{\\HT, n,\\Iw})$ transforming like functions in $V^{n-\\an}_{\\kappa_A}$; an alternative, possibly cleaner description is as a coproduct\n \\[ \\mathcal{V}^{n-\\an}_{G, \\nu_A} = \\mathcal{M}^G_{\\HT, n,\\Iw} \\times^{\\mathcal{M}^{\\square}_{G, n}} V^{n-\\an}_{G, \\kappa_A},\\]\n and similarly\n \\begin{equation}\\label{eq:Dsheafascofibreprod}\n \\mathcal{D}^{n-\\an}_{G, \\nu_A} = \\mathcal{M}^G_{\\HT, n,\\Iw} \\times^{\\mathcal{M}^{\\square}_{G, n}} D^{n-\\an}_{G, (\\kappa_A + 2\\rho_{nc})}.\n \\end{equation}\n \\end{definition}\n\n (The shift by $2 \\rho_{nc}$ is present so that the pairing between $\\mathcal{D}^{n-\\an}_{G, \\nu_A}$ and $\\mathcal{V}^{n-\\an}_{G, \\nu_A}$ lands in the dualizing sheaf of $\\mathcal{S}_G$, rather than in the structure sheaf.)\n\n \\begin{lemma}\\label{lem:sheafspec}\n The sheaves $\\mathcal{V}^{n-\\an}_{G,\\nu_A}$ and $\\mathcal{D}^{n-\\an}_{G,\\nu_A}$ are sheaves of $A$-modules, whose formation is compatible with base-change in $A$; and if $A = \\QQ_p$ and $\\nu_A = (r_1, r_2; c)$ for integers $r_1 \\ge r_2 \\ge -1$, we have classical comparison maps\n \\[ \\mathcal{V}_{G, \\kappa_A} \\hookrightarrow \\mathcal{V}^{n-\\an}_{G, \\nu_A},\n \\qquad\n \\mathcal{D}^{n-\\an}_{G, \\nu_A} \\twoheadrightarrow \\mathcal{V}_{G,(\\kappa_A + 2\\rho_{nc})^\\vee} = \\mathcal{V}_{G, \\kappa_1}. \\]\n \\end{lemma}\n\n \\begin{proof}\n See \\bp{Prop. 6.18}.\n \\end{proof}\n\n\n \\subsection{Sheaves on $H$}\n\n There are analogous constructions for sheaves for $H$. Here we use the element $\\mathrm{id} \\in {}^M W_H$ in place of $w_1$, and $w_{0, M_H}$ is the identity. So given an $n$-analytic character $\\tau_A$, we define $\\kappa_A^H = -\\tau_A-2\\rho_H$; and we set\n \\[ \\mathcal{V}^{n-\\an}_{H, \\diamondsuit, \\tau_A} = \\mathcal{M}^H_{\\HT, n,\\diamondsuit} \\times^{\\mathcal{T}^{\\diamondsuit}_n} V^{n-\\an}_{H, \\kappa_A^H},\\]\n and\n \\[ \\mathcal{D}^{n-\\an}_{H, \\diamondsuit, \\tau_A} = \\mathcal{M}^G_{\\HT, n, \\diamondsuit} \\times^{\\mathcal{T}^{\\diamondsuit}_{n}} D^{n-\\an}_{H,(\\kappa_A^H + 2\\rho_H)}.\\]\n\n Thus $\\mathcal{D}^{n-\\an}_{H,\\diamondsuit,\\tau_A}$ for a $\\QQ_p$-valued algebraic character $\\tau_A$ is simply (the restriction to $\\mathcal{U}_n^H$ of) the line bundle $\\mathcal{V}^H_{\\tau_A}$. The same definitions make sense at Iwahori level, of course, giving line bundles $\\mathcal{D}^{n-\\an}_{H,\\Iw,\\tau_A}$ and $\\mathcal{V}^{n-\\an}_{H,\\Iw,\\tau_A}$. These sheaves are in fact independent of $n$ (in the sense that $\\mathcal{D}^{(n+1)-\\an}_{H,\\Iw,\\tau_A}$ is isomorphic to the restriction of $\\mathcal{D}^{n-\\an}_{H,\\Iw,\\tau_A}$ to $\\mathcal{U}_{n+1}^H$), so we shall frequently drop the $n$ and write simply $\\mathcal{D}^{\\an}_{H,\\Iw,\\tau_A}$ etc.\n\n \\begin{remark}\\label{remark:bigsheafsmallt}\n Note that (for simplicity) we have only attempted to define the locally-analytic sheaves for $G$ when the level group at $p$ is $\\Iw(p^t)$ with $t > n$; thus our functions are defined on $\\mathcal{B}_n$ itself, rather than on a union of translates of $\\mathcal{B}_n$. (This restriction on the levels is inherited from \\bp{\\S 6.3}.)\n\n However, for $H$ the technical difficulties disappear, and we can make sense of $\\mathcal{V}^{\\an}_{H,\\Iw,\\tau_A}$ and $\\mathcal{D}^{\\an}_{H,\\Iw,\\tau_A}$ as vector bundles on $\\mathcal{U}_{n, \\Iw}^H(p^t)$ for any $n, t \\ge 1$.\n \\end{remark}\n\n\n \\subsection{Branching for sheaves}\n\n \\begin{definition}\n \\label{def:compat}\n We say the $A$-valued, $n$-analytic characters $\\nu_A$ and $\\tau_A$ of $T(\\ZZ_p)$ are \\emph{compatible} if $\\nu_A = (\\nu_1, \\nu_2; \\nu_1 + \\nu_2)$, $\\tau_A = (\\tau_1, \\tau_2; \\nu_1 + \\nu_2)$, for some characters $\\nu_i, \\tau_i$ of $\\ZZ_p^\\times$, and we have the relation\n \\[\n \\tau_1 + \\tau_2 = \\nu_1 - \\nu_2 - 2.\n \\]\n \\end{definition}\n\n\n\n Recall the kraken $\\mathscr{K}^\\lambda$ defined in \\cref{ss:kraken}. If $\\nu_A, \\tau_A$ are compatible, then taking $\\lambda = \\nu_1 - \\tau_1 = \\nu_2 + \\tau_2 + 2$, we obtain a homomorphism of $\\mathcal{T}^\\diamondsuit_n$-representations\n \\[ D^{n-\\an}_{G,(\\kappa_A + 2\\rho_{nc})}\n \\longrightarrow\n \n D^{n-\\an}_{H,-\\tau_A}\\]\n where $\\mathcal{T}^{\\diamondsuit}_n$ acts on $D^{n-\\an}_{G,(\\kappa_A + 2\\rho_{nc})}$ via $\\gamma$-conjugation. So the following result is an immediate consequence of \\cref{prop:krakenpower} and the results of \\cref{ss:analyticweights}:\n\n \\begin{propqed}\\label{prop:krakenonsheaves}\n Pairing with $\\mathscr{K}^\\lambda$ induces a morphism of sheaves over $\\mathcal{U}^H_{n}$:\n \\[ \\hat\\iota^*( \\mathcal{D}^{n-\\an}_{G,\\nu_A})\\longrightarrow \\mathcal{D}^{\\an}_{H, \\diamondsuit,\\tau_A}.\\]\n This morphism is compatible with specialisation in $A$, and if $A = \\QQ_p$ and $\\nu = (r_1, r_2; r_1+r_2)$, $\\tau = (t_1, t_2; r_1 + r_2)$ are algebraic weights with $r_1 - r_2 \\ge 0$ and $r_i, t_i \\ge -1$, then this morphism is compatible with the map of finite-dimensional sheaves $\\hat\\iota^*\\left(\\mathcal{V}_{\\kappa_1}\\right) \\to \\mathcal{V}^H_{\\tau}$ defined in \\S\\ref{sect:branchcoeffs}.\n \\end{propqed}\n\n\n\n \\subsection{Locally analytic overconvergent cohomology}\n\n Let $m,n,t$ be as in \\cref{eq:mnt}, with $n > 0$; and suppose $\\nu_A$ is an $n$-analytic $A$-valued character of $T(\\ZZ_p)$. We define cuspidal, locally analytic, overconvergent cohomology to be\n \\begin{equation}\\label{eq:cusplocanaoc}\n R\\Gamma^G_{w, \\an}(\\nu_A, \\cusp)^{-, \\fs} = R\\Gamma_{\\mathcal{I}_{mn}^G}\\left(\\mathcal{U}^G_n, \\mathcal{D}^{n-\\an}_{G,\\nu_A}(-D_G)\\right)^{-, \\fs},\n \\end{equation}\n and similarly for the non-cuspidal version. As shown in \\bp{\\S 6}, this complex is independent of $m$, $n$ and $t$, and is concentrated in degrees $[0, 1, 2]$.\n\n\n \\begin{proposition}\n \\label{prop:bigsheafpullback}\n Given $\\nu_A$ and $\\tau_A$ satisfying the compatibility condition of Definition \\ref{def:compat}, we have a morphism of complexes of $A$-modules\n \\[\n \\hat\\iota^*: R\\Gamma^G_{w, \\an}(\\nu_A, \\cusp)^{-, \\fs}\n \\to R\\Gamma_{\\mathcal{Z}^H_m}\\left(\\mathcal{U}_n^H, \\mathcal{D}^{n-\\an}_{H,\\diamondsuit,\\tau_A}(-D_H)\\right).\n \\]\n \\end{proposition}\n\n \\begin{proof}\n Immediate from \\cref{prop:krakenonsheaves}.\n \\end{proof}\n\n We have only defined this morphism for $m, n$ small relative to $t$. However, using \\cref{remark:bigsheafsmallt}, we can argue as in \\cref{sect:changelevel} and define\n \\[ R\\Gamma_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^t), \\tau_A, \\cusp)^{-, \\dag} = \\varprojlim_m R\\Gamma_{\\mathcal{Z}^H_{m, \\Iw}(p^t)}\\left( \\mathcal{U}^H_{n, \\Iw}, \\mathcal{D}^{\\an}_{H,\\Iw,\\tau_A}(-D_H)\\right).\\]\n Then we obtain a natural map\n \\[R\\Gamma^G_{w, \\an}(\\nu_A, \\cusp)^{-, \\fs} \\to R\\Gamma_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^t), \\tau_A, \\cusp)^{-, \\dag}.\\]\n\n \\begin{note}\n By construction, this morphism is compatible with derived base-change in $A$. If $A = \\QQ_p$, and $\\nu_A$ and $\\tau_A$ are algebraic weights such that $r_1 \\ge r_2 \\ge -1$ and $t_1, t_2 \\ge -1$, then this map fits into a commutative diagram with the pullback map on overconvergent cohomology defined in \\S\\ref{sect:changelevel}.\n \\end{note}\n\n \\subsection{Pairings and duality}\n\n Dually to the above, we define\n \\[\n R\\Gamma_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^t), \\tau_A)^{+, \\dag} = \\varinjlim_m R\\Gamma\\left(\\mathcal{Z}^H_{m, \\Iw}(p^t), \\mathcal{V}^{\\an}_{H,\\Iw, \\tau_A}\\right).\n \\]\n\n Note that if $\\tau_A = (t_1, t_2; c)$ then $\\mathcal{V}^{\\an}_{H,\\Iw, \\tau_A}$ is the sheaf $\\mathcal{V}^H_{-\\tau-2\\rho_H} = \\mathcal{V}_{(-2-t_1, -2-t_2; -c)}$, which is the sheaf of modular forms of weight $(t_1 + 2, t_2 + 2)$ (with the normalisation of the central character depending on $c$).\n\n \\begin{proposition}\n The above complex is concentrated in degree 0 and independent of $t$. It can be identified with the space of $p$-adic overconvergent modular forms for $H$ of tame level $K^{H, p}$ and weight $\\tau_A + (2, 2)$.\n \\end{proposition}\n\n \\begin{proof}\n For simplicity we suppose $K^{H, p}$ is the principal congruence subgroup of level $N$ for some $N$ (the general case reduces easily to this). Then the Shimura variety for $H$ is simply the fibre product (over $\\mu_N$) of two copies of the level $N$ modular curve parametrising elliptic curves with full level $N$ structure and a cyclic subgroup of order $p^t$. Then $\\pi_{HT}^{-1}(\\{\\mathrm{id}_H\\})$ is the ``canonical locus'', where the $p$-subgroups are both multiplicative; and the $\\mathcal{Z}_{m, \\Iw}^H$ are a cofinal family of neighbourhoods of this locus. Via the theory of the canonical subgroup, this space is independent of the choice of levels.\n\n Since the canonical locus is affinoid (and sufficiently small strict neighbourhoods of it also have this property), its cohomology vanishes above degree 0, and the degree 0 cohomology identifies with overconvergent sections of $\\mathcal{V}^{\\an}_{H, \\Iw, \\tau_A}$. If we choose an extension $\\tilde\\tau_A$ of $\\tau_A$ to the maximal torus of $\\GL_2 \\times \\GL_2$, then $\\mathcal{V}^{\\an}_{H, \\Iw, \\tau_A}$ decomposes as the product of two copies of the corresponding sheaves on the individual modular curves. This is precisely the construction of overconvergent modular forms described in \\cite{pilloni13} (see the discussion following Prop 6.2 of \\emph{op.cit.} for a comparison with Coleman's original approach).\n \\end{proof}\n\n \\begin{theorem}[c.f. \\bp{Theorem 6.38}]\n \\label{thm:bspairing}\n The cup product induces a pairing\n \\[ H^2_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^t), \\tau_A, \\cusp)^{-, \\dag} \\times H^0_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^t), \\tau_A)^{+, \\dag} \\longrightarrow A, \\]\n whose formation is compatible with base-change in $A$, and which is compatible with the Serre duality pairing on classical cohomology when $A = \\QQ_p$ and $\\nu$, $\\tau$ are classical weights.\n \\end{theorem}\n\n \\begin{proof}\n We define this pairing by combining the pullback map of \\ref{prop:bigsheafpullback} with the pairing between the cohomology groups $H^2_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^t), \\tau_A, \\cusp)^{-, \\dag}$ and $H^0_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^t), \\tau_A, \\cusp)^{+, \\dag}$. By construction, this is compatible with Serre duality for each classical weight.\n \\end{proof}\n\n\n\n\n\n\n\\section{Construction of the $p$-adic $L$-function}\n\n Let $L$ be a finite extension of $\\QQ_p$.\n\n \\subsection{Families of Eisenstein series}\n\n We refer to \\cite[\\S 7]{LPSZ1} for the construction of $p$-adic families of Eisenstein series $\\mathcal{E}^{\\Phi^{(p)}}(\\kappa_1, \\kappa_2; \\chi^{(p)})$, depending on a prime-to-$p$ Schwartz function $\\Phi^{(p)}$ and prime-to-$p$ Dirichlet character $\\chi^{(p)}$ (both valued in $L$) and a pair of characters $\\kappa_1, \\kappa_2$ of $\\ZZ_p^\\times$ (valued in some $p$-adically complete $L$-algebra $A$).\n\n \\begin{note}\n Note that this Eisenstein series is $p$-depleted, i.e.~lies in the kernel of $U_p$; and it is zero on any components of $\\Spec(A)$ which do not satisfy the parity condition $\\kappa_1(-1) \\kappa_2(-1) = -\\chi^{(p)}(-1)$.\n\n The construction factors through the projection of $\\Phi^{(p)}$ to the eigenspace where $\\stbt{a}{0}{0}{a}$ for $a \\in \\widehat{\\mathbf{Z}}^{(p)}$ acts as $\\widehat{\\chi}^{(p)}(a)^{-1}$, where $\\widehat{\\chi}^{(p)}$ is the adelic character attached to $\\chi^{(p)}$ as in \\cite[\\S 2.2]{LPSZ1}. We shall henceforth assume, without loss of generality, that $\\Phi^{(p)}$ lies in this eigenspace; thus $\\chi^{(p)}$ is uniquely determined by $\\Phi^{(p)}$ and we sometimes drop it from the notation.\n \\end{note}\n\n \\begin{proposition}\n If $A$ is an affinoid algebra, and one of the $\\kappa_i$ is a finite-order character, then $\\mathcal{E}^{\\Phi^{p}}(\\kappa_1, \\kappa_2)$ is an overconvergent $A$-valued cusp form of weight-character $1 + \\kappa_1 + \\kappa_2$.\n \\end{proposition}\n\n \\begin{proof}\n Since twisting by a finite-order character preserves overconvergence, it suffices to assume $\\kappa_1$ or $\\kappa_2$ is 0. Then our $p$-adic Eisenstein series is the $p$-depletion of a family of \\emph{ordinary} Eisenstein series, cf.~\\cite[\\S 2.3]{ohta99}, and it is well-known that these ordinary Eisenstein series are overconvergent (indeed, this is true by definition in Coleman's approach to overconvergent modular forms).\n \\end{proof}\n\n As noted in \\emph{op.cit.}, for $k \\ge 1$, the Eisenstein series $F^{k}_{\\Phi^p \\Phi_{\\mathrm{dep}}}$ described in \\cite[\\S 4.3]{LZ20b-regulator} is (the classical form associated to) $\\mathcal{E}^{\\Phi^p}(k-1, 0)$, and $E^{k}_{\\Phi^p \\Phi_{\\mathrm{dep}}}$ is $\\mathcal{E}^{\\Phi^p}(0, k-1)$. It also implies the following relation:\n\n \\begin{proposition}[cf.~{\\cite[Prop 16.2.1]{LZ20b-regulator}}]\n Let $t \\in \\mathbf{Z}_{\\ge 0}$. As overconvergent cusp forms of weight $-t$, we have\n \\[ \\theta^{-(1+t)}\\left(F^{(t+2)}_{\\Phi^p \\Phi_{\\mathrm{dep}}}\\right) = \\mathcal{E}^{\\Phi^p}(0, -1-t; \\Phi^{(p)}), \\]\n where $\\theta = q \\tfrac{\\mathrm{d}}{\\mathrm{d}q}$ is the Serre differential operator.\n \\end{proposition}\n\n \\subsection{Tame test data}\n\n We fix the following data:\n \\begin{itemize}\n \\item $M_0, N_0$ are positive integers coprime to $p$ with $M_0^2 \\mid N_0$, and $\\chi_0$ is a Dirichlet character of conductor $M_0$ (valued in $L$).\n \\item $M_2, N_2$ are positive integers coprime to $p$ with $M_2 \\mid N_2$, and $\\chi_2$ is a Dirichlet character of conductor $M_2$ (valued in $L$).\n \\end{itemize}\n We will consider automorphic representations $\\pi$ of $G$ with conductor $N_0$ and character $\\widehat{\\chi}_0$ up to twists by norm, and similarly $\\sigma$ of $\\GL_2$ with conductor $N_2$ and character $\\widehat{\\chi}_2$ up to twists by norm.\\footnote{This numbering of the parameters comes from the fact that the zeta-integral computations of \\cite{LPSZ1} are simpler to write down if the Eisenstein series lives on the first factor of $H$.}\n\n Let $S$ denote the set of primes dividing $N_0 N_2$. By \\emph{tame test data} we shall mean a pair $\\gamma_S = (\\gamma_{0, S}, \\Phi_S)$, where:\n \\begin{itemize}\n \\item $\\gamma_{0, S} \\in G(\\mathbf{Q}_S)$, where $\\mathbf{Q}_S = \\prod_{\\ell \\in S} \\mathbf{Q}_\\ell$;\n \\item $\\Phi_S \\in C^\\infty_c(\\mathbf{Q}_S^2, L)$, lying in the $\\left(\\widehat{\\chi}_0 \\widehat{\\chi}_2\\right)^{-1}$-eigenspace for $\\mathbf{Z}_S^\\times$.\n \\end{itemize}\n We let $K_S$ be the quasi-paramodular subgroup of $G(\\mathbf{Q}_S)$ of level $(N_0, M_0)$; and we let $\\widehat{K}_S$ be some open compact subgroup of $G(\\mathbf{Q}_S)$ such that:\n \\begin{itemize}\n \\item $\\widehat{K}_S \\subseteq \\gamma_{0, S} K_S \\gamma_{0, S}^{-1}$,\n \\item the projection of $\\widehat{K}_S \\cap H$ to the first factor of $H$ acts trivially on $\\Phi_S$,\n \\item the projection of $\\widehat{K}_S \\cap H$ to the second factor of $H$ is contained in $\\{ \\stbt \\star\\star0 1 \\bmod N_2\\}$.\n \\end{itemize}\n We define $K^p$ and $\\widehat{K}^p$ to be the products of $K_S$ and $\\widehat{K}_S$ with $G(\\AA_{\\mathrm{f}}^{pS})$, and $\\Phi^{(p)} = \\Phi_S \\cdot \\operatorname{ch}\\left((\\widehat{\\mathbf{Z}}^{S \\cup \\{p\\}})^2\\right)$.\n\n \\subsection{The correction term $Z_S$}\n\n Let $\\pi$ and $\\sigma$ be cohomological cuspidal automorphic representations of $G$ and of $\\GL_2$, both defined over some number field $E$ contained in the $p$-adic field $L$, and both globally generic and unramified outside $S$. We normalise so these are cohomological with weights $(r_1, r_2; r_1 + r_2)$ and $(t_2; t_2)$ respectively, for some integers $r_1, r_2, t_2$; and we let $\\Pi$ and $\\Sigma$ be the unitary twists of $\\pi$ and $\\sigma$ respectively, so that\n \\[ L(\\Pi \\times \\Sigma, s) = L(\\pi \\times \\sigma, s + \\tfrac{r_1 + r_2 + t_2}{2}).\\]\n\n \\begin{definition}\n For $W_0 \\in \\mathcal{W}(\\pi)_E$, $W_2 \\in \\mathcal{W}(\\sigma)_E$, and $\\Phi \\in \\mathcal{S}(\\mathbf{Q}_S^2, E)$, we consider the zeta-integral\n \\[ Z(W_0, \\Phi, W_2; s) = \\int_{(Z_G N_H \\backslash H)(\\mathbf{Q}_S)} W_0(h) f^{\\Phi}(h_1; \\omega_{\\pi}\\omega_\\sigma, s) W_2(h_2) \\, \\mathrm{d}h. \\]\n \\end{definition}\n\n We shall set\n \\[\n Z_S(\\pi \\times \\sigma, \\gamma_S; s) =\n \\frac{Z(\\gamma_{0, S} \\cdot W_0^{\\mathrm{new}}, \\Phi_S, W_2^\\mathrm{new}; s)}{G(\\chi_2^{-1})\\prod_{\\ell \\in S} L(\\pi_\\ell \\times \\sigma_\\ell, s)},\n \\]\n and\n \\[ Z_S(\\pi \\times \\sigma, \\gamma_S) = Z_S(\\pi \\times \\sigma, \\gamma_S; 1 + \\tfrac{t_1}{2})\\]\n where $t_1 = r_1 -r_2-2-t_2$ as usual. Here $G(\\chi) = \\sum_{a \\bmod N_\\chi} \\chi(a) \\exp(2\\pi i a \/ N_\\chi)$ is the Gauss sum of the character $\\chi$. One can check that this is a product of polynomials in the variables $\\ell^{\\pm s}$, for $\\ell \\in S$, with coefficients in $E$.\n\n \\begin{proposition}\n For any given $\\pi, \\sigma$, one can choose $\\gamma_S$ such that $Z_S(\\pi \\times \\sigma, \\gamma_S; s) \\ne 0$.\n \\end{proposition}\n\n \\begin{proof}\n This follows from the definition of the $L$-factor as a GCD of local zeta-integrals.\n \\end{proof}\n\n \\subsection{P-adic families for $G$}\n\n Let $U \\subset \\mathcal{W}^2$ be an open affinoid disc; and let $\\mathbf{r}_1$, $\\mathbf{r}_2: \\ZZ_p^\\times \\to \\mathcal{O}(U)^\\times$ be the universal characters associated to the two factors of $\\mathcal{W}^2$. Let $\\nu_U$ be the character $(\\mathbf{r}_1, \\mathbf{r}_2; \\mathbf{r}_1+\\mathbf{r}_2)$ of $T(\\ZZ_p)$.\n\n The theory of \\cite{boxerpilloni20} shows that there exists a rigid space $\\mathcal{E} \\xrightarrow{\\kappa} \\mathcal{W}^2$, with a map $\\mathbb{T}^- \\to \\mathcal{O}(\\mathcal{E})$ (the eigenvariety for $G$), and graded coherent sheaves $H^k(\\mathcal{M}^{\\bullet,-,\\fs}_{\\cusp, w_j})$ on $\\mathcal{E}$ for $0 \\le j, k \\le 3$, whose pushforward to any affinoid $U\\subset \\mathcal{W}^2$ as above is $H^k_{w_j,\\an}(K^p, \\nu_U, \\cusp)^{(-,\\fs)}$. By construction, the points of $\\mathcal{E}$ biject with systems of $\\mathbb{T}^-$-eigenvalues appearing in one of these modules.\n\n \\begin{definition}\n By a \\emph{family of automorphic representations} $\\underline{\\pi}$ over $U$ (of tame level $N_0$ and character $\\chi_0$), we mean the data of a finite flat covering $\\tilde{U} \\to U$, and a homomorphism $\\tilde{U} \\to \\mathcal{E}$ lifting the inclusion $U \\hookrightarrow \\mathcal{W}$, such that the following conditions hold:\n \\begin{itemize}\n \\item $\\tilde{U}$ is 2-dimensional and smooth;\n \\item the restriction of the sheaf $H^k(\\mathcal{M}^{\\bullet,-,\\fs}_{\\cusp, w_j})$ to $\\tilde{U}$ is zero if $k \\ne 3-j$, and the sheaves $S^k(\\underline{\\pi}) = H^k(\\mathcal{M}^{\\bullet,-,\\fs}_{\\cusp, w_{3-k}})$ are either free over $\\mathcal{O}(\\tilde{U})$ of rank 1 for all $k$ (a general-type family), or free of rank 1 for $k = 1,2$ and zero for $k = 0, 3$ (a Yoshida-type family);\n \\item the centre of $G(\\AA_{\\mathrm{f}}^p)$ acts on the modules $S^k(\\underline{\\pi})$ by the character $|\\cdot|^{-(\\mathbf{r}_1 + \\mathbf{r}_2)} \\widehat{\\chi}_0$.\n \\end{itemize}\n \\end{definition}\n\n Such a family determines a $\\mathcal{O}(\\tilde{U})$-valued character $\\lambda_{\\underline{\\pi}}^-$ of $\\mathbb{T}^-$, which is the system of eigenvalues by which $\\mathbb{T}$ acts on the modules $H^k(\\mathcal{M}^{\\bullet,-,\\fs}_{\\cusp, w_j})$; conversely, the character $\\lambda_{\\underline{\\pi}}^-$ and ring extension $\\mathcal{O}(\\tilde U)$ of $\\mathcal{O}(U)$ uniquely determine $\\underline{\\pi}$.\n\n \\begin{definition}\n We say a point $P \\in \\tilde{U}(L)$ is ``good for $\\underline{\\pi}$'' if the following conditions hold:\n \\begin{itemize}\n \\item the weight of $P$ is $(r_1, r_2) \\in U \\cap \\mathbf{Z}^2$ with $r_1 \\ge r_2 \\ge -1$;\n \\item the specialisation at $P$ of the system of eigenvalues $\\lambda^-_{\\underline{\\pi}}$ is the character of $\\mathbb{T}^-$ assocated to a $p$-stabilised automorphic representation $\\pi_P$, which is cuspidal, globally generic, and has conductor $N_0$ and character $\\chi_0$;\n \\item the fibre of $S^2(\\underline{\\pi})$ at $P$ maps isomorphically to the $\\pi_P$-eigenspace in the classical $H^2(K^p, \\kappa_1(\\nu), \\cusp)$; in particular, this eigenspace is 1-dimensional.\n \\end{itemize}\n \\end{definition}\n\n \\begin{remark}\n Note that we do not suppose that the $\\pi_P$ \\emph{generalised} eigenspace be 1-dimensional, and this will not hold when $\\tilde{U} \\to U$ is ramified at $P$.\n \\end{remark}\n\n By the classicity theorems for higher Coleman theory recalled above, given a family $\\underline{\\pi}$, all specialisations of integer weight $(r_1, r_2)$ with $r_1 - r_2$ and $r_2$ sufficiently large relative to the slope of $\\underline{\\pi}$ will be good; and if $\\underline{\\pi}$ is ordinary, it suffices to assume that $r_1-r_2 \\ge 3$ and $r_2 \\ge 0$.\n\n We shall choose a basis $\\underline{\\eta}$ of $S^2(\\underline{\\pi})$. Since the spaces of higher Coleman theory (of varying levels) have an action of $G(\\AA_{\\mathrm{f}}^p)$, we can make sense of $\\gamma_{0, S} \\cdot \\underline{\\eta}$ as a family of classes at tame level $\\widehat{K}^p$, which is still an eigenfamily for the Hecke operators away from $S$.\n\n \\subsection{Families for $\\GL_2$}\n\n Similarly, we choose a disc $U' \\subset \\mathcal{W}$, a finite flat covering $\\tilde{U}'\\to U'$ with $\\tilde{U}'$ smooth, and a finite-slope overconvergent $p$-adic family of modular eigenforms $\\mathcal{G}$ over $\\tilde{U}'$ (of weight $\\mathbf{t_2} + 2$ where $\\mathbf{t_2}$ is the universal character associated to $U'$). We suppose that this family is new away from $p$ of tame level $N_2$, and nebentype character $\\chi_2$.\n\n We say a point $Q \\in \\tilde{U}'$ is ``good for $\\mathcal{G}$'' if it lies above an integer $t \\in U' \\cap \\mathbf{Z}_{\\ge -1}$, and the specialisation of $\\mathcal{G}$ at $Q$, which is \\emph{a priori} an overconvergent form of weight $t + 2$, is in fact a classical form. (This is automatic if $t$ is sufficiently large compared to the slope of $\\mathcal{G}$.) We write $\\sigma_t$ for the corresponding automorphic representation (normalised to have central character $|\\cdot|^{-t} \\widehat{\\chi}_2$); and we formally write $\\underline{\\sigma}$ for the collection of the $\\sigma_t$ for varying $t$.\n\n A mildly irritating detail is that if $\\mathcal{G}$ is normalised to have $a_1(\\mathcal{G}) = 1$, and $t$ is a good specialisation, then $\\mathcal{G}_t$ has $q$-expansion coefficients in some number field $E$; but the modular form $\\mathcal{G}_t$ is not defined over $E$ as a coherent cohomology class, since the cusp $\\infty$ on $X_1(N)$ is not defined over $\\mathbf{Q}$ (with our conventions). However, the class $G(\\chi_2^{-1})\\mathcal{G}_t$ is $E$-rational. We write $S^0(\\sigma_t, E)$ for the $E$-vector space spanned by this form, and similarly $S^0(\\underline{\\sigma})$ for the $\\mathcal{O}(\\tilde{U}')$-module of overconvergent cusp forms generated by $G(\\chi_2^{-1}) \\mathcal{G}$.\n\n \\begin{remark}\n Note that by definition $S^0(\\underline{\\sigma})$ is free of rank 1, and its fibre at any good specialisation is in the image of the classical $H^0$ (because of the $q$-expansion principle for $p$-adic modular forms). Hence we do not need any auxiliary hypotheses about local freeness of sheaves.\n \\end{remark}\n\n \\subsection{Deforming eigenforms}\n\n Conversely, we say a classical ($p$-stabilised) automorphic representation $\\pi$, of some weight $\\nu$, is \\emph{deformable} if we can find a disc $U$ containing $\\nu$, a family $\\underline{\\pi}$ over some covering $\\tilde{U} \/ U$, and some $Q \\in \\tilde{U}$ above $\\nu$, such that $Q$ is good for $\\underline{\\pi}$ and the specialisation there is $\\pi$. The arguments of \\cref{sect:families} show that any generic $\\pi$ of cohomological weight, with a regular $p$-stabilisation of sufficiently small slope, will be deformable in the above sense (and we may suppose that $\\tilde{U}=U$); again, if $\\pi$ is ordinary, it suffices to suppose that $r_1 - r_2 \\ge 3$ and $r_2 \\ge 0$.\n\n For $\\GL_2$ we are in much better shape (partly because $\\GL_2$ is better understood than $\\GSp_4$, and partly because our definition of ``family'' is less restrictive): any classical $p$-stabilised newform of integer weight and Iwahori level at $p$ will be deformable, even in the worst-case scenario of non-$p$-regular weight 1 forms, since we may take $\\tilde{U}'$ to be a neighbourhood of $\\sigma$ in the normalisation of the eigencurve. Moreover, if $\\sigma$ is ordinary and has weight $\\ge 2$, we may suppose $\\tilde{U}' = U'$.\n\n \\begin{remark}\n We also expect that there exist interesting examples of deformable $\\pi$ for $G$ which do not satisfy these stringent conditions. It seems likely that the extra generality of a finite flat covering of weight space will be genuinely necessary, at least in the non-regular-weight case $r_2 = -1$. However, for simplicity of notation we shall assume $\\tilde{U} = U$ and $\\tilde{U}' = U'$ henceforth; extending these arguments to the general case is straightforward and we leave this to the reader.\n \\end{remark}\n\n\n \\subsubsection*{Families over $U \\times U'$} Let $A = \\mathcal{O}(U \\times U')$. We have two canonical $A$-valued characters of $T(\\ZZ_p)$: the canonical character $\\nu_A = (\\mathbf{r}_1, \\mathbf{r}_2; \\mathbf{r}_1 + \\mathbf{r}_2)$, and the character $\\tau_A = (\\mathbf{t}_1, \\mathbf{t}_2; \\mathbf{r}_1 + \\mathbf{r}_2)$ defined as follows: $\\mathbf{t}_2$ is the canonical character of $U'$ as above, and $\\mathbf{t}_1 = \\mathbf{r}_1 - \\mathbf{r}_2 - 2 - \\mathbf{t}_2$ and the action of the centre are determined by the requirement that $\\nu_A$ and $\\tau_A$ be ``compatible'' in the sense of \\cref{def:compat}. Then we can consider\n \\[ \\mathcal{E}^{\\Phi^{(p)}}(0, \\mathbf{t}_1 + 1) \\boxtimes G(\\chi_2^{-1}) \\mathcal{G}^{[p]} \\in H^0_{\\mathrm{id}, \\an}(\\mathcal{S}_{H, \\Iw}(p^2), \\tau_{A})^{+, \\dag},\\]\n where the tame level is taken to be $H \\cap \\widehat{K}^p$.\n\n \\begin{definition}\n We let $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}; \\underline{\\eta})$ denote the element of $A$ defined by\n \\[ \\left\\langle \\hat\\iota^*\\left(\\gamma_{0, S} \\cdot \\underline{\\eta}\\right), \\mathcal{E}^{\\Phi^{(p)}}(0, \\mathbf{t}_1 + 1) \\boxtimes G(\\chi_2^{-1}) \\mathcal{G}^{[p]}\\right\\rangle.\\]\n \\end{definition}\n\n (The product denotes the Serre duality pairing at level $\\widehat{K}^p \\cap H$, normalised by a factor $\\operatorname{vol}(\\widehat{K}^p \\cap H)$ in order to make it independent of the choice of $\\widehat{K}_p$.)\n \\begin{definition} \\\n \\begin{itemize}\n \\item We say a point $(P, Q)$ of $U \\times U'$ is \\emph{good} if $P = (r_1, r_2)$ and $Q = (t_2)$ are integer points, with $P$ good for $\\underline{\\pi}$ and $Q$ good for $\\underline{\\sigma}$.\n\n \\item We say $(P, Q)$ is \\emph{good critical} if we also have $t_2 \\le r_1 -r_2 - 1$ (i.e.~the specialisation $t_1$ of $\\mathbf{t}_1$ at $(P, Q)$ is $\\ge -1$).\n\n \\item If instead we have $r_1 - r_2 \\le t_2 \\le r_1$, we say $P$ is \\emph{good geometric}.\n \\end{itemize}\n \\end{definition}\n\n One checks easily that any integer point $(r_1, r_2, t_2)$ is the limit of a sequence of good geometric (or good critical) points, so if we exclude the pathological case when $(U \\times U') \\cap \\mathbf{Z}^3$ is empty, then the sets of good critical points and of good geometric points are both Zariski-dense in $U \\times U'$.\n\n \\subsection{Values in the critical range}\n\n \\begin{definition}\n For $(P, Q) = (r_1, r_2, t_2) \\in U \\times U'$ a good critical point, we define a degree 8 Euler factor\n \\[\n \\mathcal{E}_p(\\pi_P \\times \\sigma_Q) = \\left(1 - \\tfrac{p^{r_1 + 1}}{\\alpha \\mathfrak{a}}\\right)\\dots \\left( 1- \\tfrac{p^{r_1 + 1}}{\\beta \\mathfrak{b}}\\right) \\left(1 - \\tfrac{\\gamma \\mathfrak{a}}{p^{r_1 + 2}}\\right)\\dots\\left(1 - \\tfrac{\\delta \\mathfrak{b}}{p^{r_1 + 2}}\\right).\n \\]\n where $\\alpha, \\dots, \\delta$ are the Hecke parameters of $\\pi_P$, and $\\mathfrak{a},\\mathfrak{b}$ the Hecke parameters of $\\sigma_Q$ (so that $\\mathfrak{a}\\mathfrak{b} = p^{t_2 + 1} \\chi_2(p)$).\n \\end{definition}\n\n \\begin{proposition}\n If $\\pi_P$ is ordinary, then $\\mathcal{E}_p(\\pi_P\\times \\sigma_Q) \\ne 0$.\n \\end{proposition}\n\n \\begin{proof}\n This follows by a (somewhat tedious) explicit check from the bounds on the valuations of the Hecke parameters.\n \\end{proof}\n\n \\begin{theorem}\n The $p$-adic $L$-function $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}, \\underline{\\eta})$ has the following interpolation property: if $(P, Q)$ is good critical, then\n \\[ \\frac{\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}, \\underline{\\eta})(P, Q)}{\\Omega_p(\\pi_P, \\eta_P)}\n =Z_S(\\pi_{P} \\times \\sigma_{Q}, \\gamma_S) \\cdot \\mathcal{E}_p(\\pi_P \\times \\sigma_Q) \\cdot\n \\frac{G(\\chi_2^{-1})^2 \\Lambda\\left(\\Pi_P \\times \\Sigma_Q, 1 + \\tfrac{t_1}{2}\\right)}{\\Omega_\\infty(\\pi_P, \\eta_P)}, \\]\n with both sides lying in the field of rationality of $\\pi_P\\times \\sigma_Q$.\n\n Here $\\Pi_P$ and $\\Sigma_Q$ are the (unitary) automorphic representations generated by the specialisations of $\\underline{\\eta}$ and $\\mathcal{G}$ at $P$; and $\\Lambda(\\Pi_P\\times \\Sigma_Q, s)$ denotes the $L$-function of these automorphic representations, with its archimedean $\\Gamma$-factors included.\n \\end{theorem}\n\n \\begin{remark}\n Note that $s = 1 + \\tfrac{t_1}{2}$ is the upper endpoint of the interval of critical values (in the sense of Deligne) for the degree 8 $L$-function $L\\left(\\Pi_P \\times \\Sigma_Q, s\\right)$. This critical interval is symmetric about $s = \\tfrac{1}{2}$, so unless $t_2 = r_1 - r_2 - 1$ (so that $t_1 = -1$), there are other critical values which we do not see by this method.\n\n We optimistically hope that there should be a $p$-adic $L$-function on the 4-dimensional space $U \\times U' \\times \\mathcal{W}$ which interpolates the full range of critical values, and that both the above $p$-adic $L$-function on $U \\times U'$, and the 2-variable $p$-adic $L$-function on $U' \\times \\mathcal{W}$ (for fixed $\\pi$) considered in \\cite[\\S 5]{LZ20b-regulator}, should be ``slices'' of this more general construction. However, this seems beyond reach with our present methods.\n \\end{remark}\n\n \\begin{proof}\n By construction, we have\n \\[ \\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}; \\underline{\\eta})(P, Q) = G(\\chi_2^{-1}) \\left\\langle \\hat\\iota^*\\left(\\gamma_{0, S} \\cdot \\eta_P\\right), \\mathcal{E}^{\\Phi^{(p)}}(0, t_1 + 1) \\boxtimes \\mathcal{G}_P^{[p]}\\right\\rangle. \\]\n This expands as the product of $G(\\chi_2^{-1})\\Lambda\\left(\\Pi_P \\times \\Sigma_Q, 1 + \\tfrac{t_1}{2}\\right)$ and a product of normalised local zeta-integrals, exactly as in \\cite{LPSZ1}. The local integrals away from $pS$ are all 1. The local zeta-integral at $p$ is evaluated in \\cite{LZ21-zeta2}, and gives the Euler factor $\\mathcal{E}_p(-)$. The product of zeta-integrals at the bad primes is by definition $G(\\chi_2^{-1}) Z_S(\\dots)$ and the result follows.\n \\end{proof}\n\n \\subsection{Values in the geometric range}\n\n Suppose $(P, Q)\\in U \\times U'$ is a point in the good geometric range; and let us set $t_1' = -2-t_1 = t_2 - r_1 + r_2$. Then the ``geometric'' condition implies that $0 \\le t_1' \\le r_2$, and the quadruple $(r_1, r_2, t_1', t_2)$ satisfies the branching law for algebraic representations defined in \\cite[Proposition 6.4]{LPSZ1}, which is the condition needed to define motivic cohomology classes associated to $\\pi_P \\otimes \\sigma_Q$, using the pushforward of a $\\GL_2$ Eisenstein class of weight $t_1'$ (see \\cite{HJS20}).\n\n \\begin{remark}\n Note that this Euler system class lands in the Galois representation $V_p(\\pi \\times \\sigma)^*(-1-r_1)$, and corresponds to the complex $L$-function $L(\\pi \\times \\sigma, s)$ at $s = -\\tfrac{t_1'}{2} = 1 + \\tfrac{t_1}{2}$; but this is no longer a critical value, and the Archimedean $\\Gamma$-factors force the $L$-function to vanish here to degree exactly one (except in some exceptional cases when $t_1' = 0$ and $\\pi$ is a Yoshida lift, when it can happen that the completed $L$-function has a simple pole at $s = 0, 1$).\n\n The $L$-values having this property are an interval (disjoint from the critical interval, if any) and the value $s = -\\tfrac{t_1'}{2}$ is the \\emph{upper} end of this interval. So our restriction to using only overconvergent, rather than nearly-overconvergent, Eisenstein series pegs us to the the upper endpoint of the critical interval when $P$ is critical, and to the upper endpoint of the geometric interval when $P$ is geometric.\n \\end{remark}\n\n In \\cite[\\S 4]{LZ20b-regulator}, we defined an object $\\operatorname{Per}_{\\eta}(\\pi_P \\times \\sigma_Q)$ associated to $\\pi_P \\times \\sigma_Q$, the choice of twist $t_1'$, and the basis vector $\\eta_P \\in S^2(\\pi_P, L)$. This was an $H(\\AA_{\\mathrm{f}}^p)$-equivariant map $\\mathcal{T}^p \\to L$, where $\\mathcal{T}^p = \\mathcal{W}(\\pi_{P, \\mathrm{f}}^p) \\otimes C^\\infty_c( (\\AA_{\\mathrm{f}}^p)^2) \\otimes \\mathcal{W}(\\sigma_{Q, \\mathrm{f}}^p)$. Our choice of $\\gamma_S$ defines a choice of vector\n \\[ (\\gamma_{0, S} W^{\\mathrm{new}}_{\\pi_P}) \\otimes \\Phi_S \\otimes W^{\\mathrm{new}}_{\\sigma_Q} \\in \\mathcal{T}^p \\]\n and we write $\\operatorname{Per}_{\\eta}(\\pi \\times \\sigma, \\gamma_S) \\in L$ for the value of $\\operatorname{Per}_{\\eta}(\\pi \\times \\sigma)$ on this vector.\n\n \\begin{remark}\n If $t_1 \\ne 0$, then one can check that the space of $H(\\AA_{\\mathrm{f}}^p)$-equivariant maps in which $\\operatorname{Per}_{\\eta}(\\pi_P \\times \\sigma_Q)$ lies is in fact 1-dimensional and spanned by the product of zeta integrals used to define $Z_S(\\dots)$. It follows that there is a quantity $\\operatorname{Per}_{\\eta}(\\pi_P \\times \\sigma_Q)^{\\mathrm{univ}} \\in L$ such that for all $\\gamma_S$ we have\n \\[ \\operatorname{Per}_{\\eta}(\\pi_P \\times \\sigma_Q, \\gamma_S) = Z_S(\\pi_{P} \\times \\sigma_{Q}, \\gamma_S) \\operatorname{Per}_{\\eta}(\\pi_P \\times \\sigma_Q)^{\\mathrm{univ}}. \\]\n Similar results also hold for $t_1 = 0$ under some mild additional conditions on $\\pi_P$ and $\\sigma_Q$; compare Theorem 6.6.2 of \\cite{LZ20} in the $\\GSp_4$ case. However, we do not need this for the proof of our main theorem, so we shall not pursue it further here.\n \\end{remark}\n\n \\begin{proposition}\n \\label{prop:coherentperiod}\n We have\n \\[\n \\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}; \\underline{\\eta})(P, Q) = \\operatorname{Per}_{\\eta_P}(\\pi_P \\times \\sigma_Q, \\gamma_S).\n \\]\n \\end{proposition}\n\n \\begin{proof}\n By construction, we have\n \\[\n \\operatorname{Per}_{\\eta_P}(\\pi_P \\times \\sigma_Q, \\gamma_S) = \\left\\langle \\iota^*_{\\Kl}\\left(\\gamma_{0, S}\\cdot \\eta_{P, \\Kl}\\right), \\theta^{-{t_1'}}\\left(F^{(t_1' + 2)}_{\\Phi^p \\Phi_{\\mathrm{dep}}}\\right) \\boxtimes G(\\chi_2^{-1}) \\mathcal{G}^{[p]}\\right\\rangle, \\]\n where $\\iota_{\\Kl}$ is an embedding of Shimura varieties at Klingen level. The term on the right-hand side is exactly the specialisation at $P$ of our family of $p$-adic modular forms for $H$. From the zeta-integral computations of \\cite{LZ21-zeta2}, we may replace $\\iota^*_{\\Kl}\\left(\\eta_{\\Kl}\\right)$ with $\\hat\\iota^*\\left(\\eta_{\\Iw}\\right)$ without changing the value of the pairing.\n \\end{proof}\n\n\n\n\\section{Families of cohomology classes}\n\n We persist with the notation and assumptions of the previous section. We also suppose that the family $\\underline{\\pi}$ is not of Yoshida type, so that for each classical specialisation $P$, the $\\lambda_{P}$-eigenspace in \\'etale cohomology of $S_{G, K^p\\Iw(p)}$ is 4-dimensional. We suppose furthermore that $\\underline{\\pi}$ and the $\\GL_2$ family $\\underline{\\sigma}$ are ordinary at $p$.\n\n \\subsection{Galois representations}\n\n Associated with the family $\\underline{\\pi}$ we have a family of Galois representations $V(\\underline{\\pi})$, which is a rank 4 $\\mathcal{O}(U)$-module with an action of $\\Gal(\\overline{\\mathbf{Q}}\/\\mathbf{Q})$, unramified outside $pN_0$ and satisfying $\\operatorname{tr}(\\operatorname{Frob}_\\ell^{-1} | V(\\underline{\\pi})) = \\lambda(T_{1,\\ell})$ for $\\ell \\nmid pN_0$.\n\n The existence of this family is a consequence of the results of \\cite{tilouineurban99}, who also give a canonical realisation of the dual representation $V(\\underline{\\pi})^*$ as a localisation of the module\n \\[ e'_B \\cdot \\varprojlim_t H^3_{\\text{\\textup{\\'et}}, c}\\left(\\mathcal{S}_{G, K^pK_{p, t}, \\overline{\\mathbf{Q}}}, \\ZZ_p\\right) \\otimes_{\\ZZ_p[[\\ZZ_p^{\\times 2}]]} \\mathcal{O}(U), \\]\n where $K_{p, t}$ is some family of subgroups of $G(\\ZZ_p)$ and $e^-_B$ is the ordinary projector associated to $\\mathcal{U}'_B$. Similarly, there is a 2-dimensional family of Galois representations over $U'$ associated to $\\underline{\\sigma}$.\n\n \\begin{remark}\n If the family $\\underline{\\pi}$ has a classical specialisation whose weight is sufficiently regular, but small relative to $p$ (and some additional hypotheses hold regarding the image of the residual Galois representation), then the results of \\cite{mokranetilouine02} and \\cite{rockwood-control} imply that $V(\\underline{\\pi})$ is free of rank 4 over $\\mathcal{O}(U)$.\n\n Without this condition, we can only deduce that $V(\\underline{\\pi})$ is locally free in a neighbourhood of each good \\emph{cohomological} weight, but not necessarily elsewhere. One can work around this by replacing $V(\\underline{\\pi})^*$ with its double dual (reflexive hull), which does not change its specialisations in cohomological weights.\n \\end{remark}\n\n\n \\begin{definition}\n We set\n \\[\\mathbb{V}^* = V(\\underline{\\pi})^* \\times V(\\underline{\\sigma})^*(-1-\\mathbf{r}_1),\\]\n which is an 8-dimensional family of Galois representations over $U \\times U'$.\n \\end{definition}\n\n \\subsection{Ordinary filtrations at $p$}\n\n The Galois representation $V(\\underline{\\pi})$ has a decreasing filtration by $\\mathcal{O}(U)$-submodules stable under $\\Gal(\\overline{\\mathbf{Q}}_p\/\\QQ_p)$ (via results of Urban \\cite{urban05}; see \\cite[Theorem 17.3.1]{LZ20} for the formulation we use). We write $\\mathcal{F}^i V(\\underline{\\pi})$ for the codimension $i$ subspace, and similarly for its dual $V(\\underline{\\pi})^*$. Note that $\\Gr^0 V(\\underline{\\pi})^*$ is unramified, with arithmetic Frobenius acting as the $U_{\\Sieg}$-eigenvalue. Abusing notation slightly\\footnote{What we really mean is that $\\Gr^1 V(\\underline{\\pi})^*$ is isomorphic to the tensor product of $\\chi_{\\mathrm{cyc}}^{(1 + \\mathbf{r}_2)}$ and an unramified character.}, we may say that $\\Gr^1 V(\\underline{\\pi})^*$ has ``Hodge--Tate weight $1 + \\mathbf{r}_2$''.\n\n Similarly, there is a 2-step filtration of $V(\\underline{\\sigma})^*$, with $\\Gr^1V(\\underline{\\sigma})^* = \\mathcal{F}^1 V(\\underline{\\sigma})^*$ having Hodge--Tate weight $1 + \\mathbf{t}_2$.\n\n \\begin{definition}\n We set\n \\[\\mathbb{V}^* = V(\\underline{\\pi})^* \\times V(\\underline{\\sigma})^*(-1-\\mathbf{r}_1);\\]\n and we let\n \\[ \\mathcal{F}^{(f)} V(\\underline{\\pi} \\times \\underline{\\sigma})^* = \\mathcal{F}^2 V(\\underline{\\pi}) \\otimes V(\\underline{\\sigma})^*,\\]\n and\n \\[ \\mathcal{F}^{(e)} V(\\underline{\\pi} \\times \\underline{\\sigma})^* = \\left(\\mathcal{F}^2 V(\\underline{\\pi})^* \\otimes V(\\underline{\\sigma})^*\\right) + \\left(\\mathcal{F}^1 V(\\underline{\\pi})^* \\otimes \\mathcal{F}^1 V(\\underline{\\sigma})^*\\right). \\]\n For a good weight $(P, Q)$ we write $\\mathbb{V}_{P, Q}^*$ for the specialisation of $\\mathbb{V}^*$ at $(P, Q)$, so $\\mathbb{V}_{P, Q}^* = V(\\pi_P)^* \\otimes V(\\sigma_Q)^*(-1-r_1)$ if $P = (r_1, r_2)$.\n \\end{definition}\n\n (For the significance of the labels (e) and (f), see Figure 2 of \\cite{LZvista}.) Thus $\\mathcal{F}^{(e)}$ has rank 5, $\\mathcal{F}^{(f)}$ has rank 4, and the quotient $\\Gr^{(e\/f)} \\cong \\left(\\Gr^1 V(\\pi)^*\\right)\\otimes\\left(\\mathcal{F}^1 V(\\underline{\\sigma})^*\\right)(-1-\\mathbf{r}_1)$ has Hodge--Tate weight $\\mathbf{t}_1' = -2-\\mathbf{t}_1$.\n\n \\begin{remark}\n Note that\n \\[ \\mathcal{E}_p(\\pi_P\\times \\sigma_Q) = \\det \\left(1 - \\varphi: \\mathbf{D}_{\\mathrm{cris}}(\\mathcal{F}^{(f)} \\mathbb{V}_P^*)\\right) \\cdot \\det\\left( 1 - p^{-1}\\varphi^{-1}: \\mathbf{D}_{\\mathrm{cris}} \\left(\\mathbb{V}_P^* \/ \\mathcal{F}^{(f)}\\right) \\right).\\qedhere\\]\n \\end{remark}\n\n\n \\subsection{P-adic periods}\n\n The representations $\\Gr^1V(\\underline{\\pi})^*(-1-\\mathbf{r_2})$ and $\\Gr^1 V(\\underline{\\sigma})^*(-1-\\mathbf{t}_2)$ are unramified, and hence crystalline as $\\mathcal{O}(U)$ (resp.~$\\mathcal{O}(U')$)-linear representations. Since $\\mathbf{D}_{\\mathrm{cris}}(\\QQ_p(1))$ is canonically $\\QQ_p$, we can therefore define $\\mathbf{D}_{\\mathrm{cris}}(\\Gr^{(e\/f)} \\mathbb{V}^*)$ to be an alias for the rank 1 $\\mathcal{O}(U \\times U')$-module\n \\[\n \\mathbf{D}_{\\mathrm{cris}}\\left(\\Gr^1 V(\\pi)^*(-1-\\mathbf{r_2})\\right) \\mathop{\\hat\\otimes} \\mathbf{D}_{\\mathrm{cris}}\\left( \\Gr^1 V(\\underline{\\sigma})^*(-1-\\mathbf{t}_2)\\right).\n \\]\n As in \\cite[\\S 8.2]{KLZ17}, we can define a Coleman\/Perrin-Riou big logarithm map for $\\Gr^{(e\/f)} \\mathbb{V}^*$, which is a morphism of $\\mathcal{O}(U \\times U')$-modules\n \\[ \\mathcal{L}^{\\mathrm{PR}}: H^1(\\QQ_p, \\Gr^{(e\/f)} \\mathbb{V}^*) \\to \\mathbf{D}_{\\mathrm{cris}}(\\Gr^{(e\/f)} \\mathbb{V}^*). \\]\n By construction, for good geometric weights $P$, this specialises to the Bloch--Kato logarithm map, up to an Euler factor; and for good critical weights it specialises to the Bloch--Kato dual exponential.\n\n \\subsection{P-adic Eichler--Shimura isomorphisms}\n\n Let $P$ be a good weight. Then the Faltings--Tsuji comparison isomorphism of $p$-adic Hodge theory gives an identification between $\\mathbf{D}_{\\mathrm{cris}}(V(\\pi_P))$ and the $\\pi_P$-eigenspace in de Rham cohomology (compatibly with the Hodge filtration); and the graded pieces of this filtration are identified with the coherent cohomology groups $S^i(\\pi_P, L)$.\n\n Since the Hodge and Newton filtrations on $\\mathbf{D}_{\\mathrm{cris}}$ must be complementary to each other (by weak admissibility), we deduce that there is an \\emph{Eichler--Shimura} isomorphism\n \\[\n \\ES^2_{\\pi_P}: S^2(\\pi_P, L)\\cong \\Gr^{(r_2 + 1)}_{\\mathrm{Hdg}} \\mathbf{D}_{\\mathrm{cris}}(V(\\pi_P)) \\cong \\mathbf{D}_{\\mathrm{cris}}(\\Gr^2 V(\\pi_P)).\n \\]\n Concretely, the isomorphism is given by mapping an element in $\\Gr^{(r_2 + 1)}_{\\mathrm{Hdg}} \\mathbf{D}_{\\mathrm{cris}}(V(\\pi_P))$ to its unique lifting to $\\Fil^{(r_2 + 1)}_{\\mathrm{Hdg}}\\mathbf{D}_{\\mathrm{cris}}(V(\\pi_P)) \\cap \\ker( (\\varphi - \\alpha_P)(\\varphi - \\beta_P))$.\n\n \\begin{remark}\n More generally, we have isomorphisms $\\ES^i: S^i(\\pi_P, L)\\cong \\mathbf{D}_{\\mathrm{cris}}(\\Gr^i V(\\pi_P))$ for each $0 \\le i \\le 3$, where $S^i(\\pi_P, L)$ is the $\\pi_P$-eigenspace in coherent $H^i$.\n\n We caution the reader that although the source and target of $\\ES^i_{\\pi_P}$ are the specialisations at $P$ of rank-one $\\mathcal{O}(U)$-modules, it is \\textbf{by no means obvious} that the isomorphisms $\\ES^i_{\\pi_P}$ for varying $P$ are the specialisations of a single $\\mathcal{O}(U)$-module isomorphism ``$\\ES^i_{\\underline{\\pi}}$''. We shall establish (a slightly weakened form of) this below, under some additional hypotheses, as a by-product of our main Euler system argument.\n\n It would be very interesting to have a direct construction of the maps $\\ES^i_{\\underline{\\pi}}$ by methods of arithmetic geometry. For $i = 0$ (corresponding to classical holomorphic Siegel modular forms) this has been achieved in the recent preprint \\cite{diao-rosso-wu21}. One can also obtain $\\ES^3_{\\underline{\\pi}}$ from this via Serre duality; but it seems to be more difficult to construct the ``intermediate'' filtration steps $i = 1, 2$.\n \\end{remark}\n\n \\subsubsection{Analogue for $\\GL_2$} Similarly, for $\\GL_2$ we have an isomorphism\n \\[ \\ES^0_{\\sigma_Q}: S^0(\\sigma_Q, L) \\cong \\mathbf{D}_{\\mathrm{cris}}(\\Gr^0 V(\\sigma_Q)). \\]\n In this setting the existence of comparison isomorphisms in families is known:\n\n \\begin{theorem}[Ohta, Kings--Loeffler--Zerbes]\n There exists an isomorphism of $\\mathcal{O}(U')$-modules\n \\[ \\ES^0_{\\underline{\\sigma}}: S^0(\\underline{\\pi}) \\cong \\mathbf{D}_{\\mathrm{cris}}(\\Gr^0 V(\\underline{\\sigma}))\\]\n interpolating the isomorphisms $\\ES^0_{\\sigma_Q}$ for varying $P$, where $\\mathcal{S}^0(\\underline{\\pi})$ is the $\\mathcal{O}(U')$-module spanned by $\\underline{\\omega} = G(\\chi_2^{-1}) \\cdot \\mathcal{G}$.\n \\end{theorem}\n \\begin{proof}\n This is a restatement of \\cite[Proposition 10.1.1(1)]{KLZ17}, where it is derived from results of Ohta \\cite{ohta00}. For an alternative derivation applying to possibly non-ordinary Coleman families, see \\cite{andreattaiovitastevens,loefflerzerbes16}.\n \\end{proof}\n\n \\subsection{Euler system classes}\n Let us suppose that the character $\\chi_0 \\chi_2$ is non-trivial (this allows us to get rid of a ``smoothing factor'' $c$ appearing in the Euler system constructions). Then, associated to the data $\\gamma_S$, we also have a family of cohomology classes\n \\[ \\mathbf{z}_{m}(\\underline{\\pi} \\times \\underline{\\sigma}, \\gamma_S) \\in H^1(\\mathbf{Q}(\\mu_{m}), \\mathbb{V}^*), \\]\n for all square-free integers coprime to some finite set $T \\supseteq S \\cup \\{p\\}$.\n By construction, the image of $ \\mathbf{z}_{m}(\\underline{\\pi} \\times \\underline{\\sigma}, \\gamma_S)$ under localisation at $p$ lands in the image of the (injective) map from the cohomology of $\\mathcal{F}^{(e)} \\mathbb{V}^*$. So we may make sense of\n \\[ \\mathcal{L}^{\\mathrm{PR}}\\left( \\mathbf{z}_{m}(\\underline{\\pi} \\times \\underline{\\sigma}, \\gamma_S) \\right) \\in \\mathbf{D}_{\\mathrm{cris}}(\\Gr^{(e\/f)} \\mathbb{V}^*). \\]\n We denote its image under specialisation at $(P,Q)$ by $ \\mathcal{L}^{\\mathrm{PR}}\\left( \\mathbf{z}_{m}(\\underline{\\pi} \\times \\underline{\\sigma}, \\gamma_S) \\right)(P,Q)$.\n Combining \\cref{prop:coherentperiod} with the main result of \\cite{LZ20b-regulator}, which relates the periods $\\operatorname{Per}_\\eta(\\dots)$ to the Euler system classes, we have the following result:\n\n \\begin{theorem}\\label{prop:geomreg}\n For each $P$ in the good geometric range, we have\n \\[ \\left\\langle \\mathcal{L}\\left( \\mathbf{z}_{1}(\\underline{\\pi} \\times \\underline{\\sigma}, \\gamma_S) \\right)(P,Q), \\mathrm{ES}^2_{\\pi_P}(\\eta_P) \\otimes \\mathrm{ES}^0_{\\sigma_Q}(\\omega_P)\\right\\rangle = \\mathcal{L}_{p,\\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}; \\underline{\\eta})(P, Q). \\]\n \\end{theorem}\n\n \\subsection{Reciprocity laws and meromorphic Eichler--Shimura}\n\n \\begin{definition}\n Let $\\mathfrak{S}(\\underline{\\pi}; \\underline{\\sigma})$ denote the set of points $P = (r_1, r_2) \\in U \\cap \\mathbf{Z}^2$ which are good for $\\underline{\\pi}$, and satisfy the following condition: there exists some $t_2 \\in U' \\cap \\mathbf{Z}_{\\ge 0}$, and some local data $\\gamma_S$, such that $(P, Q) = (r_1, r_2, t_2)$ is good geometric and $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma};\\underline{\\eta})$ is non-vanishing at $(P, Q)$.\n \\end{definition}\n\n \\begin{lemma}\n Let $\\underline{\\sigma}, \\underline{\\sigma}'$ be two Hida families satisfying our running hypotheses (possibly of different tame levels and characters). Then the set $\\mathfrak{S}(\\underline{\\pi},\\underline{\\sigma}) \\cap \\mathfrak{S}(\\underline{\\pi},\\underline{\\sigma}')$ is Zariski-dense. In particular, $\\mathfrak{S}(\\underline{\\pi},\\underline{\\sigma})$ is itself Zariski-dense.\n \\end{lemma}\n\n \\begin{proof}\n We first note that there exists $\\gamma_S$ for which $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma};\\underline{\\eta})$ is not identically zero. To see this, we choose some good \\emph{critical} point $(P, Q)$ having $t_1 = r_1 - r_2 - t_2 - 2 \\ge 0$, so that $\\Lambda(\\pi_P \\times \\sigma_Q, 1 + \\tfrac{t_1}{2})$ lies outside the strip $0 < \\Re(s) < 1$ and hence cannot vanish. We can then choose $\\gamma_S$ such that $Z_S(\\pi_P \\times \\sigma_Q, \\gamma_S) \\ne 0$ (which is always possible). Thus $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma})$ is non-vanishing at $(P, Q)$, and hence generically non-vanishing on $U \\times U'$.\n\n Repeating the construction, we can find local data $\\gamma_{S}'$ for $\\underline{\\pi} \\times \\underline{\\sigma}'$ such that $\\mathcal{L}_{p, \\gamma_S'}(\\underline{\\pi} \\times \\underline{\\sigma}')$ is generically non-vanishing. So there is an open subset $V \\subset U$ such that for all $v \\in V$, neither $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma})$ nor $\\mathcal{L}_{p, \\gamma_S'}(\\underline{\\pi} \\times \\underline{\\sigma}')$ vanishes identically along $\\{ v \\} \\times U'$.\n\n Since $V$ is open, it must contain some $(r_1, r_2) \\in V \\cap \\mathbf{Z}^2$; and we can therefore find an integer $t$ such that both $p$-adic $L$-functions are non-vanishing at $P = (r_1, r_2, t)$. We consider the sequence of weights $P_k = (r_1 + 3(p-1)p^{k}, r_2 + (p-1)p^{k}, t_2 + 2(p-1)p^k)$ for $k \\to \\infty$. For all but finitely many $k$ the weight $P_k$ will be good geometric, and $P_k$ tends to $P$, so $\\mathcal{L}_{p, \\gamma_S}(P_k) \\ne 0$ for sufficiently large $k$. Thus the projection of $P_k$ to $U$ lies in $\\mathfrak{S}(\\underline{\\pi} \\times \\underline{\\sigma})$, and also in $\\mathfrak{S}(\\underline{\\pi} \\times \\underline{\\sigma}')$. It follows that $(r_1, r_2)$ is a limit point of $\\mathfrak{S}(\\underline{\\pi} \\times \\underline{\\sigma}) \\cap \\mathfrak{S}(\\underline{\\pi} \\times \\underline{\\sigma}')$ in the analytic topology. Thus the Zariski-closure of this intersection contains all points of $U \\cap \\mathbf{Z}^2$ outside a proper closed subset, and hence must be all of $U$.\n \\end{proof}\n\n Let us write $\\mathcal{Q}(U)$ for the fraction field of $\\mathcal{O}(U)$ (and similarly for $U \\times U'$ etc).\n\n \\begin{theorem}\n There exists an isomorphism of $\\mathcal{Q}(U)$-modules\n \\[ \\ES^2_{\\underline{\\pi}}: S^2(\\underline{\\pi}) \\otimes_{\\mathcal{O}(U)} \\mathcal{Q}(U) \\cong \\mathbf{D}_{\\mathrm{cris}}(\\Gr^2 V(\\underline{\\pi})) \\otimes_{\\mathcal{O}(U)} \\mathcal{Q}(U), \\]\n depending only on $\\underline{\\pi}$, characterised uniquely by the following property: for all Hida families $\\underline{\\sigma}$ as above, and all $P = (r_1, r_2) \\in \\mathfrak{S}(\\underline{\\pi}, \\underline{\\sigma})$, the morphism $\\ES^2_{\\underline{\\pi}}$ is non-singular at $P$ and its fibre at $P$ coincides with the Eichler--Shimura morphism $\\ES^2_{\\pi_P}$. Moreover, we have the explicit reciprocity law\n \\[\n \\left\\langle\n \\mathcal{L}\\left( \\mathbf{z}_{m}(\\underline{\\pi} \\times \\underline{\\sigma}, \\gamma_S)\\right),\n \\mathrm{ES}^2_{\\underline{\\pi}}(\\underline{\\eta}) \\otimes \\mathrm{ES}^0_{\\underline{\\sigma}}(\\underline{\\omega})\\right\\rangle = \\mathcal{L}_{p,\\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}; \\underline{\\eta}).\n \\]\n \\end{theorem}\n\n \\begin{proof}\n We start by choosing a ``random'' isomorphism $\\jmath$ between $S^2(\\underline{\\pi})$ and $\\mathbf{D}_{\\mathrm{cris}}(\\Gr^2 V(\\underline{\\pi}))$, which is possible since both are free rank 1 $\\mathcal{O}(U)$-modules.\n\n As in the proof of the preceding lemma, we choose local data $\\gamma_S$ such that $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi}, \\underline{\\sigma}; \\underline{\\eta})$ is not identically zero, and consider the ratio\n \\[ \\mathsf{R} = \\frac{1}{\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}, \\underline{\\eta})} \\left\\langle\\mathcal{L}\\left( \\mathbf{z}_{m}(\\underline{\\pi} \\times \\underline{\\sigma}, \\gamma_S)\\right),\n \\jmath(\\underline{\\eta}) \\otimes \\mathrm{ES}^0_{\\underline{\\sigma}}(\\underline{\\omega})\\right\\rangle \\in \\mathcal{Q}(U \\times U').\n \\]\n\n If we now take a $(P, Q)$ that is good geometric, and such that $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi}, \\underline{\\sigma}; \\underline{\\eta})$ does not vanish at $(P, Q)$, it follows from the \\cref{prop:geomreg} that $\\mathsf{R}$ is regular at $(P, Q)$ and its value there is equal to the ratio $\\jmath_P \/ \\ES^2_{\\pi_P}$ (independent of $Q$).\n\n We claim that $\\mathsf{R} \\in \\mathcal{Q}(U)$; that is, as a meromorphic function on $U \\times U'$, it is independent of the $U'$ variable. To justify this, we argue as in Proposition 17.7.3 of \\cite{LZ20}: we consider the meromorphic function $\\mathsf{R}(\\mathbf{r}_1, \\mathbf{r}_2, \\mathbf{t}_2) - \\mathsf{R}(\\mathbf{r}_1, \\mathbf{r}_2, \\hat{\\mathbf{t}}_2)$ on $U \\times U' \\times U'$, where $\\hat{\\mathbf{t}}_2$ is the coordinate on a second copy of $U'$. Because of \\cref{prop:geomreg}, this function has to vanish at all points $(r_1, r_2, t_2, \\hat{t}_2)$ such that $(r_1,r_2, t_2)$ and $(r_1, r_2, \\hat{t}_2)$ are both good geometric and neither is in the vanishing locus of $\\mathcal{L}_{p, \\gamma_S}(\\underline{\\pi} \\times \\underline{\\sigma}, \\underline{\\eta})$; this set is easily seen to be Zariski-dense in $U \\times U' \\times U'$. The same argument also shows that $\\mathsf{R}$ doesn't depend on $\\gamma_S$.\n\n Thus $\\mathsf{R}$ is an element of $\\mathcal{Q}(U)^\\times$, regular at all points $P \\in \\mathfrak{S}(\\underline{\\pi}, \\underline{\\sigma})$ and coinciding at each such point with the ratio $j_P \/ \\ES^2_{\\pi_P}$. So if we define $\\ES^2_{\\underline{\\pi}} = \\mathsf{R}^{-1} \\jmath$, then $\\ES^2_{\\underline{\\pi}}$ is regular at all points in $\\mathfrak{S}(\\underline{\\pi}, \\underline{\\sigma})$ and coincides at such points with $\\ES^2_{\\pi_P}$. By the preceding lemma, this interpolating property uniquely determines $\\ES^2_{\\underline{\\pi}}$, and is independent of $\\underline{\\sigma}$; and the reciprocity law holds by construction.\n \\end{proof}\n\n \\begin{remark}\n Note that there could, \\emph{a priori}, be points where $\\ES^2_{\\underline{\\pi}}$ is 0 or $\\infty$; or where it is a well-defined isomorphism but this isomorphism does not coincide with $\\ES_{\\pi_P}^2$.\n \\end{remark}\n\n \\subsection{Application to the Bloch--Kato conjecture}\n\n Let us now consider the following situation:\n \\begin{itemize}\n \\item $\\pi$ and $\\sigma$ are cohomological cuspidal automorphic representations of $\\GSp_4 \\times \\GL_2$, with $p$-stabilisations which are ordinary and $p$-regular, which are ``deformable'' in the above sense.\n\n \\item If $t_2 = r_1 - r_2 - 1$ (so that $t_1 = -1$), then we suppose that $L(\\Pi \\times \\Sigma, \\tfrac{1}{2}) \\ne 0$. (In all other cases the non-vanishing of $L(\\Pi \\times \\Sigma, 1+\\tfrac{t_1}{2})$ is automatic.)\n\n \\item The Galois representation $V = V_p(\\pi)^* \\otimes V_p(\\sigma)^*(-1-r_1)$ satisfies the ``big image'' conditions of \\cite[\\S 3.5]{mazurrubin04}.\n\n \\item None of the eight characters appearing as graded pieces of $V$ as a $\\Gal(\\overline{\\mathbf{Q}}_p \/ \\QQ_p)$-representation are congruent mod $p$ to the trivial character, or to the $p$-adic cyclotomic character (``$p$-distinction'').\n \\end{itemize}\n\n (Note that the ``big image'' hypothesis can only be satisfied if $\\chi_0\\chi_2 \\ne 1 \\bmod p$, but is frequently satisfied when this condition does hold; compare the discussion in \\S 11.1 of \\cite{KLZ17} in the Rankin--Selberg case.)\n\n\n \\begin{theorem}\n \\label{thm:BKconj}\n In the above setting, we have\n \\[ H^1_{\\mathrm{f}}(\\mathbf{Q}, V(\\pi)^* \\otimes V(\\sigma)^*(-1-r_1)) = 0,\\]\n as predicted by the Bloch--Kato conjecture.\n \\end{theorem}\n\n \\begin{proof}\n If $(r_1, r_2)$ is in the set $\\mathfrak{S}(\\underline{\\pi}, \\underline{\\sigma})$ defined above (or more generally in $\\mathfrak{S}(\\underline{\\pi}, \\underline{\\sigma}')$ for some possibly different Hida family $\\underline{\\sigma}'$), then the theorem of the previous section implies that we have an Euler system for $V(\\pi)^* \\otimes V(\\sigma)^*(-1-r_1)$ whose bottom class is non-zero. Hence we may apply the machinery of ``Euler systems with local conditions'' developed in \\cite[\\S 12]{KLZ17} to deduce the finiteness of the Selmer group.\n\n The exceptional case which we need to deal with is when the ``family'' Eichler--Shimura isomorphism degenerates at $(r_1, r_2)$. We expect that this never occurs, but we cannot yet rule it out. In this situation, we use a version of the ``leading term argument'' from \\cite{LZ20, LZ20-yoshida}). The construction of the $p$-adic $L$-function (and the proof of the reciprocity law) extend immediately to equivariant $p$-adic $L$-functions over $\\mathbf{Q}(\\zeta_m)$, for all $m$ coprime to $T$. If the Eichler--Shimura isomorphism degenerates at $(r_1, r_2)$, then not only the class $\\mathbf{z}_1(\\pi \\times \\sigma)$, but all the classes $\\mathbf{z}_m$, must satisfy the stronger local condition defined by $\\mathcal{F}^{(f)}$; and this forces all the classes to be zero, as in \\ and this forces all of the classes to be zero. So we may replace the whole Euler system by its first derivative (in some arbitrarily chosen direction in weight space) and rescale the Eichler--Shimura isomorphism accordingly. Proceeding inductively, we eventually obtain an Euler system with non-trivial bottom class, and the argument proceeds as before.\n\n (A slight complication here is that in the exceptional case, the Euler system we obtain for $V$ does not necessarily extend to classes over the $p$-cyclotomic tower satisfying the extra-strong local condition $\\mathcal{F}^{(f)}$, since our explicit reciprocity law does not ``see'' the cyclotomic variable. Hence we cannot use the arguments of \\cite[\\S 12]{KLZ17} to prove the crucial lemma that this local condition is preserved by the passage from Euler to Kolyvagin systems, as these arguments rely on the presence of the $p$-cyclotomic tower. This is the reason for imposing the rather stringent $p$-distinction hypothesis, which allows us to use the alternative, slightly more direct approach given in the appendix of \\cite{leiloefflerzerbes14b}, in which the cyclotomic extension is not needed.)\n \\end{proof}\n\n\n\n\\newlength{\\bibitemsep}\n\\setlength{\\bibitemsep}{0.75ex plus 0.05ex minus 0.05ex}\n\\newlength{\\bibparskip}\n\\setlength{\\bibparskip}{0pt}\n\\let\\oldthebibliography\\thebibliography\n\\renewcommand\\thebibliography[1]{%\n \\oldthebibliography{#1}%\n \\setlength{\\parskip}{\\bibparskip}%\n \\setlength{\\itemsep}{\\bibitemsep}%\n}\n\n\\newcommand{\\noopsort}[1]{}\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}[1]{}\n\\renewcommand{\\MR}[1]{%\n MR \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#1}.\n}\n\\providecommand{\\href}[2]{#2}\n\\newcommand{\\articlehref}[2]{\\href{#1}{#2}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe extensive study of Kummer surfaces is explained by their rich\ngeometry and their multiple roles in the theory of K3 surfaces and\nbeyond \\cite{H1,torelli, B3}.\n\nLet $A$ be an abelian surface and consider the involution which\nmaps $a$ to $-a$ for any $a$ in $A$. This involution has sixteen\nfixed points, namely the sixteen two-torsion points of $A$. The\nquotient surface has sixteen ordinary double points and its\nminimal resolution is a smooth K3 surface called the Kummer\nsurface associated to $A$ and denoted by $\\mathrm{Kum}(A)$. Nikulin\nproved that any K3 surface containing sixteen disjoint smooth\nrational curves is a Kummer surface \\cite{N2}.\n\nGiven a Kummer surface $\\mathrm{Kum}(A)$, there is a natural way of\nconstructing new Kummer surfaces from it. One takes the minimal\nmodel of the double cover of $\\mathrm{Kum}(A)$ branched along eight\ndisjoint smooth rational curves $C_1, \\dots, C_8,$ that are even\n(see section 2) and that are orthogonal in $\\mathrm{Pic(Kum}(A))$ to\neight other smooth rational curves. We obtain in this way a new\nKummer surface $\\mathrm{Kum}(B)$ together with a rational map\n$\\mathrm{Kum}(B) \\stackrel{\\tau}\\dashrightarrow \\mathrm{Kum}(A)$.\n\nIn the second section of the paper, we explain this construction\nin details and show that the abelian surface associated to the new\nKummer surface $\\mathrm{Kum}(B)$ is isogenous to $A$. In fact we prove\nthat the map $\\tau$ is induced by an isogeny of degree two on the\nassociated abelian surfaces.\n\nIn section 3, we describe the geometry of a generic jacobian\nKummer surface and explain its classical double plane model. We\nalso recall a theorem of Naruki \\cite{Naruki} giving explicit\ngenerators of the N\\'eron-Severi lattice of a generic jacobian\nKummer surface.\n\nIn section 4, we apply the construction of section 2 to the\ngeneric jacobian Kummer surface. We obtain in this way, fifteen\nnon isomorphic Kummer surfaces which are associated to\n$(1,2)$-polarized abelian surfaces.\n\nFinally in section 5, we show that the Kummer surfaces of section\n4 admit an elliptic fibration with twelve singular fibers of the\ntype $I_2$. We also prove that these Kummer surfaces are double\ncover of a week Del Pezzo surface (i.e. the blowup of $\\mathbb\nP^2$ at seven points) and that for each of our Kummer surfaces\nthere exists a decomposition of a very degenerate sextic $\\mathcal\nS$ (see figure \\ref{plane sextic}) into a quartic $Q$ and a conic\n$C$ for which we have the theorem\n\n\\begin{theorem}\\label{San11}\nThe rational double cover $\\mathrm{Kum}(B) \\stackrel{\\tau}\\dashrightarrow \\mathrm{Kum}(A)$ decomposes as\n$$\\xymatrix{\\mathrm{Kum}(B) \\ar@{-->}[d]^{\\tau}\\ar[r]^{\\varphi}&\nT \\ar@{->}[d]^{\\zeta}\\\\ \\mathrm{Kum}(A) \\ar@{->}[r]^{\\phi}& \\mathbb P^2}$$ where\n$\\phi$ is the canonical resolution of the double cover of $\\mathbb\nP^2$ branched along $\\mathcal S$. The maps $\\zeta$ and $\\varphi$\nare the canonical resolutions of the double covers branched along $Q$ and $\\zeta^*(C)$ respectively.\n\\end{theorem}\n\n\n\\section{Even Eight and Kummer surface}\n\nWe will now introduce the notion of an even eight and the double\ncover construction associated to it. By applying this construction\nto special even eights of a Kummer surface, we obtain new Kummer\nsurfaces.\n\\begin{definition}\\label{eveneight}\nLet $Y$ be a K3 surface, an \\textit{even eight} on $Y$ is a set of\neight disjoint smooth rational curves $C_1, \\dots, C_8,$ for which\n$C_1 + \\cdots + C_8 \\in 2S_Y.$ Here $S_Y$ denotes the\nN\\'eron-Severi group of $Y.$\n\\end{definition}\n\n\nIf $C_1, \\dots, C_8,$ is an even eight on a K3 surface $Y$, then\nthere is a double cover $Z \\stackrel{p}\\to Y$, branched on $C_1 +\n\\dots +C_8$. If $E_i$ denotes the inverse image of $C_i$, then\n$p^*(C_i)=2E_i$ and $E_i^2=-1$. Hence, we may blowdown the $E_i$'s\nto the surface $X$ and obtain the diagram $$\\xymatrix{Z\n\\ar@{->}[d]_{p }\\ar[r]^{\\epsilon} & X \\ar@{-->}^{2:1}[ld]\n\\\\Y & }$$\nIt turns out that the surface $X$ is again a K3 surface and the\ncovering involution $\\iota: X \\to X$ is symplectic with eight\nfixed points \\cite{N2}.\n\n\nSuppose now that the K3 surface $Y$ is a Kummer surface, we want\nto exhibit natural even eights lying on it. For this purpose, we\nrecall a central lemma of Nikulin.\n\\begin{lemma}\\cite{N2}\\label{nikulin}\nLet $Y$ be a Kummer surface and let $E_1, \\dots, E_{16} \\subset Y$\nbe sixteen smooth disjoint rational curves. Denote by $I=\\{ 1,\n\\dots, 16 \\}$ the set of indices for the curves $E_{i}$'s and by\n$Q = \\{ M \\subset I \\vert \\quad \\frac{1}{2} \\sum_{i\\in M}E_{i}\\in\nS_Y\\}$; then for every $M$ in $Q$, we have $\\# |M|=8 \\textrm{ or }\n16$ and there exists on $I$ a unique 4-dimensional affine geometry\nstructure over $\\mathbb F_{2}$, whose hyperplanes consist of the\nsubsets $M \\in Q$ containing eight elements.\n\\end{lemma}\n\nThe existence of such a 4-dimensional affine geometry implies that\n$I \\in Q $ or equivalently that $\\sum_{i=1}^{16}E_{i}\\in 2 S_Y.$\nWe can proceed exactly as for an even eight and take the double\ncover $V \\stackrel{p}\\to Y$ branched along $E_1+ \\dots + E_{16}$.\nAgain we blowdown the preimage of the $E_i$'s to a surface $A$ and\nobtain the diagram $$\\xymatrix{V \\ar@{->}[d]_{p }\\ar[r]^{\\epsilon}\n& A \\ar@{-->}^{\\pi_A }[ld]\n\\\\Y & }$$The difference with this diagram and the one above is\nthat now the surface $A$ is an abelian surface and that the map\n$\\pi_A$ realizes $Y$ as the Kummer surface associated to $A$. We\npoint out that by uniqueness, the affine geometry on $I$\ncorresponds to the one existing on $A_{2}$, the set of 2-torsion\npoints on $A$. \\cite{N2}.\n\nIt follows also from the lemma that there exist on $Y$ ($\\simeq\n\\mathrm{Kum}(A)$) thirty even eights, denoted by $M_1, \\cdots,\nM_{30}$, i.e. the thirty affine hyperplanes of $I$.\n\nLet $M \\in \\{ M_1, \\dots, M_{30} \\}$ be one of these even eights.\nWe can assume that $M$ consists of the curves $E_{1}, \\dots,\nE_{8}$. The curves $E_{9}, \\dots, E_{16}$ are then orthogonal to\n$M$, i.e. $$E_i \\cdot E_j =0 \\textrm{ if } 1 \\leq i \\leq 8\n\\textrm{ and } 9\\leq j \\leq 16.$$ If $X\n\\stackrel{\\tau}\\dashrightarrow \\mathrm{Kum}(A) $ is the double cover\nassociated to $M$, then the K3 surface $X$ contains again sixteen\ndisjoint smooth rational curves. Indeed since the curves $E_{9},\n\\dots, E_{16}$ do not intersect the branch locus of the double\ncover $p: Z \\to Y$, they split under $p$ and define sixteen\ndisjoint smooth rational curves on $Z$. These sixteen curves are\nthen isomorphically mapped by the blowdown $Z \\stackrel{\\epsilon}\n\\to X$ to sixteen curves on $X$. It follows that $X$ contains\nsixteen disjoint smooth rational curves and hence it is a Kummer\nsurface.\n\n\\begin{prop}\\label{prop}\nLet $M$ be an even eight on a Kummer surface $\\mathrm{Kum}(A)$ such as\nabove, then the K3 surface $X$ associated to $M$ is a Kummer\nsurface. Moreover there is an abelian surface $B$ associated to\n$X$ for which we have the commutative diagram\n$$\\xymatrix{B\\ar@{-->}[d]_{\\pi_B } \\ar[r]^{p} & A\n\\ar@{-->}[d]_{\\pi_A}\n\\\\X=\\mathrm{Kum}(B) \\ar@{-->}[r]^{\\tau}& \\mathrm{Kum}(A)}$$where $B\\stackrel{p}\\to\nA$ is an isogeny of degree two.\n\\end{prop}\n\n\\begin{proof}\n\\noindent Since we have already shown that $X$ is a Kummer\nsurface, we only have to prove that $B$ is degree two isogenous to\n$A$. Write the abelian surface $A$ as the complex torus $\\mathbb\nC^2 \/ \\Lambda$ and let $E_{9}, \\dots, E_{16}\\subset \\mathrm{Kum}(A)$\nbe the eight disjoint smooth rational curves orthogonal to $M$.\nThese curves also form an even eight and hence they correspond to\nan affine hyperplane $H$ in $A_2$. Up to translation we can fix\nthe origin on $A$ in $H$. Let $\\frac{[v]}{2}$ be the generator of\n$A_2\/H$, it defines a sublattice $\\Lambda' \\subset \\Lambda$.\nExplicitly we have that $\\Lambda'= \\mathbb Z h_1 \\oplus \\mathbb Z\nh_2 \\oplus \\mathbb Z h_3 \\oplus \\mathbb Z 2v$, where $H=\\langle\n\\frac{[h_1]}{2}, \\frac{[h_2]}{2}, \\frac{[h_3]}{2} \\rangle \\subset\nA_2$.\nThe canonical inclusion $\\Lambda' \\hookrightarrow \\Lambda$,\ninduces the following commutative diagram : $$\\xymatrix{\\mathbb\nC^2\/\\Lambda'\\ar@{-->}[d]_{\\pi'} \\ar[r]^{p} & \\mathbb C^2\/\\Lambda\n\\ar@{-->}[d]_{\\pi}\\\\\n \\mathrm {\\kum(\\mathbb C^2\/\\Lambda')} \\ar@{-->}[r]^{q}& \\mathrm {\\kum(\\mathbb C^2\/\\Lambda)}}$$\nwhere $p$ is an isogeny of degree two. The covering involution of\n$p$ is given by the translation by the 2-torsion point $[v]$ in\n$\\mathbb C^2\/\\Lambda'$.\nIt induces the symplectic involution on $\\mathrm {\\kum(\\mathbb\nC^2\/\\Lambda')}$ $$\\sigma: \\mathrm {\\kum(\\mathbb C^2\/\\Lambda')} \\to \\mathrm\n{\\kum(\\mathbb C^2\/\\Lambda')}$$which has exactly eight fixed points\n\\cite{N5}, namely\nthe projection of the sixteen points on $\\mathbb C^2\/\\Lambda'$ satisfying$$[z]+[v]=-[z] \\textrm{, or equivalently } 2[z]=[v].$$ \nThe isogeny $p$ maps the set $\\{ [z] \\in \\mathbb C^2\/\\Lambda'\n\\textrm { }| \\textrm { }2[z]=[v]\\}$ to $\\mathrm {A_2-H}$. In other\nwords, the affine hyperplane $A_2-H$ corresponds to the even\neight $M$ in $\\mathrm {Kum}(\\mathbb C^2\/\\Lambda)$. Hence the\nresolution of the rational map $q$ is exactly the double cover of\n$\\mathrm{Kum}(A)$ branched along $M$ and the abelian surface $\\mathbb\nC \/\\Lambda'$ is $B$.\n\n \n\\end{proof}\n\n\n\\section{Jacobian Kummer surface}\nIn this section we briefly expose the classical geometry of a\njacobian Kummer surface and its beautiful $16_6$-configuration. We\ndescribe its double plane model and give explicit generators for\nits N\\'eron-Severi lattice. This description follows a paper of N\naruki \\cite{Naruki}.\n\nA Kummer surface $\\kum(A)$ is said to be a jacobian Kummer surface\nif the surface $A$ is the jacobian of a curve $C$ of genus two.\nMoreover, it is a generic jacobian Kummer surface if its Picard\nrank is 17.\n\nRecall that the degree two map given by the linear system $|2C|$,\n$A \\stackrel{|2C|} \\to \\mathbb P^3$, factors through the\ninvolution $a \\stackrel{i} \\mapsto -a$, and hence defines an\nembedding $A\/\\{ 1, i\\} \\hookrightarrow \\mathbb P^3$. The image of\nthis map is a quartic $Y_0 \\subset \\mathbb P^3$ with sixteen\nnodes. Denote by $L_0$ the class of a hyperplane section of $Y_0$.\nProjecting $Y_0$ from a node defines a rational map $Y_0\n\\stackrel{2:1}\\dashrightarrow \\mathbb P^2$. We blowup the center\nof projection\n\n$$\\xymatrix{Y_1\\ar@{->}[d] \\ar[rd] & \\\\\n Y_0 \\ar@{-->}[r]& \\mathbb P^2}$$and we call $E_1\\subset Y_1$ the\n exceptional divisor and $L_1\\subset\n Y_1$ the pullback of a line in $\\mathbb P^2$.\n Finally we resolve the remaining fifteen\n singularities of $Y_1$ and obtain the Kummer surface\n $\\mathrm{Kum}(A)$ and a map of degree two\n$\\mathrm{Kum}(A) \\stackrel{\\phi} \\to \\mathbb P^2$. The map $\\phi$ is\ngiven by the linear system $|L-E_0|$, where $L$ and $E_0$ are the\npullback of $L_1$ and $E_1$ respectively.\n\nThe branch locus of the map $\\phi$ is a reducible\n plane sextic $\\mathcal S$, which is the union of six lines, $l_1, \\cdots, l_6$, all tangent to a conic $W$\n\n\\begin{figure}[h]\n$$\n \\begin{xy}\n <0cm,0cm>;<1.5cm,0cm>:\n (2,-.3)*++!D\\hbox{$l_1$},\n (1.1,0.25)*++!D\\hbox{$l_2$},\n (1.1,1.3)*++!D\\hbox{$l_3$},\n (2,1.8)*++!D\\hbox{$l_4$},\n (2.8,1.3)*++!D\\hbox{$l_5$},\n (2.9,0.2)*++!D\\hbox{$l_6$},\n (1.5,1)*++!D\\hbox{$W$},\n \n (.5,0.18);(3.5,0.18)**@{-},\n (0.5,2);(1.6,0)**@{-},\n (3.1,2);(2.6,0)**@{-},\n (0.7,.5);(2,2.5)**@{-},\n (0.5,1.82);(3.5,1.82)**@{-},\n (2,2.7);(3.45,0)**@{-},\n \n \n (2,1)*\\xycircle(.8,.8){},\n \\end{xy}\n $$\n\\caption{The sextic $\\mathcal S$\\label{plane sextic}}\n\\end{figure}\n\n\\noindent Let $p_{ij}=l_i \\cap l_j \\in \\mathbb P^2$, where $1\\leq\ni < j \\leq 6$. Index the ten $(3,3)$-partitions of the set $\\{ 1,\n2, \\dots ,6 \\}$, by the pair $(i,j)$ with $2 \\leq i < j \\leq 6$.\nEach pair $(i,j)$ defines a plane conic $l_{ij}$ passing through\nthe sixtuplet $p_{1i}, p_{1j}, p_{ij},p_{lm},p_{ln},p_{mn}$, where\n$\\{ l,m,n \\}$ is the complement of $\\{ 1,i,j\\}$ in $\\{ 1, 2, \\dots\n,6 \\}$ and where $l < m < n$. The map $\\phi$ factors as\n$$\\mathrm{Kum}(A) \\stackrel{\\tilde{\\phi}} \\longrightarrow\n\\tilde{\\mathbb P}^2\\stackrel{\\eta}\\longrightarrow \\mathbb\nP^2$$where $\\eta$ is the blowup of $\\mathbb P^2$ at the $p_{ij}$'s\nand where $\\tilde{\\phi}$ is the double cover of $\\tilde{\\mathbb\nP}^2$ branched along the strict transform of the plane sextic\n$\\mathcal S$ in $\\tilde{\\mathbb P}^2$. Denote by $E_{ij} \\subset\n\\mathrm{Kum}(A)$ the preimage of the exceptional curves of\n$\\tilde{\\mathbb P}^2$. The ramification of the map $\\tilde{\\phi}$\nconsists of the union of six disjoint smooth rational curves $C_0+\nC_{12} + C_{13}+ C_{14}+ C_{15}+ C_{16}$. The preimage of the ten\nplane conics $l_{ij}$ defines ten more smooth disjoint rational\ncurves $C_{ij}\\subset \\mathrm{Kum}(A), 2 \\leq i < j \\leq 6$. Finally,\nnote that $\\phi(E_0)=W$. The sixteen curves $E_0, E_{ij} \\quad 2\n\\leq i < j \\leq 6$ are called the \\textit{nodes} of $ \\mathrm{Kum}(A)$\nand the sixteen curves $C_0, C_{ij}$, $2 \\leq i < j \\leq 6$ are\ncalled the \\textit{tropes} of $ \\mathrm{Kum}(A)$. These two sets of\nsmooth rational curves satisfy a beautiful configuration called\nthe $16_6$-configuration, i.e. each node intersects exactly six\ntropes and vice versa.\n\n\\noindent It is now possible to fully describe the N\\'eron-Severi\nlattice $S_{ \\mathrm{Kum}(A)}$ of a general jacobian Kummer surface.\n\n\\begin{theorem}\\cite{Naruki} Let $\\mathrm{Kum}(A)$ be a generic jacobian Kummer surface. Its N\\'eron-Severi lattice $S_{ \\mathrm{Kum}(A)}$\nis generated by the classes of $E_{0}, E_{ij}$, $C_{0}, C_{ij}$\nand $L$, with the relations:\n\n\\begin{enumerate}\n\\item $C_{0}= \\frac{1}{2}(L- E_{0} -\n\\sum_{i=2}^{6} E_{1i}),$\n\\item $C_{1j}= \\frac{1}{2}(L - E_{0} - E_{1j}- \\cdots - E_{j-1j}- E_{jj+1}- \\cdots E_{j6}),$ where $2\\leq j \\leq 6,$\n\\item $C_{jk}= \\frac{1}{2}(L - E_{1j} -E_{1k}- E_{jk}- E_{lm}-E_{ln}-\nE_{mn})$ where $2\\leq i < j \\leq 6,$ and $\\{ l, m, n\\}$ are as\ndescribed above.\n\n\\end{enumerate}\n\n\\noindent The intersection pairing is given by:\n\n\\begin{enumerate}\n\n\\item the $E_{0}, E_{ij}$ are mutually orthogonal,\n\n\\item $\\langle L, L \\rangle =4, \\langle L, E_{0} \\rangle =\\langle L, E_{ij}\\rangle =0,$\n\n\\item $\\langle E_{0}, E_{0} \\rangle= \\langle E_{ij}, E_{ij} \\rangle=-2$,\n\n\\item the $C_{0}, C_{ij}$ are mutually orthogonal,\n\n\\item $\\langle L, C_{0} \\rangle= \\langle L, C_{ij} \\rangle=2$.\n\n\\end{enumerate}\n\n\n\\noindent The action on $S_{ \\mathrm{Kum}(A)}$ of the covering\ninvolution $\\alpha$ of the map $\\phi$ is given by:\n\n\\begin{tabular}{lrrlr}\n\n$\\alpha (C_{0})=C_{0}$ & &&\n\n$\\alpha (C_{1j})=C_{1j}$ & $2 \\leq j \\leq 6$\\\\\n\n$\\alpha (E_{ij})=E_{ij}$ & $1 \\leq i < j \\leq 6$ &&\n\n$\\alpha (L)=3 \\mathrm L - 4 E_{0}$ & \\\\\n\n$\\alpha (E_{0})=2 L - 3E_{0}$& &&\n\n$\\alpha (C_{ij})= C_{ij} + L - 2E_{0}$ & $2 \\leq i < j \\leq 6$.\\\\\n\n\\end{tabular}\n\n\\end{theorem}\n\\begin{remark}\nThe minimal resolution of the double cover of $\\mathbb P^2$\nbranched along the sextic $\\mathcal S$ in figure \\ref{plane\nsextic} is a Kummer surface (see \\cite{H1} for a proof).\\\\\n\\end{remark}\n\n\\section{(1,2)-polarized Kummer surfaces}\n\nIn this section, we apply the construction of section 2 to a\ngeneric jacobian Kummer surface. We identify all the even eights\nmade out of its nodes and study the associated Kummer surfaces.\nFirst we recall some standard facts about the polarization of\nabelian varieties.\n\nA polarization on a complex torus $\\mathbb C^g\/ \\Lambda$ is the\nclass of an ample line bundle $L$ in its the N\\'eron-Severi group.\nAs the latter group is equal, for abelian varieties, to the group\nof hermitian forms $H$ on $\\mathbb C^g$, satisfying\n$E=\\mathrm{Im}H(\\Lambda, \\Lambda) \\subset \\mathbb Z$, the ample line\nbundle $L$ corresponds to a positive definite hermitian forms\n$E_L$. According to the elementary divisor theorem, there exists a\nbasis $\\lambda_1, \\dots, \\lambda_g, \\mu_1, \\dots, \\mu_g$ of\n$\\Lambda$, with respect to which $E_L$ is given by the matrix\n$$\\left(\n\\begin{array}{cc}0 & D \\\\ -D & 0\\end{array}\\right) \\textrm{ with } D=\\left(\n\\begin{array}{cccc} d_1 & 0 & 0 & \\ldots \\\\ 0& d_2 & 0& \\ldots \\\\ \\vdots & 0& \\ddots& 0 \\\\ \\vdots & \\vdots & 0 &d_g \\end{array}\\right)$$\nwhere $d_i \\ge 0$ and $d_i | d_{i+1}$ for $i=1, \\dots, g-1.$ The\nvector $(d_1, d_2, \\dots, d_g)$ is the type of the line bundle\n$L$.\n\n\\begin{example}\\cite{BL}\\label{example}\n\n\\begin{enumerate}\n\\item If $J(C)$ is the Jacobian of a curve $C$ of genus two,\nthen\nthe line bundle associated to the divisor $C$ is a polarization of\ntype $(1,1)$.\n\n\\item If $L$ is a polarization of type $(d_1, \\dots, d_g)$ on a complex torus, then\n$\\chi(L)=d_1 \\cdot \\cdot \\cdot d_g$.\n\n\\item If $X_1 \\stackrel{p} \\to X_2$ is an isogney of degree 2 of\nabelian surfaces and $L$ is a polarization of type $(1,1)$ on\n$X_2$, then $\\chi(p^*(L))=2 \\chi(L)=2\\cdot 1$. Hence $p^*(L)$ is a\npolarization of type $(1,2)$ on $X_1$.\n\\end{enumerate}\n\\end{example}\n\n\n\n\\begin{prop}\\label{fifteen}\nLet $\\mathrm{Kum}(A)$ be a generic jacobian Kummer surface and let\n$E_0, E_{ij},$ $1\\leq i}[r]^{g^*}\\ar[d]^{i^*_1}&\nH^2(X_2, \\mathbb Z) \\ar@{->}[d]^{i^*_2}\\\\ H^2(X_1, \\mathbb Z)\n\\ar@{->}[r]^{g^*}& H^2(X_2, \\mathbb Z).}$$\n\n\n\\noindent \\textit{Proof of the claim:} Suppose that the above\ndiagram does not commute. Then the surface $X_1$ would admit two\ndistinct symplectic involutions, namely $i_1$ and $g \\circ\ni_2\\circ g^{-1}$. Moreover the quotient of $X_1$ by both of these\ninvolutions would be birational to the same Kummer surface $Y$. In\n\\cite{AM}, it is shown that the rational double cover of a Kummer\nsurface $\\mathrm{Kum}(A)$ is determined by an embedding $T_X\n\\hookrightarrow T_A$ preserving the Hodge decomposition of $T_X$\nand $T_A$. Since there is an unique embedding of $T_X$ into $T_A$\nwhich preserves the Hodge decomposition, it follows that\n$i_1=g^{-1} \\circ i_2\\circ g.$\n\nHence $i_2 \\circ g = g \\circ i_1$ and the isomorphism $g$ descends\nto an isomorphism on the quotients $$X_2\/ i_2 \\stackrel{g}\\to\nX_1\/i_1.$$ Since this isomorphism maps the eight singular points\nof $X_2\/ i_2$ to the eight singular points of $X_1\/ i_1$, it\nextends to an automorphism $Y \\stackrel{f}\\to Y$, for which\n$f^*(N_1)=N_2.$\n\nConversely, let $Y \\stackrel{f} \\to Y$ be an automorphism of $Y$\nfor which $ f^*(N_1)=N_2$. Denote by $Z_i \\stackrel{p_i}\\to Y$ the\ndouble cover of $Y$ branched along the even eight $N_i$ for\n$i=1,2$. Consider the fiber product $$\\xymatrix{Z_1 \\times_Y Y\n\\ar@{->}[r]^{q}\\ar[d]^{p}& Z_1 \\ar@{->}[d]^{p_1}\\\\Y\n\\ar@{->}[r]^{f}&Y.}$$ The map $Z_1 \\times_Y Y \\stackrel{p} \\to Y$\nis a double cover of $Y$ branched along the even set $N_2$ or\nequivalently $Z_1 \\times_Y Y=Z_2$. Similarly, by considering the\nfiber product $$\\xymatrix{Z_2 \\times_Y Y\n\\ar@{->}[r]^{h}\\ar[d]^{r}& Z_2 \\ar@{->}[d]^{p_2}\\\\Y\n\\ar@{->}[r]^{f}&Y,}$$ we see that $Z_2 \\times_Y Y =Z_1$. The maps\n$h$ and $q=h^{-1}$ define an isomorphism between $Z_1$ and $Z_2$\nwhich induces the required isomorphism between $X_1$ and $X_2$.\n\\end{proof}\n\nUsing the same notation as in the proposition \\ref{fifteen}, we\nprove the following theorem\n\n\\begin{prop}\nLet $\\Delta_{ij}$ and $\\Delta_{i'j'}$ be two even eights defined\nas in propostion \\ref{fifteen}. $$\\mathrm{Kum}(B_{ij}) \\simeq\n\\mathrm{Kum}(B_{i'j'}) \\Leftrightarrow \\{i,j\\}=\\{i',j'\\}.$$\n\\end{prop}\n\\begin{proof}\nIt is clear that if $\\{i,j\\}=\\{i',j'\\}$, then the corresponding\nKummer surfaces are equal. Thus we only have to prove the other\ndirection. Without loss of generality, we may assume that\n$\\Delta_{i'j'}=\\Delta_{12}$ and we suppose that there exists $f$\nan automorphism of $\\mathrm{Kum}(A)$ for which\n$f^*(\\Delta_{12})=\\Delta_{ij}$.\n\n\\noindent \\textit{Claim}: $$\\{\nf^*(E_{13}),f^*(E_{14}),f^*(E_{15}),f^*(E_{16}),f^*(E_{23}),f^*(E_{24}),f^*(E_{25}),f^*(E_{26})\\}=$$\n$$ \\{ E_{1i},\\cdots, \\hat{E_{ij}}, \\cdots ,E_{i6},E_{1j},\\cdots,\n\\hat{E_{ij}},\\cdots, E_{j6}\\}$$\n\n\\noindent\\textit{Proof of the claim}: Let $N$ be a Nikulin lattice\nand let $D \\in N$ be a divisor represented by a smooth rational\ncurve. Note that since $D$ is an effective reduced divisor and\n$N$ is negative definite, then $D^2=-2$. It is therefore\nsufficient to show that the only $-2$-classes in $N$ are the\n$c_i$'s and the claim will follow. We write $D$ as $D=\n\\sum_{i=1}^8 \\lambda_i c_i+\\epsilon d$ where $\\lambda_j \\in\n\\mathbb Z$ and $\\epsilon=0$ or $1$. If $\\epsilon=1$, then the\nequality $$D^2=-2\\sum_{i=1}^8 \\lambda_i^2-2\\sum_{i=1}^8\n\\lambda_i-4=-2$$implies that $\\sum_{i=1}^8\n\\lambda^2_i+\\lambda_i=-1.$ Since the latter equation has no\ninteger solution, we conclude that $\\epsilon=0$. Hence\n$$D^2=-2\\sum_{i=1}^8 \\lambda^2_i=-2$$or equivalently,\n$\\sum_{i=1}^8 \\lambda_i^2=1.$ Therefore there exists an unique\n$\\lambda_k$ for which $\\lambda_k=1$ and $\\lambda_i=0$ for $i\\ne\nk$.\n\nIn \\cite{Keum2}, it is proven that any automorphism of a jacobian\ngeneric Kummer surface induces $\\pm \\textrm{identity}$ on\n$D_{S_{\\mathrm{Kum}(A)}}$ where $D_{S_{\\mathrm{Kum}(A)}}$ is the\ndiscriminant group $S_{\\mathrm{Kum}(A)}^*\/S_{\\mathrm{Kum}(A)}$. We want to\napply this fact to the automorphism $f$. We consider the action of\n$f^*$ on the following two independent elements of\n$D_{S_{\\mathrm{Kum}(A)}}$ $$\\frac{1}{2}(E_{13}+E_{14}+E_{23}+E_{24})\n\\textrm{ and } \\frac{1}{2}(E_{12}+E_{23}+E_{15}+E_{35}).$$ From\nthe claim, we deduce that $$f^*(E_{13}+E_{14}+E_{23}+E_{24})=\nE_{i_1i}+E_{i_2i}+E_{j_1j}+E_{j_2j}$$for some classes\n$E_{i_1i},E_{i_2i},E_{j_1j},E_{j_2j} \\in \\Delta_{ij}$.\n\n\\noindent From the identity $f^*_{D_{S_{\\mathrm{Kum}(A)}}}=\\pm\n\\textrm{id}_{D_{S_{\\mathrm{Kum}(A)}}},$ we also deduce that\n$$f^*(\\frac{1}{2}(E_{13}+E_{14}+E_{23}+E_{24}))= \\pm\n\\frac{1}{2}(E_{13}+E_{14}+E_{23}+E_{24}).$$ Putting these two\ninformations together we find that $$E_{13}+E_{14}+E_{23}+E_{24}+\nE_{i_1i}+E_{i_2i}+E_{j_1j}+E_{j_2j} \\in 2S_Y.$$Since the only even\neights containing $E_{13}, E_{14},E_{23},E_{24}$ are $\\Delta_{12}$\nand $\\Delta_{34}$, we deduce that $\\Delta_{ij}=\\Delta_{34}$. We\nproceed similarly for $f^*(E_{12}+E_{23}+E_{13}+E_{35})$ and find\nthat $\\Delta_{ij}$ must be equal to $\\Delta_{25}$ which yields to\na contradiction.\n\\end{proof}\n\n\\begin{corollary}\nThe fifteen Kummer surfaces $\\mathrm{Kum}(B_{ij})$ are not isomorphic.\n\\end{corollary}\n\n\\section{Elliptic Fibration and weak del Pezzo surface}\n\nIn this section, we provide an alternate description of the Kummer\nsurfaces $\\mathrm{Kum}(B_{ij})$ as the double covers of a weak del\nPezzo surface. We relate this construction to the projective\ndouble plane model of the generic jacobian Kummer surface of\nsection 3. First we note the existence on $\\mathrm{Kum}(B_{ij})$ of an\nelliptic fibration that will be useful later. For simplicity, we\nwill always argue for the Kummer surface $\\mathrm{Kum}(B_{12})$.\n\n\\begin{prop}\\label{fibration}\nLet $\\mathrm{Kum}(B_{12})$ be the Kummer surface constructed in the\nproposition \\ref{fifteen}. The surface $\\mathrm{Kum}(B_{12})$ admits a\nWeierstrass elliptic fibration with exactly twelve singular fibers\nof the type $I_2$.\n\\end{prop}\n\\begin{proof}\nLet $\\mathrm{Kum}(A) \\stackrel{\\phi} \\to \\mathbb P^2$ be the double\nplane model of the generic jacobian Kummer surface introduced in\nsection 3. Consider the pencil of lines passing through the point\n$p_{12}$ in $\\mathbb P^2$. Its preimage in $\\mathrm{Kum}(A)$ defines\nan elliptic fibration, given by the divisor class\n$F=L-E_0-E_{12}.$ The divisors $$F_1=E_{15}+ E_{16}+2C_{0}+\nE_{13}+ E_{14}, \\quad \\textrm{and} \\quad F_2=E_{25}+\nE_{26}+2C_{12}+ E_{23}+ E_{24}$$ define two fibers of type $I^*_0$\nof this fibration. Moreover, the six divisors\n\\begin{center}\n$F_3= L-E_0-E_{12}-E_{45}+E_{45}$,\n\n$F_4= L-E_0-E_{12}-E_{46}+E_{46}$,\n\n$F_5= L-E_0-E_{12}-E_{35}+E_{35}$,\n\n$F_6= L-E_0-E_{12}-E_{36}+E_{36}$,\n\n$F_7= L-E_0-E_{12}-E_{34}+E_{34}$,\n\n$F_8= L-E_0-E_{12}-E_{56}+E_{56}$\n\\end{center}\n\\noindent define six $I_2$ fibers. Since the Euler characteristics\nof the $F_i$'s add up to 24, which is equal to the Euler\ncharacteristic of a K3 surface, we conclude by Shioda's formula\n\\cite{Shioda} that the $F_i$'s are the only singular fibers of the\nelliptic fibration defined by the linear system $|F|$. Note also\nthat the curves $C_{13}$, $C_{14}$, $C_{15}$ and $C_{16}$ are\nsections of this fibration.\n\nWe now analyze the induced fibration $\\tau^*F$ on\n$\\mathrm{Kum}(B_{12})$, where\n$\\mathrm{Kum}(B_{12})\\stackrel{\\tau}\\dashrightarrow \\mathrm{Kum}(A)$ is\nthe rational double cover defined by the even eight $\\Delta_{12}.$\nWe remark that the even eight $\\Delta_{12}$ satisfies\n$$\\Delta_{12} = F_1 +F_2 -2(C_0 +C_{12})$$ which mean that the\neight components of $\\Delta_{12}$ are exactly the eight components\nof the fibers $F_1$ and $F_{2}$ that appear with multiplicity one.\nHence $\\tau^*F_1$ and $\\tau^*F_2$ are just smooth elliptic curves.\nHowever the six fibers $F_3, \\dots, F_8$ split under the cover and\ndefine twelve $I_2$ fibers of the elliptic fibration on\n$\\mathrm{Kum}(B_{12})$ defined by $\\tau^*F$. Again a computation of\nEuler characteristics shows that these twelve $I_2$ fibers are the\nonly singular fibers of the linear system $|\\tau^*F|$. Also the\nsections $C_{13}$, $C_{14}$, $C_{15}$ and $C_{16}$ of $|F|$ pull\nback to sections of $\\tau^*F$, which is therefore a Weierstrass\nelliptic fibration.\n\n\\end{proof}\n\nWe now proceed to the realization of the surface\n$\\mathrm{Kum}(B_{12})$ as a double cover of a weak del Pezzo surface.\nWe decompose the sextic $\\mathcal S$ (see figure \\ref{plane\nsextic}) into the quartic $Q=l_3+l_4+l_5+l_6$ and the conic\n$C=l_1+l_2$.\n\\begin{theorem}\\label{San11}\nThe rational double cover associated to $\\Delta_{12}$,\n$\\mathrm{Kum}(B_{12}) \\stackrel{\\tau}\\dashrightarrow Y$ decomposes as\n$$\\xymatrix{\\mathrm{Kum}(B_{12}) \\ar@{-->}[d]^{\\tau}\\ar[r]^{\\varphi}&\nT \\ar@{->}[d]^{\\zeta}\\\\ Y \\ar@{->}[r]^{\\phi}& \\mathbb P^2}$$ where\n$\\phi$ is the canonical resolution of the double cover of $\\mathbb\nP^2$ branched along $\\mathcal S$. The maps $\\zeta$ and $\\varphi$\nare the canonical resolutions of the double covers branched along\n$Q$ and $\\zeta^*(C)$ respectively.\n\\end{theorem}\n\\begin{proof}\n\nLet $T_0 \\to \\mathbb P^2$ be the double cover of $\\mathbb P^2$\nramified over the reducible quartic $Q$. Its canonical resolution\ninduces the diagram $$\\xymatrix{T \\ar@{->}[d]\n\\ar[r]\\ar[rd]^{\\zeta}& T_0 \\ar@{->}[d]\\\\ \\tilde{\\mathbb P}^2\n\\ar@{->}[r]& \\mathbb P^2}$$ where $\\tilde{\\mathbb P}^2 \\to \\mathbb\nP^2$ is the the blowup of $\\mathbb P^2$ at the six singular points\nof $Q$. The surface $T$ is a non-minimal rational surface\ncontaining six disjoint smooth rational curves. Indeed by Hurwitz\nformula, the canonical divisor of $T$ is given by\n\n$$K_{T}= \\zeta^*(K_{\\mathbb\nP^2}+\\frac{1}{2}(l_3+l_4+l_5+l_6))=-\\zeta^*(H)$$where $H$ is a\nhyperplane section. Thus $K_T^2=2, H^2 =2$ and $P_2(T)=0$. Denote\nby $\\tilde{Q}$ the proper transform of $Q$ in $T$. Using the\nadditivity of the topological Euler characteristic and the Noether\nformula, we have that $$e(T)= e(T- \\tilde{Q}) + e(\\tilde{Q})=10\n\\Rightarrow \\mathcal X(\\mathcal O_T)=1 \\Rightarrow q(T)=0$$ By\nCastelnuovo's rationality criterion, $T$ is a rational surface. In\nfact, we show that $T$ is a weak del Pezzo surface of degree two,\ni.e. the blow up of $\\mathbb P^2$ at seven points with nef\ncanonical divisor. Indeed we successively blown down the preimages\nin $T$ of the four lines $l_3, l_4, l_5$ and $l_6$ as well as the\npreimages in $T$ of the three ``diagonals'' of the complete\nquadrangle formed by $l_3, l_4, l_5, l_6$. The surface obtained\nafter these seven blow down is a projective plane.\n\nConsider the following curves of $T$\n$$\\zeta^*(C)=\\zeta^*(l_1+l_2)= E_{1} + E_{2}, \\textrm{ where } E_1\n\\textrm{ and } E_2 \\textrm{ are smooth elliptic curves}$$ and\n$$\\zeta^*(W)=W_1+W_2 \\textrm{ where } W_1 \\textrm{ and } W_2\n\\textrm{ are smooth rational curves}$$ with the following\nintersection properties:\n$$E_i^2=2, \\quad W_i^2=0, \\quad E_1 \\cdot E_2=2, \\quad W_1 \\cdot\nW_2=4, \\quad W_i \\cdot E_j=2 \\quad \\textrm{ for}\\quad i\\neq j.$$\n(recall that $W$ is the plane conic tangent to the six lines $l_1,\n\\cdots,l_6$). The linear system $|E_1|$ defines an elliptic\nfibration on $T$ with six singular fibers of type $I_2$. Take the\ndouble cover branched along the two fibers $E_1 + E_2 \\in\n2\\mathrm{Pic(T)}$. It induces the canonical resolution commutative\ndiagram: $$\\xymatrix{X \\ar@{->}[d]\\ar[r] \\ar[dr]^{\\varphi}& X_0\n\\ar@{->}[d]\\\\ \\tilde{T} \\ar@{->}[r]& T}$$ where $\\tilde{T} \\to T$\nis the blowup of $T$ at the two singular points of $E_1+E_2$.\n\n\\textit{Claim:} X is a Kummer surface.\n\n\\textit{Proof of the Claim:} Clearly $\\mathcal\nK_{X}=\\varphi^*(\\zeta^*(-H)+ \\frac{1}{2}(E_1+E_2))=\\mathcal O_X$.\n\n\n1) The pullback by $\\varphi$ of the six exceptional curves on $T$ define\ntwelve disjoint smooth rational curves on $X$.\n\n2) The two exceptional curves of $X$ give two more rational curves\ndisjoint from 1).\n\n3) Let $\\varphi^*(W_1)=W'_1+W''_1$ and $\\varphi^*(W_2)=W'_2+W''_2$\nand let $\\sigma$ be the lift on $X$ of the covering involution of\n$\\zeta$, then $\\sigma(W'_1)=W'_2$ or $\\sigma(W'_1)=W''_2$. Without\nloss of generality, we can assume that $\\sigma(W'_1)=W'_2$ and\nhence get the following intersection numbers $$W'^2_i=W''^2_i=-2,\n\\quad W'_i\\cdot W''_i=2 \\quad \\textrm{for } i=1,2 $$ $$ W'_1\\cdot\nW'_2=W''_1\\cdot W''_2=4 \\quad \\textrm{and} \\quad W'_1\\cdot\nW''_2=W''_1\\cdot W'_2=0.$$ One easily checks that $W'_1$ and\n$W''_2$ do not intersect the fourteen curves from 1) and 2).\n\n\\noindent In particular, the $K3$ surface $X$ contains sixteen\ndisjoint smooth rational curves. Consequently $X$ is a Kummer\nsurface.\n Moreover, the surface $X$ contains an elliptic fibration with twelve $I_2$ fibers. It\n also admits two non symplectic involutions $\\theta$ and $\\sigma$ where $\\theta$ is the covering involution of\n the map $\\varphi$ and $\\sigma$ is the lift of the covering involution of $\\zeta$ on $T$ encountered earlier.\n The composition $\\iota = \\varphi \\circ \\sigma$ defines a symplectic involution on $X$ whose quotient is a $K3$\n surface admitting an elliptic fibration with singular fibers identical to the one defined by $F$ on $Y$ in the proposition\n \\ref{fibration}.\n\n In fact, we can now recover sixteen disjoint rational curves on the quotient and conclude that\n it is our original general Kummer surface $Y$ and that $X \\simeq \\mathrm{Kum}(B_{12})$.\n\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nLow power consumption is a key requirement for modern computational devices.\nNon-volatility is one of the core concepts to reduce \npower consumption for logics and memories\nin normally-off computing\n\\cite{ando_fed_2001, ando_spin-transfer_2014, nakada_normally-off_2017}. \nMagnetoresistive random access memory (MRAM) is a promising\nnon-volatile memory that stores information \nassociated with the direction of\nmagnetization in magnetic tunnel junctions (MTJs)\n\\cite{yuasa_giant_2004, parkin_giant_2004, djayaprawira_230_2005, yuasa_giant_2007,\nkishi_lower-current_2008, kitagawa_impact_2012,\napalkov_magnetoresistive_2016, sbiaa_recent_2017, cai_high_2017}. \nIn order to reduce\npower consumption of MRAM, \nseveral types of writing schemes have been\ndeveloped. The currently used writing scheme is based on the\nspin-transfer-torque (STT) switching phenomena, which were\nproposed by\nSlonczewski \\cite{slonczewski_conductance_1989,slonczewski_current-driven_1996}\nand independently by Berger \\cite{berger_emission_1996}. The write\nenergy of STT-MRAM is of the order of 100 fJ\/bit\n\\cite{kitagawa_impact_2012, cai_high_2017}, which is\nstill 2 orders of magnitude larger than that of static random-access\nmemory.\n\nDiscovery of the voltage control of magnetic anisotropy (VCMA) effect\n\\cite{weisheit_electric_2007, maruyama_large_2009, duan_surface_2008,\n nakamura_giant_2009, tsujikawa_finite_2009, nozaki_voltage-induced_2010,\n endo_electric-field_2010,\n nozaki_magnetization_2014, skowronski_perpendicular_2015, nozaki_large_2016,\n li_enhancement_2017}\npaved the way for further reduction of write energy in MRAM.\nThe mechanism of VCMA in an MgO-based MTJ is considered to be the\ncombination of the selective electron or hole doping into the\n$d$-electron orbitals and the induction of a magnetic dipole moment,\nwhich affect the electron spin through spin-orbit interaction\n\\cite{duan_surface_2008,nakamura_giant_2009,tsujikawa_finite_2009,miwa_voltage_2017}.\nThe MRAM which uses the VCMA effect to switch magnetization is called\nthe voltage controlled MRAM (VC-MRAM)\n\\cite{shiota_induction_2012, shiota_pulse_2012,\nkanai_electric_2012,shiota_evaluation_2016, grezes_ultra-low_2016, kanai_electric-field-induced_2016,\nshiota_reduction_2017, \nmatsumoto_voltage-induced_2018, yamamoto_thermally_2018, yamamoto_write-error_2019, \nmatsumoto_voltage-induced_2019, imamura_impact_2019, matsumoto_methods_2019}.\nThe writing procedure of a conventional VC-MRAM is as follows.\nThe perpendicularly magnetized MTJ is subjected to an in-plane\nexternal magnetic field ($H_{\\rm ext}$) as shown in Fig. \\ref{fig:schem}(a).\nThe magnetic anisotropy (MA) constant of the free layer can be controlled by\napplying voltage ($V$) as shown in Fig. \\ref{fig:schem}(b).\nHere, $K_{\\rm eff}$ is the effective perpendicular anisotropy constant\nwhere the demagnetization energy is subtracted from the perpendicular anisotropy constant.\nThroughout the paper, the superscript (0) indicates the quantities at $V = 0$.\nThe voltage pulse with critical amplitude $V_{\\rm c}$\neliminates the MA and induces the precession of the magnetization around\nthe external magnetic field. By turning off the voltage at one half\nperiod of precession, the magnetization switching completes.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics [width=\\columnwidth] {fig1.eps}}\n \\caption{\n \\label{fig:schem} \n (a) Magnetic tunnel junction with circular cylinder shape,\n and definitions of Cartesian coordinates $(x, y, z)$, polar angle\n ($\\theta$), and azimuthal angle ($\\phi$).\n The $x-$axis is parallel to the direction of \n the external in-plane magnetic field, ${\\bm H}_{\\rm ext}$.\n The unit vector ${\\bm m} = (m_{x}$, $m_{y}$, $m_{z})$ \n represents the direction of the magnetization in the free layer.\n The magnetization in the reference layer (ref.) is fixed to align in the\n positive $z-$direction.\n (b) The voltage ($V$) dependence of the effective perpendicular anisotropy constant ($K_{\\rm eff}$).\n The effective anisotropy constant at $V=0$ is represented by\n $K_{\\rm eff}^{(0)}$. It takes the value of\n $K_{\\rm eff} = K_{\\rm eff}^{\\rm p}$ at $V=V_{\\rm p}$.\n }\n\\end{figure}\n\nThe write energy of VC-MRAM is estimated from the Joule heating energy loss\nduring the pulse. Assuming that the voltage\npulse with amplitude $V$ and duration $t_{\\rm p}$ is applied to the MTJ with\nresistance $R$, the write energy is given by \n\\begin{equation}\n E_{\\rm J} = \\frac{V^{2}}{R}t_{\\rm p},\n\\end{equation}\nTo reduce the write energy, \nthe VC-MRAM should be designed to have large\nresistance and short pulse duration. The pulse duration is given by\na half period of precession as\n\\begin{align}\n \\label{eq:period_Hext}\n t_{\\rm p} = \\frac{\\pi (1+\\alpha^{2}) }{\\gamma H_{\\rm ext}},\n\\end{align}\nwhere $\\alpha$ is the Gilbert damping constant and $\\gamma$ is the\ngyromagnetic ratio. For example, $t_{\\rm p}$ = 0.18 ns for $\\alpha$ =\n0.1 and $\\mu_{0} H_{\\rm ext}$ = 100 mT, where $\\mu_{0}$ is the vacuum\npermeability. Recently, Grezes $et$ $al$. demonstrated \na very small write energy of 6 fJ\/bit for the VC-MRAM with $R=330$ k$\\Omega$ \nat $V$ = 1.96 V and $t_{\\rm p}$ = 0.52 ns\n\\cite{grezes_ultra-low_2016}. The similar results were also obtained\nindependently by Kanai $et$\n$al$. \\cite{kanai_electric-field-induced_2016}.\n\n\nIt is difficult to use MTJ with huge $R$ to further reduce\nwrite energy because the read time of the VC-MRAM increases \nwith increase of $R$. \nAdopting a scheme of decreasing \npulse duration\nby increasing external magnetic field \nshould also be avoided \nsince the application of a\nstrong in-plane magnetic field $H_{\\rm ext}$ deteriorates the thermal stability\nfactor defined as\n\\begin{align}\n \\label{eq:delta}\n \\Delta^{(0)}\n =\n \\frac{\n \\left(2 K_{\\rm eff}^{(0)}\n - \\mu_{0}M_{\\rm s} H_{\\rm ext}\n \\right)^{2}\n V_{\\rm F}\n }{4K_{\\rm eff}^{(0)} k_{\\rm B}T},\n\\end{align}\nwhere \n$k_{\\rm B}$ is the Boltzmann constant, $T$ is the temperature,\n$M_{\\rm s}$ is the saturation magnetization, \nand $V_{\\rm F}$ is the volume of the free layer.\n\n\n\n\nIn this paper, we propose another switching scheme which could reduce the pulse\nduration and therefore the write energy of a VC-MRAM. The main\ndifference between the conventional scheme and the proposed switching scheme is\nthe polarity of the voltage pulse. Application of the voltage pulse\nwith the polarity that is opposite to the conventional switching can\nenhance the magnetic anisotropy and induce the precession around the\naxis close to the easy axis. After turning off the voltage \nproximately at a\nhalf of precession period, the magnetization relaxes to the opposite\nequilibrium direction and the switching completes.\nWe perform numerical simulations and demonstrate that the pulse\nduration of the proposed switching scheme is as short as a few tens\nof pico seconds. We also evaluate the write error rate (WER) and\nshow that the WER is minimized if the pulse duration is about half the\nperiod of precession similar to the conventional switching scheme.\n\n\n\\section{Theoretical model}\nThe system we consider is schematically shown in Fig. \\ref{fig:schem}(a).\nThe macrospin model is employed to describe the magnetization dynamics.\nThe direction of the magnetization in the free layer is\nrepresented by the unit vector ${\\bm m} = (m_{x}$, $m_{y}$, $m_{z})\n= (\\sin \\theta \\cos \\phi$, $\\sin \\theta \\sin \\phi$, $\\cos \\theta$), \nwhere $\\theta$ and $\\phi$ are the polar and azimuthal angles.\nThe $x$ axis is parallel to the direction of external in-plane\nmagnetic field ${\\bm H}_{\\rm ext}$.\n\nThe energy density of the free layer is given by\n\\begin{align}\n \\label{eq:energy_density}\n {\\cal E} (m_{x}, m_{y}, m_{z})\n = \n -K_{\\rm eff} m_{z}^{2} - \\mu_{0} M_{\\rm s} H_{\\rm ext} m_{x},\n\\end{align}\nThe first term of Eq. \\eqref{eq:energy_density} is the sum of the\nshape, the bulk crystalline and the interfacial\nanisotropies. Owing to the VCMA effect, \n$K_{\\rm eff}$ can be controlled\nby application of $V$ as shown in Fig. \\ref{fig:schem}(b).\nHere $K_{\\rm eff}^{(0)}$ represents the effective anisotropy constant\nwithout the voltage application. We assume that $K_{\\rm eff}$\ndecreases with increase of $V$ and vanishes at $V=V_{\\rm c}$.\nApplying the voltage $V_{\\rm p} ( < 0)$ increases\n$K_{\\rm eff}$ to $K_{\\rm eff}^{\\rm p}$ and induces\nthe precessional motion of $\\bm{m}$ around the effective magnetic\nfield. The effective field is given by \n$H_{\\rm eff} = (H_{\\rm ext},$ $0,$ $H_{\\rm\n K} m_{z})$, where ${H}_{\\rm K} = 2 K_{\\rm eff}^{\\rm p} \/ ( \\mu_{0}\nM_{\\rm s} )$ is the anisotropy field.\n\n\nThe magnetization dynamics is simulated by solving \nthe following Landau-Lifshitz-Gilbert equation \\cite{brown_thermal_1963},\n\\begin{equation}\n \\label{eq:LLG}\n \\frac{{\\rm d} {\\bm m}}{{\\rm d}t}\n = -\\gamma_{0} {\\bm m}\\times\n \\left(\\bm{H}_{\\rm eff} + \\bm{h}\\right)\n +\\alpha\n \\bm{m}\\times\n \\frac{{\\rm d} {\\bm m}}{{\\rm d}t},\n\\end{equation}\nwhere $\\bm{h}$ represents the thermal agitation field satisfying the\nfollowing relations:\n\\begin{align}\n &\\langle h_{\\iota}(t)\\rangle=0\n \\\\\n & \\langle\nh_{\\iota}(t)h_{\\kappa}(t') \n\\rangle\n= \\frac{2\\alpha k_{\\rm B} T }{ \\gamma_{0} \\mu_{0} M_{\\rm s} V_{\\rm F} }\\delta_{\\iota\\kappa}\\delta(t-t').\n\\end{align}\nHere $\\iota,\\kappa=x,y,z$, \nand $\\langle X \\rangle$ denotes the statistical average of $X$.\n\nThroughout this paper, we assume that the external field is $\\mu_{0}\nH_{\\rm ext}$ = 100 mT and the saturation magnetization of the free\nlayer is $M_{\\rm s}= 1400$ kA\/m.\nAlso the radius of the junction area is assumed as $r=50$ nm and the\nthickness of the free layer, $t_{\\rm F}=1$ nm, and therefore the volume of\nthe free layer as $V_{\\rm F}=\\pi r^{2} t_{\\rm F}=7854$ nm$^{3}$. \nThe initial states are prepared by 10 ns relaxation from the\nequilibrium direction at $T=0$, that is \n$(\\theta^{(0)},$ $\\phi^{(0)}) = \\left( \\sin^{-1} \\left[ \\mu_{0} M_{\\rm s} H_{\\rm ext} \/ (2 K_{\\rm eff}^{(0)} ) \\right] \\right.,$ \n$ 0 \\Bigr)$\n\\cite{matsumoto_voltage-induced_2019}. \nThe write error rates are calculated from\n$10^{6}$ trials with 10 ns relaxation after the pulse.\n\n\n\n\\section{Results and discussions}\nFirst we show the difference between the mechanisms of the conventional\nvoltage controlled switching and the proposed switching that utilizes\nthe enhancement of the magnetic anisotropy.\nThis can be accomplished by analyzing the switching\ntrajectories at $T=0$.\nFigures \\ref{fig:t0}(a) and (b) show the shape of the voltage\npulse and the corresponding time dependence of the effective\nanisotropy constant for the conventional voltage controlled switching.\nThe induced switching dynamics of $\\bm{m}$ at $T=0$ is shown in\nFigs. \\ref{fig:t0}(c) together with the color map of the energy\ndensity of Eq. \\eqref{eq:energy_density} at $V=0$.\nThin black dotted curves represent energy contours. Thick black curves\nrepresent the energy contour crossing\n$\\bm{m}=(1,$ $0,$ $0)$. The initial direction of the magnetization is the\nequilibrium direction with $m_{z}>0$ indicated by the black circle,\nwhich we call as the ``up state''. \n\nIn Figs. \\ref{fig:t0}(a), (b) and (c),\napplication of the voltage pulse with $V_{\\rm c}$\neliminates the magnetic anisotropy and induce the precession of $\\bm{m}$\naround the external magnetic field as represented by the red\ncurve. After turning off the voltage at one-half period of precession, \nthe magnetization starts to relax from the point indicated by the orange\ncircle\nto the other equilibrium\ndirection with $m_{z}<0$, i. e. the ``down state'', indicated by the\nblack circle. \nNote that the black circle at $m_{z}<0$ is illustrated \nunder the green curve. \nThe switching is thus completed as represented by the green\ncurve.\n\n\\begin{figure}[H]\n \\centerline{\n \\includegraphics [width=0.8\\columnwidth] {fig2.eps}\n }\n \\caption{\n \\label{fig:t0} \n (a) The shape of the voltage pulse for the {\\em conventional switching\n scheme}. The amplitude including the polarity of the pulse and duration of the pulse are \n $V_{\\rm c}$ (positive value) and $t_{\\rm p}$, respectively.\n (b) The corresponding time dependence of the effective anisotropy\n constant $K_{\\rm eff}$. At $V=0$, it takes the value $K_{\\rm eff}^{\\rm (0)}$. \n During the pulse, $K_{\\rm eff} = 0$ because $V=V_{\\rm c}$.\n (c) The color map of the energy density at $V=0$ on the\n $\\phi-m_{z}$ plane. Thin black dotted curves represent energy\n contours. Thick black curves represent the energy contour crossing\n $\\bm{m}=(1,$ $0,$ $0)$.\n The trajectory of $\\bm{m}$ during and after the pulse\n are shown by the red and green curves, respectively. The direction\n of the trajectory is indicated by the triangle. The orange circle\n represents the direction of ${\\bm m}$ at the end of the pulse.\n We assume that $\\alpha$ = 0.1.\n (d) The shape of the voltage pulse for the {\\em proposed switching scheme}.\n The polarity is negative, i. e. $V_{\\rm p}<0$, to enhance $K_{\\rm eff}$.\n (e) The corresponding time dependence of the effective anisotropy\n constant. During the pulse, it is enhanced to $K_{\\rm eff}^{\\rm p}$.\n (f) The color map of the energy density at $V=0$ on the\n $\\phi-m_{z}$ plane. We assume that $K_{\\rm eff}^{\\rm p}$ = 400\n kJ\/m$^{3}$ and $\\alpha$ = 0.21. The symbols are the same as those\n in Panel (c).\n Please note that the left and right boundaries at $\\phi =\n \\pm \\pi$ represent the same direction of $\\bm{m}$.\n}\n\\end{figure}\n\nFigures \\ref{fig:t0}(d) and (e) show the shape of the voltage pulse\nand the corresponding time dependence of the effective anisotropy\nconstant for the switching using the enhanced $K_{\\rm eff}$.\nThe induced switching dynamics of $\\bm{m}$ at $T=0$ is shown in\nFig. \\ref{fig:t0}(f) together with the color map of the energy\ndensity at $V$ = 0. \nThe initial state is the up state indicated by the\nblack circle at $m_{z}>0$. Application of the voltage pulse with $V_{\\rm p} (< 0)$ \nenhances the effective anisotropy constant from $K_{\\rm eff}^{(0)}$ to\n$K_{\\rm eff}^{\\rm p}$ and induce the precession of $\\bm{m}$\naround the effective magnetic field as represented by the red curve.\nThe value of $K_{\\rm eff}^{\\rm p}$ is assumed to be 400\nkJ\/m$^{3}$, which gives the anisotropy field of $\\mu_{0} H_{\\rm K}$ = 570\nmT. The effective field is nearly parallel to the easy axis or the $z$\naxis because the directional cosine of the effective field relative to the\neasy axis is 0.98. \nThe voltage is turned off at about a half period\nof the precession, and the magnetization reaches the point, \n$\\phi \\simeq \\pi$ indicated by the orange circle. As will be shown later, \nthe write error rate (WER) is minimized if the pulse duration is set about\nhalf the period of precession. After turning off the pulse, the\nmagnetization relaxes to the down state and completes the switching as\nshown by the green curve. The proposed switching scheme does not\nreduce the thermal stability factor of Eq. \\eqref{eq:delta} \nbecause it\njust enhances $K_{\\rm eff}$ during the voltage pulse.\n\n\n\\begin{figure}[H]\n\\includegraphics [width=\\columnwidth] {fig3.eps}\n\\caption{\\label{fig:mm} \n (a) The Cartesian components of ${\\bm m} = (m_{x}$, $m_{y}$,\n $m_{z})$ of a typical switching trajectory \n are plotted as a function of time\n during the\n pulse at $T$ = 300 K.\n $K_{\\rm eff}^{\\rm p}$ = 400 kJ \/m$^{3}$ and $\\alpha$ = 0.21.\n The unit of the horizontal axis is ps. \n (b) The same as (a) after the pulse. The unit of the horizontal axis is ns. \n (c) The pulse duration dependence of the WER at $T$ = 300 K for\n $K_{\\rm eff}^{\\rm p}$ = 300 kJ \/m$^{3}$ and $\\alpha$ = 0.18.\n (d) The same as (c) for $K_{\\rm eff}^{\\rm p}$ = 400 kJ \/m$^{3}$ and\n $\\alpha$ = 0.21.\n }\n\\end{figure}\n\nNext we discuss the switching properties of the proposed switching\nscheme at $T$ = 300 K by analyzing the numerical simulations results.\nThe time evolution of the Cartesian components of ${\\bm m}$ for a\ntypical switching trajectory during the pulse are\nshown in Fig. \\ref{fig:mm}(a). The value of $K_{\\rm eff}^{(0)} = 100$ kJ\/m$^{3}$\nand $\\alpha$ are \nthe same as in Fig. \\ref{fig:t0}(f),\n$K_{\\rm eff}^{\\rm p}$ = 400 kJ\/m$^{3}$ and $\\alpha$ = 0.21.\nDuring the pulse duration, \n$m_{z}$ increases with the\nincrease of time because the effective anisotropy constant is\nenhanced. The shapes of $m_{x}$ and $m_{y}$ are very similar to\nthe cosine and sine functions, respectively, \nbecause \n$\\bm{m}$ precesses around the effective field which is almost parallel\nto the $z$ axis. \nFigure \\ref{fig:mm}(b) shows the time evolution of $m_{x}$, $m_{y}$\nand $m_{z}$ after the pulse. Please note that the horizontal axis is\nin unit of ns. $m_{z}$ monotonically decreases with the increase of time\nand the switching completes at around 0.4 ns.\n\n\nFigures \\ref{fig:mm}(c) and (d) show the pulse duration, $t_{\\rm p}$,\ndependence of the write error rate (WER) for different\nvalues of $K_{\\rm eff}^{\\rm p}$ and $\\alpha$.\nThe parameters are $K_{\\rm eff}^{\\rm p}$ = 300 kJ \/m$^{3}$ and\n$\\alpha$ = 0.18 for Fig. \\ref{fig:mm}(c), \nand $K_{\\rm eff}^{\\rm p}$ =\n400 kJ \/m$^{3}$ and $\\alpha$ = 0.21 for Fig. \\ref{fig:mm}(d).\nIn Fig. \\ref{fig:mm}(c), the WER takes a minimum value of $7.6 \\times\n10^{-3}$ at $t_{\\rm p}$ = 46 ps.\nIn Fig. \\ref{fig:mm}(d), the WER takes a minimum value of $3.2 \\times\n10^{-3}$ at $t_{\\rm p}$ = 36 ps. These optimal values of $t_{\\rm p}$ at which\nthe WER is minimized are almost the same as one half period of precession around\n$H_{\\rm eff}$.\n\n\n\n\\section{conclusion}\nIn summary, we propose a low power switching scheme of magnetization using\nenhanced magnetic anisotropy by applying a short voltage pulse.\nThe proposed switching scheme can reduce the pulse duration and\ntherefore the write energy substantially without deteriorating \nthermal stability.\nWe perform numerical simulations and show that the pulse duration of\nthe proposed switching scheme is as short as a few\ntens of pico seconds. We also calculated the pulse duration\ndependence of the WER, \nand showed that the optimal values of $t_{\\rm\n p}$ at which the WER is minimized are nearly half the period of\nprecession around the effective field.\n\n\\acknowledgements\nThis work was partly supported by JSPS KAKENHI Grant No. JP19K05259\nand 19H01108.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}