Overconvergent Eichler-Shimura Isomorphisms on Shimura Curves Over a Totally Real Field

Overconvergent Eichler-Shimura Isomorphisms on Shimura Curves over a Totally Real Field Shan Gao AThesis In the Department of Mathematics and Statistics Presented in Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy (Mathematics) at Concordia University Montreal, Quebec, Canada September 2016 c Shan Gao, 2016 CONCORDIA UNIVERSITY SCHOOL OF GRADUATE STUDIES This is to verify that the thesis prepared By: Shan Gao Entitled: Overconvergent Eichler-Shimura Isomorphisms on Shimura Curves over a Totally Real Field and submitted in partial fulfilment of requirements for the degree of Docotor of Philosophy (Mathematics) complies with the regulations of the University and meets the accepted standards with respect to originality and quality. Signed by the final examining committee: Dr. Marcie Frank Chair Dr. Henri Darmon External Examiner Dr. Pablo Bianucci External to Program Dr. Hershy Kisilevsky Examiner Dr. Christopher Cummins Examiner Dr. Adrian Iovita Thesis Supervisor Approved by Dr. Arusharka Sen Chair of Department or Graduate Program Director September 8, 2016 Dr. AndréRoy Dean of Faculty ABSTRACT Overconvergent Eichler-Shimura Isomorphisms on Shimura Curves over a Totally Real Field Shan Gao, Ph.D. Concordia University, 2016 In this work we construct overconvergent Eichler-Shimura isomorphisms on Shimura curves over a totally real field F . More precisely, for a prime p>2 and a wide open disk U in the weight space, we construct a Hecke-Galois-equivariant morphism from the space of families of overconvergent modular symbols over U to the space of families of overconvergent modular forms over U. In addition, for all but finitely many weights λ ∈ U, this morphism provides a description of the finite slope part of the space of overconvergent modular symbols of weight λ in terms of the finite slope part of the space of overconvergent modular forms of weight λ + 2. Moreover, for classical weights these overconvergent isomorphisms are compatible with the classical Eichler-Shimura isomorphisms. iii Acknowledgements Fist, I would like to thank my supervisors Dr. Adrian Iovita greatly, for the support, guidance, patience and encouragement throughout my study at Concordia University. Dr. Iovita is an enthusiastic and creative researcher who taught and led me to the field of Number Theory, and I have been really enjoying it. I wish him nothing but the best. I thank the members of the Number Theory Seminars holding at McGill University, Con- cordia University,University of Montreal and UQAM for helpful discussions. The great and interesting presentations helped my mathematical study a lot. Thank Dr. D. Barrera sincerely for the help and fruitful discussions. It has been nice to be working with him. I am grateful to all the members of my thesis examining committee, thank you for your time and work. I really appreciate the courses and advice from the members in the Department of Math- ematics and Statistics. I thank Concordia University for admitting me to the PhD program and for the financial support. I thank my parents and all other family members, for their care and selfless support. I would like to thank my husband Zhenyang, for the love and support all these years. iv Contents 1 Introduction 1 2 Log Schemes and Log Smoothness 11 2.1Monoids...................................... 12 2.2LogSchemes.................................... 19 2.2.1 Logstructures............................... 19 2.2.2 Charts................................... 21 2.2.3 Fiberedproductsoflogschemes..................... 23 2.3 Log smooth and log étalemorphisms...................... 25 2.4 Kummer étaletopology.............................. 30 3 P-adic Modular Forms over Shimura Curves 33 3.1ShimuraCurves.................................. 33 3.1.1 Quaternionic Shimura curves MK (G, X) ................ 33 3.1.2 Unitary Shimura curves MK (G ,X ) .................. 35 3.1.3 ModuliproblemsforunitaryShimuracurves.............. 40 3.1.3.1 Thecanonicalmodeloverthereflexfield........... 40 v 3.1.3.2 Thecanonicalmodeloveralocalfield............ 44 3.1.3.3 Speciallevelstructures..................... 52 3.1.3.4 Integralmodels......................... 53 3.2HasseInvariantandCanonicalsubgroups.................... 55 3.2.1 Hasseinvariant.............................. 55 3.2.2 Quotient of (A,ι,θ,k¯P) by a finite flat subgroup of A ......... 59 3.2.3 Canonicalsubgroups........................... 60 3.3ThemapdlogandtheHodge-Tatesequence.................. 64 3.3.1 Group schemes with strict OP-action.................. 64 3.3.2 Themapdlog............................... 66 3.3.3 TheHodge-Tatesequence........................ 68 3.4ModularSheaves................................. 71 3.4.1 Theweightspace............................. 71 3.4.2 Atorsor.................................. 72 3.4.3 Modularsheaves............................. 73 3.5HeckeOperators................................. 76 3.5.1 TheUoperator.............................. 76 3.5.2 OtherHeckeoperators.......................... 80 4 Faltings’ Sites 82 4.1Faltings’topos:thesmoothcase......................... 82 4.1.1 Thealgebraicsetting........................... 82 4.1.2 Theformalsetting............................ 91 vi 4.2Faltings’topos:thesemistablecase....................... 96 4.2.1 Assumptions................................ 96 4.2.2 Faltings’sites............................... 99 4.2.2.1 The site Xket .......................... 99 4.2.2.2 Presheaves on Xket ....................... 100 4.2.2.3 Faltings’sites ......................... 102 4.3Faltings’siteassociatedtoShimuracurves................... 103 4.3.1 Logstructures............................... 103 4.3.2 The sites M(w)andM(H, π) ...................... 105 4.3.3 Inducedsites............................... 106 4.4Continuousfunctors............................... 115 4.5Localizationfunctors............................... 117 5 Distributions and Overconvergent Cohomology 119 5.1Distribution.................................... 119 5.1.1 Definitions................................. 119 5.1.2 Filtration................................. 127 5.1.3 Specialization............................... 132 5.2Overconvergentcohomology........................... 135 5.2.1 Definitions................................. 135 5.2.2 Groupcohomology............................ 138 5.2.3 Slopedecomposition........................... 141 5.2.4 SheavesonFaltings’Site......................... 146 vii 6 The Morphism 148 6.1 Modular sheaves on M(w)............................ 149 6.1.1 Themapdlog............................... 149 6.1.2 Atorsor.................................. 152 6.1.3 Action of Gr ................................ 154 6.1.4 Modularsheaves............................. 157 λ 6.2 Cohomology of the sheaf ωM(w) ......................... 158 6.3Themorphism................................... 162 6.3.1 Notations................................. 162 6.3.2 Constructionofthemorphism...................... 164 6.4Heckeoperators.................................. 173 7 Eichler-Shimura isomorphisms 179 7.1Classicalweights................................. 180 7.2Mainresult.................................... 188 7.2.1 Assumptionsandnotations....................... 188 7.2.2 Theproofofthemaintheorem..................... 192 Bibliography 202 viii Chapter 1 Introduction Let us review the classical Eichler-Shimura isomorphism on modular curves. Fix a prime p ≥ 3, an integer N ≥ 3 such that (p, N) = 1 and let Γ := Γ1(N) ∩ Γ0(p) ⊆ SL2(Z). Let X := X(N,p) be the modular curve for the group Γ over Spec(Z[1/(Np)]), E → X ∗ 1 the universal semi-abelian scheme and ω := ωE/X = e (ΩE/X) the invertible sheaf on X of invariant 1-differentials, where e : X → E is the zero section. We have the following theorem. Theorem 1.0.1. (Deligne [1971b]) For every nonnegative integer k, we have a natural isomorphism: 1 ∼ 0 k+2 ⊕ 0 k ⊗ 1 H (Γ,Vk,C) = H (XC,ω ) H (XC,ω ΩX/ C), k 2 where Vk,C := Sym (C ) and the overline on the second term of the right hand side is the complex conjugation. Moreover, this isomorphism is compatible with the action of the Hecke operators. 1 The elements of H (Γ,Vk,C) are called classical weight k modular symbols. The elements 1 0 k+2 0 k ⊗ 1 of H (XC,ω ), respectively H (XC,ω ΩX/ C) are called classical modular, respectively cusp forms of weight k + 2. The classical Eichler-Shimura isomorphism describes the space of weight k modular symbols in terms of elliptic modular forms of weight k +2.InFaltings [1987] a more arithmetic version of this isomorphism is presented. Now fix a complete discrete valuation field L of characteristic 0, ring of integers OL and residue field L, a perfect field of characteristic p. WedenotebyCp the p-adic completion of an algebraic closure L of L. Let us consider the modular curve X over the p-adic field L and for a nonnegative integer k Q2 · k,letVk := Sym ( p)(1), where ( )(k) is the Tate twist, with the natural action of Γ and GL =Gal(L/L). The following theorem is obtained in Faltings [1987]. Theorem 1.0.2. With the above notations we have a canonical isomorphism compatible with the actions of GL and all Hecke operators 1 ∼ 0 k+2 1 −k H Γ,Vk ⊗L Cp = H (X, ω ) ⊗L Cp ⊕ H (X, ω ) ⊗L Cp(k +1) . In Coleman [1997] and Coleman and Mazur [1998], the authors show that modular eigenforms of finite slope can be p-adically interpolated. In fact there exists a geometric object parameterizing such modular eigenforms called the eigencurve. On the other hand, modular symbols have interesting p-adic properties. The work in Stevens [2015] defines overconvergent

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