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UNIVERSITY of CALIFORNIA Los Angeles Structure of Various UNIVERSITY OF CALIFORNIA Los Angeles Structure of various Lambda-adic arithmetic cohomology groups A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Jaehoon Lee 2019 c Copyright by Jaehoon Lee 2019 ABSTRACT OF THE DISSERTATION Structure of various Lambda-adic arithmetic cohomology groups by Jaehoon Lee Doctor of Philosophy in Mathematics University of California, Los Angeles, 2019 Professor Haruzo Hida, Chair In this thesis, we study the structure of various arithmetic cohomology groups as Iwasawa modules, made out of two different p-adic variations. In the first part, for an abelian va- riety over a number field and a Zp-extension, we study the relation between structure of the Mordell-Weil, Selmer and Tate-Shafarevich groups over the Zp-extension as Iwasawa modules. In the second part, we consider the tower of modular curves, which is an analogue of the Zp-extension in the first part. We study the structure of the ordinary parts of the arithmetic cohomology groups of modular Jacobians made out of this tower. We prove that the ordinary parts of Λ-adic Selmer groups coming from one chosen tower and its dual tower have almost the same Λ-module structures. This relation of the two Iwasawa modules explains well the functional equation of the corresponding p-adic L-function. We also prove the cotorsionness of Λ-adic Tate-Shafarevich group under mild assumptions. ii The dissertation of Jaehoon Lee is approved. Romyar Thomas Sharifi Chandrashekhar Khare Don M. Blasius Haruzo Hida, Committee Chair University of California, Los Angeles 2019 iii To my mother iv TABLE OF CONTENTS Acknowledgments :::::::::::::::::::::::::::::::::::: vii Vita ::::::::::::::::::::::::::::::::::::::::::::: viii 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 1.1 Iwasawa theory of abelian varieties.......................1 1.2 Towers of modular curves............................3 1.3 Functors F and G .................................5 1.4 Organization...................................5 1.5 Notations.....................................6 2 Structure of cohomology groups over cyclotomic towers :::::::::: 8 2.1 Structure of limit Mordell-Weil groups.....................9 2.2 Selmer groups and Tate-Shafarevich groups................... 15 2.3 Estimates on the size of W1 ........................... 18 2.4 Algebraic functional equation.......................... 20 3 Tower of modular curves and Hecke algebra :::::::::::::::::: 26 3.1 Various modular curves.............................. 26 3.2 Big Ordinary Hecke algebra........................... 28 3.3 Limit Mordell-Weil groups and limit Barsotti-Tate groups........... 30 4 Twisted Pairings ::::::::::::::::::::::::::::::::::: 35 4.1 Various actions on Jacobians........................... 35 4.2 Twisted Weil pairing............................... 38 v 4.3 Local Tate duality................................ 40 4.4 Poitou-Tate duality................................ 41 4.5 Twisted pairings of Flach and Cassels-Tate................... 41 5 Cohomology groups arising from the Barsotti-Tate group g :::::::: 44 5.1 Local cohomology groups............................. 44 5.2 Global cohomology groups............................ 51 6 Structure of global Λ-adic cohomology groups :::::::::::::::: 57 6.1 Λ-adic Selmer groups............................... 57 6.2 Λ-adic Tate-Shafarevich groups......................... 64 7 Functors F and G :::::::::::::::::::::::::::::::::: 66 7.1 The functor F ................................... 66 7.2 The functor G ................................... 70 References ::::::::::::::::::::::::::::::::::::::::: 73 vi ACKNOWLEDGMENTS I am grateful to Don Blasius, Chandrashekhar Khare, Romyar Sharifi and Haruzo Hida for being part of the thesis committee. I am indebted to my advisor Haruzo Hida for his continuous guidance, encouragement and support. I've had the privilege of meeting with him every week and sitting in his classes and seminars. My view towards math has been heavily influenced by conversations with him. Without his patient guidance over the years, I wouldn't even be close to finishing this thesis. He has been very generous in sharing his ideas and intuition. It has been a great honour for being his student. My sincere thanks also go to Chandrashekhar Khare and Romyar Sharifi, who taught me a great deal of mathematics and provided me not only mathematical ideas, but also advice and encouragement. I also want to thank Ashay Burungale, Francesc Castella, Jaclyn Lang, Haesang Sun, Chan-Ho Kim and Gyujin Oh for their encouragement, advice and for the many interesting conversations. Most importantly, I would like to thank my mother, for her unconditional love. vii VITA 2013 B.S. (Mathematics), Seoul National University. 2015{2018 Teaching Assistant, Mathematics Department, UCLA. PUBLICATIONS Structure of Mordell-Weil groups over Zp-extensions. 2018, Submitted. viii CHAPTER 1 Introduction 1.1 Iwasawa theory of abelian varieties The Iwasawa theory of abelian varieties (elliptic curves) was initiated by Mazur in his sem- inal paper [Maz72], where he proved that the p1-Selmer groups of an abelian variety A defined over a number field K are \well-controlled" over Zp-extensions if A satisfies a certain reduction condition at places dividing p. More precisely, let K1 be a Zp-extension of K and Kn be the n-th layer. We identify Zp[[Gal(K1=K)]] with Λ := Zp[[T ]] by fixing a generator γ 1 of Gal(K1=K) and identifying it with 1+T: Let SelKn (A)p be the classical p -Selmer group over Kn attached to A. For the natural restriction map A pn Sn : SelKn (A)p ! SelK1 (A)p[(1 + T ) − 1]; we have the following celebrated theorem. Theorem 1.1.1 (Control Theorem). If either • (Mazur,[Maz72]) A has a good ordinary reduction at all places of K dividing p or • (Greenberg,[Gre99, Proposition 3.7]) K = Q and A is an elliptic curve having multi- plicative reduction at p A then Coker(Sn ) is finite and bounded independent of n. 1 Here the word \control" means that the Selmer group SelKn (A)p at each layer Kn can be described by the one object. Hence we can expect that the each Selmer group over Kn should behave in a certain regular way governed by the \limit" Selmer group SelK1 (A)p. For instance, we have the following consequence of the above theorem. Proposition 1.1.2. Assume either one of the condition of Theorem 1.1.1 and also assume 1 that both A(Kn) and WKn (A)p are finite for all n. Then there exist µ, λ, ν such that 1 en jSelKn (A)pj = jWKn (A)pj = p (n >> 0) where n en = p µ + nλ + ν: The value pnµ + nλ + ν in the Proposition 1.1.2 naturally appears from the structure theory of Λ-modules. For a finitely generated Λ-module M, there is a Λ-linear map n ! m ! r M Λ M Λ M ! Λ ⊕ e ⊕ g i pfj i=1 i j=1 with finite kernel and cokernel where r; n; m ≥ 0, e1; ··· ; en; f1; ··· ; fm are positive integers, and g1; ··· ; gn are distinguished irreducible polynomial of Λ. The quantities r; e1; ··· ; en; f1; ··· ; fm; g1; ··· ; gn are uniquely determined, and we call n ! m ! r M Λ M Λ E(M) := Λ ⊕ e ⊕ g i pfj i=1 i j=1 as an elementary module of M following [NSW00, Page 292]. The λ-invariant λ(M) is defined n X as ei · deggi and the µ-invariant µ(M) is defined as f1 + ··· + fm. i=1 In Proposition 1.1.2, the constants λ and µ are indeed the λ and µ-invariants of the _ _ module SelK1 (A)p , respectively and hence we can say that the module E SelK1 (A)p gives the information about the arithmetic of the A at each finite level at once. Hence it is _ quite natural to ask questions about the shape of the Λ-module E SelK1 (A)p . We can 2 also consider the same question for the Mordell-Weil group and the Tate-Shafarevich group, which fit into a natural exact sequence 1 0 ! A(K1) ⊗ Qp=Zp ! SelK1 (A)p ! WK1 (A)p ! 0: Instead of the characteristic ideals of the modules above which are usually studied because of their connection with the Iwasawa Main Conjecture, we study their Λ-module structure. Naively, we can ask the following question: Question 1.1.3. Can we describe the structure of _ _ 1 _ E (A(K1) ⊗Zp Qp=Zp) ;E SelK1 (A)p ;E WK1 (A)p ? Or can we find some relations among these three modules? In this thesis, under suitable assumptions, we will prove an isomorphism _ t _ι E SelK1 (A)p ' E SelK1 (A )p 1 of Λ-modules where ι is an involution of Λ satisfying ι(T ) = 1+T − 1. (Theorem 2.4.3) 1.2 Towers of modular curves In this thesis, we consider one more p-adic variation of arithmetic cohomologies which we briefly introduce here. For simplicity, in this introduction let Xr=Q be the (compactified) modular curve classifying the triples φ r φN p r (E; µN ,−! E; µpr ,−−! E[p ])=R where E is an elliptic curve defined over a Q-algebra R. We then have a natural covering map Xr+1 ! Xr for all r, which gives rise to the tower ··· Xr+1 ! Xr !··· X2 ! X1 of modular curves, which is an analogue of Zp-extension introduced in the last section. 3 Let Jr=Q be the Jacobians of Xr. For a number field K, we have the Mordell-Weil group Jr(K) ⊗ Qp=Zp, (geometric) p-adic Selmer group SelK (Jr)p and the p-adic Tate-Shafarevich 1 n! group WK (Jr)p of the Jacobian Jr. We can apply the idempotent e := lim U(p) to n!1 ord ord 1 ord those groups to take ordinary parts (Jr(K) ⊗ Qp=Zp) , SelK (Jr)p and WK (Jr)p . (See Definition 3.3.3 for the precise definition.), which gives rise to • (J (K) ⊗ = )ord := lim (J (K) ⊗ = )ord 1 Qp Zp −! r Qp Zp r • Sel (J )ord := lim Sel (J )ord K 1 p −! K r p r • 1 (J )ord := lim 1 (J )ord WK 1 p −! WK r p r via Picard functoriality.
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