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Selmer Groups in Degree $\Ell$ Twist Families $\ell^{\infty}$-Selmer Groups in Degree $\ell$ Twist Families The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Smith, Alexander. 2020. $\ell^{\infty}$-Selmer Groups in Degree $\ell$ Twist Families. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37365902 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA `1-Selmer groups in degree ` twist families A dissertation presented by Alexander Smith to The Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics Harvard University Cambridge, Massachusetts April 2020 © 2020 – Alexander Smith All rights reserved. Dissertation Advisor: Professor Elkies and Professor Kisin Alexander Smith `1-Selmer groups in degree ` twist families Abstract Suppose E is an elliptic curve over Q with no nontrivial rational 2-torsion point. Given p a nonzero integer d, take Ed to be the quadratic twist of E coming from the field Q( d). For every nonnegative integer r, we will determine the natural density of d such that Ed has 2-Selmer rank r. We will also give a generalization of this result to abelian varieties defined over number fields. These results fit into the following general framework: take ` to be a rational prime, take F to be number field, take ζ to be a primitive `th root of unity, and take N to be an `- divisible Z`[ζ]-module with an action of the absolute Galois group GF of F . Given a homo- χ morphism χ from GF to hζi, we can define a twist N , and we can define a (1 − ζ)-Selmer group for each of these twists. Under some hypotheses, we determine the distribution of (1 − ζ)-Selmer ranks in this family of degree ` twists. To give a non-geometric example, this framework allows us to determine some aspects of the distribution of `-primary part of the class groups in families of degree ` extensions. One of the main goals of this dissertation is to prove these results in a way that stream- lines the approach to finding the distribution of higher Selmer groups given by the author in [48]. Along the way, we generalize the Cassels-Tate pairing to a certain pairing between Selmer groups defined from finite Galois modules. On the analytic side, we give a general bilinear character sum estimate that is suitable both for the base-case work of this disserta- tion and as a replacement for the Chebotarev density theorem in a future generalization of [48]. iii CONTENTS Acknowledgements vi 1. Introduction 1 1.1. Outline of this work and its connection to ongoing work 11 Part 1. The Cassels-Tate pairing, Theta Groups, and 1-Cocycle Parameterization 19 2. Legendre symbols and spin 19 3. Theta groups 33 3.1. Theta groups of abelian varieties 37 3.2. Involution spin 40 4. The Cassels-Tate pairing 44 4.1. Antisymmetry 55 5. Appendix: Review of Galois cohomology 59 5.1. Review of group cohomology 59 5.2. Local bookkeeping 64 5.3. Poitou-Tate duality 71 Part 2. Bilinear character sums 72 6. Main results for bilinear character sums 72 7. Proof of Theorem 6.3 79 Part 3. The Base Case I: Linear Algebra 82 8. Twistable modules and Selmer groups 82 8.1. Tamagawa ratios 89 8.2. Dual modules and base-case Selmer groups 91 8.3. Tuple sets of twists and moment estimates 95 9. Favorable twists and the main theorems 96 9.1. Main base-case results 103 iv 10. Base-case Selmer conditions 114 10.1. Dual modules 115 10.2. Selmer conditions 120 10.3. Some cancellable pairs 126 10.4. Main-term pairs 129 10.5. The space Q((qs)s) 132 11. Ignorable pairs 134 11.1. Characterizing non-ignorable pairs 135 11.2. Counting non-ignorable pairs 140 11.3. Bounding moments using non-ignorable pairs 144 11.4. Estimating moments using non-ignorable pairs 149 Part 4. The Base Case II: Gridding 153 12. Grids of twists 153 12.1. Grids of ideals 153 12.2. Grids of twists 158 12.3. The Chebotarev Density Theorem 164 13. Bad grids and the proof of Proposition 12.5 167 13.1. Preliminary results 167 13.2. The proof of Proposition 12.5 169 14. Proofs of the base-case theorems 177 14.1. The parity of Selmer ranks in grids 177 14.2. Moment estimates on good grids of twists 180 14.3. The proofs of the base-case Selmer rank theorems 185 References 191 v ACKNOWLEDGEMENTS I would like to thank my advisors, Noam Elkies and Mark Kisin, for their support in my development as a mathematician. I would also like to thank Melanie Matchett Wood, Shou-Wu Zhang, Dorian Goldfeld, and Andrew Granville for their encouragement and guidance from outside the Harvard mathematics department. I want to acknowledge Dimitris Koukoulopoulos, Peter Koymans, Adam Morgan, Jesse Thorner, and David Yang for specific insights that had a major impact on this work. I would like to thank the many people who have had comments on previous versions of this work, including Stephanie Chan, Brian Conrad, Djordjo Milovic, Carlo Pagano, Ye Tian, and Boya Wen. I would like to thank George Boxer, Jordan Ellenberg, Wei Ho, Barry Mazur, Hector Pasten, Karl Rubin, Bjorn Poonen, Peter Sarnak, Frank Thorne, Richard Taylor, Nicholas Triantafillou, Ila Varma, and Xinyi Yuan for helpful discussion over the course of this project. I would also like to thank Morgan, Geoff, Sarah, Greg, and Jill. vi 1. INTRODUCTION Goldfeld’s Conjecture. Given an abelian variety A defined over a number field F , the Mordell-Weil theorem states that the group of F -rational points of A is a finitely-generated abelian group. There is then a well-defined nonnegative integer r called the rank of A over F so that there some isomrphism ∼ r A(F ) = Z ⊕ A(F )tor of abelian groups, where A(F )tor denotes the subgroup of points of finite order in A(F ). There is no proven algorithm for calculating the rank of an abelian variety, and under- standing the behavior of the ranks of abelian varieties, and in particular of elliptic curves, constitutes one of the most fundamental problems in arithmetic geometry. There are two close analogues of rank that are of interest to us: • We define the analytic rank of A=F , denoted ran(A=F ), to be the order of vanishing of the L-function associated to E=F at s = 1. For F = Q and A an elliptic curve, this rank is well-defined; for some higher number fields and higher dimensional abelian varieties, its existence is conjectural. 1 • For any rational prime p, we define the p -Selmer corank of A=F , denoted rp1 (A=F ), to be the limit of the sequence rp(A=F ); rp2 (A=F ); rp3 (A=F );:::; where rn denotes the n-Selmer rank of A=F , a nonnegative integer we will define in greater detail later. Conjecturally, we should always have ran(A=F ) = rank(A=F ) = r21 (A=F ) = r31 (A=F ) = r51 (A=F ) = :::: 1 The equality of rank and analytic rank is the celebrated conjecture of Birch and Swinnerton- Dyer (the BSD conjecture), and the equality of rank and the p1-Selmer coranks would fol- low as a consequence of the conjectured finiteness of the Shafarevich-Tate group X(A=F ) (the Shafarevich-Tate conjecture). For now, we note the following, easily-provable esti- mate; for p a rational prime and k ≥ 1, we always have (1.1) rp(A=F ) ≥ rp2 (A=F ) ≥ · · · ≥ rpk (A=F ) ≥ rp1 (A=F ) ≥ rank(A=F ): In 1979, Goldfeld conjectured the following: Conjecture 1.1 ([13]). Given an elliptic curve E=Q with narrow Weierstrass form y2 = x3 + ax + b; and given a nonzero integer d, take Ed to be the elliptic curve over Q with narrow Weier- strass form Ed : y2 = x3 + d2ax + d3b: p This curve is a quadratic twist of E, and is isomorphic to E over Q( d). Then, for r ≥ 0, 8 > >1=2 for r = 0 d > # d : 0 < jdj ≤ N and ran(E =Q) = r < lim = 1=2 for r = 1 N!1 2N > > :>0 for r ≥ 2: In particular, if both the BSD conjecture and Goldfeld’s conjecture hold, 100% of the curves in the quadratic twist family of a given E=Q will have rank at most one. Between this dissertation and some work in preparation, we will prove an analogue of Goldfeld’s conjecture for 21-Selmer coranks. For technical reasons, we need to place certain restrictions on the elliptic curves we consider. These restrictions vary depending on the structure of the 2-torsion subgroup E(Q)[2] of E(Q). 2 Assumption 1.2. An elliptic curve E=Q obeys the assumptions of Theorem 1.3 if one of the following holds: (1) E(Q)[2] = 0; or ∼ (2) E(Q)[2] = Z=2Z and, writing φ : E ! E0 for the unique isogeny of degree 2 over Q, we have Q(E0[2]) 6= Q and Q(E0[2]) 6= Q(E[2]); or (3) E(Q)[2] ∼= (Z=2Z)2 and E has no cyclic degree 4 isogeny defined over Q. We note that a “generic” elliptic curve over Q satisfies E(Q)tor = 0, fitting it into the first case given above.
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