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Asymptotics of the Tate-Shafarevich Group by Shelly Manber A

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Asymptotics of the Tate-Shafarevich Group by Shelly Manber A Asymptotics of the Tate-Shafarevich Group by Shelly Manber A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Xinyi Yuan, Chair Professor Martin Olsson Professor Lisa Wymore Spring 2018 Asymptotics of the Tate-Shafarevich Group Copyright 2018 by Shelly Manber 1 Abstract Asymptotics of the Tate-Shafarevich Group by Shelly Manber Doctor of Philosophy in Mathematics University of California, Berkeley Professor Xinyi Yuan, Chair For any elliptic curve E with everywhere good or split multiplicative reduction over a finite e (pn) extension of Fq(t) with q = p , p 6= 2, we consider the family of elliptic curves fE gn2Z+ consisting of pn-power frobenius automorphism twists of E. Assuming BSD for function fields, we compute the asymptotics of the order of the Tate-Shafarevich group of E(pn) as n approaches infinity. We prove that there is a twist E(pk) such that for any n greater than k, we can explicitly compute the order of the Tate-Shafarevich group of E(pn) as a power of p (given in terms of n) times the order of the Tate-Shafarevich group of E(pk). In particular, as n approaches infinity, the size of the Tate-Shafarevich group of E(pn) grows on the order of O(qδpn ) for some constant δ determined by E(pk). More precisely, the size of the Tate- Shafarevich group of E(pn) grows as θ(peδpn−n(m+r)) where r is the rank of E(pk) and m is the number of places of bad reduction of E(pk). 1 Chapter 1 Introduction Let K be a global field and let E be an elliptic curve over K. The set of K-points of E, denoted E(K), is a finitely generated abelian group under the group law of E, so E(K) ∼= G × Zr where G is a finite group and r is a nonnegative integer. We call r the rank of E. Much remains unknown about the ranks of elliptic curves. We do not know how, in general, to compute the rank of an elliptic curve. For elliptic curves over Q, Bhargava and Shankar [1] proved that the average rank of elliptic curves ordered by height is bounded above by 1:5, but it remains unknown if there exist elliptic curves of arbitrarily high rank. The Birch and Swinnerton-Dyer Conjecture, proposed in the 1960s, asserts that the rank of an elliptic curve is equal to the order of vanishing at s = 1 of its L-function, and remains unproven to this day, despite being one of the Clay Institute million dollar problems. The n-descent formula for E 0 ! E(K)=nE(K) ! Sel(E)(n) ! X(E)[n] ! 0 gives some hope of understanding the rank of E by understanding the n-Selmer group Sel(E)(n) and the n-torsion of the Tate-Shafarevich group X(E) of E for any positive integer n ≥ 2. This gives some motivation for the topic of this thesis: computing the order of the Tate-Shafarevich group, defined as the kernel of the natural map 1 Y 1 H (K; E) ! H (Kν;E) ν 1 1 sep sep where H (K; E) := H (GK ;E(K )) denotes Galois cohomology, considering E(K ) as a GK -module. There are many results in the literature on the order of the Tate-Shafarevich group. Kloosterman proves in [8] that for every prime p and every positive integer k, there exists a number field K and an elliptic curve E=K such that jX(E)[p]j ≥ pk, i.e. that there exist elliptic curves over number fields of arbitrarily high Tate-Shafarevich group p-torsion. Dabrowski and Szymaszkiewicz [4] conjecture that if E is an elliptic curve over Q then for any positive integer k, the number of quadratic twists E(d) of E for which jX(E(d))j = k2 2 and d < X grows on the order of 3=4 c2 c1X (log(X)) as X approaches infinity, where c1 and c2 are constants dependent on k. Goldfeld and Szpiro [6] prove that, if we assume that X(E) is finite for all elliptic curves over function fields, and if we order the set of elliptic curves over a fixed function field by conductor N, then the average size of X grows on the order of N 1=2+ as N approaches infinity, where > 0 is a fixed constant. Note that, although we talk about the size of X(E), it is not yet proven that X(E) is always finite, though it is conjectured and widely believed to be true. Indeed, the Birch and Swinnerton-Dyer Conjecture, in addition to asserting that the rank of an elliptic curve is equal to the order of vanishing at s = 1 of its L-function, conjectures that the Tate- Shafarveich group X(E) is finite, and conjectures the following relationship between the leading coefficient of the L-function of E and the size of the X(E): 1 (r) jX(E)jR(E)τ(E) L (E=K; 1) = 2 : (1.1) r! jE(K)torsj Here r := ords=1L(E=K; s), R(E) is the regulator of E, and τ(E) is the Tamagawa number of E. Each of these values will be defined in detail later on. The Birch and Swinnerton-Dyer Conjecture has been proved for some families of elliptic curves, including elliptic curves over Q with complex multiplication and isotrivial elliptic curves over function fields. Tate and Milne proved (see [16], Theorem 12.1) that when K is a function field, the rank of E(K) is less than or equal to the order of vanishing of the L-function at s = 1, and that the following are equivalent: 1. rank(E(K)) = ords=1L(E; s) 2. X(E) is finite 3. X(E)`1 - the `-primary part of X(E) - is finite for any one prime `. Furthermore, if any of those equivalent conditions hold, then so does the full formula (1.1). This very powerful result allows us to use the full force of the BSD conjecture with only the assumption that X(E) is finite. In this thesis, we fix a function field K of characteristic p and, assuming that X(E) is finite for all elliptic curves E over K, use formula (1.1) to construct a sequence of elliptic n curves with Tate-Shafarevich group growing at the rate of O(pc1p −c2n) for positive integers n c1 and c2. In particular, for any positive integer n, let φpn : K ! K denote the p -power (pn) Frobenius and E the twist of E by φpn . The precise result is the following. Theorem 1.0.1. Let K be the function field of a smooth, projective, geometrically integral e curve over Fq with q = p , p 6= 2, and let E be an elliptic curve over K with split multiplicative 3 reduction at m places and good reduction everywhere else. Assuming BSD, there exists a positive integer k such that for all positive integers n, (n+k) n k jX(E(p ))j = peδ(p −1)−n(m+r)jX(E(p ))j where r is the algebraic rank of E(pk) and 1 X δ := c deg(ν); 12 ν ν2Mbad (pk) where cν is the local Tamagawa number at ν of E and Mbad the set of places at which E has bad reduction. This theorem tells us that after a certain point, the size of the Tate-Shafarevich group grows on the order of magnitude of O(peδpn ). The theorem follows from a result of Ghioca [5], which says that E(Kper) is finitely gener- ated, where Kper := [K1=pn . Since the Frobenius induces a bijection between E(K1=pn ) and (pn) (pn) (pn) (pn) E (K) for all n, this result allows us to compute how R(E ); τ(E ); and jE (K)torsj behave as a function of n. Finally, we note that E and E(pn) are isogenous for all n and thus have the same L-function, from which we can obtain our main result. The paper is organized as follows. In the background section, we define the regulator R(E), the Tamagawa number τ(E), and the L-function L(E; s), giving all necessary back- ground to do so. Next, we compute R(E(pn)) as a function of n, then we compute τ(E(pn)) as a function of n, and finally, we put together the pieces to prove Theorem 1.0.1. 4 Chapter 2 Background 2.1 Height and Regulator The goal of this section is to define the regulator R(E) of an elliptic curve E. To do this, we will begin with the definition of the Néron-Tate height function. We follow the exposition of [2], [3], and [13]. Let K be the function field of a smooth, projective, geometrically integral curve C over e Fq with q = p and let MK be the set of places of K, i.e., the set of discrete valuations on K. There is a bijective correspondence between the set of places of K and the set of closed points of C such that the discrete valuation ring OKν associated to ν 2 MK is the local ring of the corresponding point of C. For each valuation ν 2 MK , we define an absolute value on −ν(x) K by jxjν := q for all x 2 K.
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