Elliptic curves with large Tate-Shafarevich groups over \mathbbF_q(t)

2019/07/30 by Griffon, Richard, de Wit, Guus
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1907.13038

Abstract

Let \mathbbF_q be a finite field of odd characteristic p. We exhibit elliptic curves over the rational function field K = \mathbbF_q(t) whose Tate-Shafarevich groups are large. More precisely, we consider certain infinite sequences of explicit elliptic curves E, for which we prove that their Tate-Shafarevich group III(E) is finite and satisfies |III(E)| = H(E)^1+o(1) as H(E)→∞, where H(E) denotes the exponential differential height of E. The elliptic curves in these sequences are pairwise neither isogenous nor geometrically isomorphic. We further show that the p-primary part of their Tate-Shafarevich group is trivial. The proof involves explicitly computing the L-functions of these elliptic curves, proving the BSD conjecture for them, and obtaining estimates on the size of the central value of their L-function.

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