Elliptic curves with large Tate-Shafarevich groups over $\mathbb{F}_q(t)$

Abstract

Let F q be a finite field of odd characteristic p. We exhibit elliptic curves over the rational function field K = F 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 X(E) is finite and satisfies |X(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.