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

Abstract

Let ${\mathbb{F}}_q$ be a finite field of odd characteristic $p$. We exhibit elliptic curves over the rational function field $K={\mathbb{F}}_q(t)$ 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 $\mathrm{III}(E)$ is finite and satisfies $|\mathrm{III}(E)|=H(E{)}^{1+o(1)}$ as $H(E)\to \infty$, 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.

22 pages, Comments are very welcome!