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

TL;DR

This paper constructs explicit elliptic curves over function fields with large Tate-Shafarevich groups, demonstrating their size grows with the height, and verifies the BSD conjecture for these curves.

Contribution

It provides explicit examples of elliptic curves over $_q(t)$ with large Tate-Shafarevich groups and proves the BSD conjecture for these cases.

Findings

Tate-Shafarevich groups grow as the height increases.

The p-primary part of the Tate-Shafarevich group is trivial.

Explicit computation of L-functions confirms BSD for these curves.

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.