Elliptic curves with large Tate-Shafarevich groups over $\mathbb F_q(t)$

Abstract: Tate-Shafarevich groups are important arithmetic invariants of elliptic curves, which remain quite mysterious: for instance, it is conjectured that they are finite, but this is only known in a limited number of cases. Assuming finiteness of $\operatorname{Sha}(E)$, work of Goldfeld and Szpiro provides upper bounds on $\#\operatorname{Sha}(E)$ in terms of the conductor or the height of $E$. I will talk about a recent work (joint with Guus de Wit) where we investigate whether these upper bounds are optimal, in the setting of elliptic curves over $\mathbb F_q(t)$. More specifically, we construct an explicit family of elliptic curves over $\mathbb F_q(t)$ which have ``large'' Tate-Shafarevich groups. In this family, $\operatorname{Sha}(E)$ is indeed essentially as large as it possibly can, according to the above mentioned bounds. In contrast with similar results for elliptic curves over $\mathbb Q$, our result is unconditional. We also provide additional information about the structure of the Tate-Shafarevich groups under study. The proof combines various interesting intermediate results, including an explicit expression for the relevant $L$-functions, a detailed study of the distribution of their zeros, and the proof of the BSD conjecture for the elliptic curves in the sequence.

algebraic geometrynumber theory

Audience: researchers in the topic

MAGIC (Michigan - Arithmetic Geometry Initiative - Columbia)

Series comments: Description: Research seminar in arithmetic geometry

(Zoom password = order of the alternating group on six letters)