Richard Griffon, August 13, 2020
Title : Elliptic curves with large Tate-Shafarevich groups over 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 Sha(E), work of Goldfeld and Szpiro provides upper bounds on #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 F_q(t). More specifically, we construct an explicit family of elliptic curves over F_q(t) which have ``large'' Tate-Shafarevich groups. In this family, 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 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.