Number Theory Seminar
Speaker: Richard Griffon (Basel).
Title: Elliptic curves with large Tate-Shafarevich groups over function fields.
Abstract: Tate-Shafarevich groups of elliptic curves are arithmetic objects which remain mysterious: they are conjectured to be finite but the conjecture is not known in general. Even assuming finiteness of Sha(E), the size of this group is only partially understood: some upper bounds on $\#Sha(E)$ in terms of the height of E are known, and some heuristics suggest that the group $Sha(E)$ should often be ''small''.
I will report on a recent work with Guus de Wit, where we exhibit a family of elliptic curves (over the function field F_q(t)) with ''large'' Tate-Shafarevich groups. For these curves, $\#Sha(E)$ is essentially as large as it is possibly allowed to be by the above-mentioned bounds. Our result is unconditional and quite explicit, it also provides additional information about the structure of the Tate-Shafarevich groups under study. We use various techniques, including the computation of the relevant L-functions, a detailed study of the distribution of their zeros, and the proof of the BSD conjecture for these curves.