Title: Elements of order p in Tate-Shafarevich groups
Abstract:
I'll discuss a result with Ariel Weiss, where we show that for
every prime p there exist (absolutely simple) abelian varieties over Q
with an element of order p in their Tate-Shafarevich groups. The abelian
varieties are related to the Jacobian of the modular curve X_0(q),
where q is a prime which is 1 mod p. These results are inexplicit, in
the sense that we don't write the examples down, we only prove that
they exist. If there is time, I'll also say a bit about work in
progress on finding explicit examples (again for every p) among
Jacobians of superelliptic curves.