Amod Agashe

Let $J$ be an abelian variety and $A$ be any sub-abelian variety of $J$, both defined over $\Q$. An element of the Tate-Shafarevich group of $A$ is said to be visible in $J$ if the corresponding torsor is isomorphic over $\Q$ to a subvariety of $J$. This concept was introduced by Mazur in the context of optimal modular elliptic curves. We show, based on calculations, assuming the Birch-Swinnerton-Dyer conjecture, that for certain primes $p$, there are elements of the Tate-Shafarevich group of certain sub-abelian varieties of $J_0(p)$ and $J_1(p)$ that are not visible.