Mod-p reducibility, the torsion subgroup, and the Shafarevich-Tate group

Abstract

Let $E$ be an optimal elliptic curve over $\Q$ of prime conductor $N$. We show that if for an odd prime $p$, the mod $p$ representation associated to $E$ is reducible (in particular, if $p$ divides the order of the torsion subgroup of $E(\Q)$), then the $p$-primary component of the Shafarevich-Tate group of $E$ is trivial. We also state a related result for more general abelian subvarieties of $J_0(N)$ and mention what to expect if $N$ is not prime.