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

TL;DR

This paper establishes a link between mod p reducibility of elliptic curves over Q and the triviality of their p-primary Shafarevich-Tate groups, with implications for abelian subvarieties and non-prime conductors.

Contribution

It proves that mod p reducibility implies the triviality of the p-primary Shafarevich-Tate group for optimal elliptic curves over Q with prime conductor, extending to more general abelian varieties.

Findings

If the mod p representation is reducible, then the p-primary Shafarevich-Tate group is trivial.

The result applies to optimal elliptic curves with prime conductor and extends to certain abelian subvarieties.

Expectations are discussed for cases where the conductor N is not prime.

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.