Mod-p Reducibility, the Torsion Subgroup, and the Shafarevich-Tate Group

Let N be a prime and let A be an abelian subvariety of J₀(N) that is stable under the action of the Hecke algebra T. We show that if m is a maximal ideal of T with odd residue characteristic such that A[m] is reducible as a Gal(Q/Q) representation over T/m, then the m torsion subgroup of the Shafarevich-Tate group of A is trivial. In particular, this implies that if E is an optimal elliptic curve over Q of prime conductor such that 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 discuss what one may expect when the level N (or the conductor) is not prime.