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

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

2009-05-26

arxiv.org/abs/0905.4217v1

Mathematics

Number Theory

Scientific paper

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.

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