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

Amod Agashe

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

Number Theory 2009-05-27 v1 Algebraic Geometry

Authors: Amod Agashe

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.

Keywords

elliptic curves mod p reducibility torsion subgroup Shafarevich-Tate group optimal curves prime conductor

Cite

@article{arxiv.0905.4217,
  title  = {Mod-p reducibility, the torsion subgroup, and the Shafarevich-Tate group},
  author = {Amod Agashe},
  journal= {arXiv preprint arXiv:0905.4217},
  year   = {2009}
}

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