On abelian varieties whose torsion is not self-dual

TL;DR

This paper constructs infinitely many abelian surfaces over rationals with specific non-self-dual torsion properties for small primes, analyzing Galois actions on Tate modules.

Contribution

It provides explicit examples of abelian surfaces with non-self-dual torsion for primes up to 7, using Galois module analysis.

Findings

Existence of infinitely many such abelian surfaces.

Explicit Galois action analysis on Tate modules.

Identification of non-isomorphic torsion subgroups for primes ≤ 7.

Abstract

We construct infinitely many abelian surfaces A defined over the rational numbers such that, for a prime ell <= 7, the ell-torsion subgroup of A is not isomorphic as a Galois module to the ell-torsion subgroup of its dual. We do this by explicitly analyzing the action of the Galois group on the ell-adic Tate module and its reduction modulo ell.