K-varieties and Galois representations

TL;DR

This paper explores attaching Galois representations to abelian varieties with special properties, constructing pairings to control their images, and applying these methods to prove modularity of certain abelian surfaces.

Contribution

It introduces a method to attach irreducible Galois representations to abelian K-varieties and constructs pairings to restrict their images, advancing understanding of their Galois actions.

Findings

Successfully attaches Galois representations to abelian K-varieties.

Constructs Galois-equivariant pairings to control representation images.

Proves modularity of abelian surfaces over Q with potential quaternionic multiplication.

Abstract

In a remarkable article Ribet showed how to attach rational $2$-dimensional representations to elliptic ${\mathbb Q}$-curves. An abelian variety $A$ is a (weak) $K$-variety if it is isogenous to all of its $\text{Gal}_K$-conjugates. In this article we study the problem of attaching an absolutely irreducible $\ell$-adic representation of $\text{Gal}_K$ to an abelian $K$-variety, which sometimes has smaller dimension than expected. When possible, we also construct a Galois-equivariant pairing, which restricts the image of this representation. As an application of our construction, we prove modularity of abelian surfaces over ${\mathbb Q}$ with potential quaternionic multiplication.