K-varieties and Galois representations

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 $\mathrm{\ell }$-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.