Galois action on the $\bar\mathbb{Q}$-isogeny classes of abelian surfaces with quaternionic multiplication

TL;DR

This paper constructs projective Galois representations for abelian surfaces with quaternionic multiplication, extending classical representations from elliptic curves to higher dimensions, and analyzes their action on isogeny classes.

Contribution

It introduces new projective Galois representations for abelian surfaces with quaternionic multiplication, generalizing known representations from elliptic curves.

Findings

Constructed projective Galois representations for abelian L-surfaces with quaternionic multiplication.

Proved these representations describe the Galois action on isogeny classes.

Extended classical Galois representation concepts from elliptic curves to abelian surfaces.

Abstract

An abelian variety over a number field is called L-abelian variety if, for any element of the absolute Galois group of a number field L, the conjugated abelian variety is isogenous to the given one by means of an isogeny that preserves the Galois action on the endomorphism rings. We can think of them as generalizations of abelian varieties defined over L with endomorphisms also defined over L. In the one dimensional case, an elliptic curve defined over L gives rise to a Galois representation provided by the Galois action on its Tate module. This classical Galois representation has been a central object of study in Number Theory over the last decades. Besides, given an elliptic L-curve one can construct a projective analogue of the previous Galois representation. In this work we construct similar projective representations in the two-dimensional case, namely, attached to abelian L-surfaces with quaternionic multiplication or fake elliptic curves. Moreover, we prove that such projective representations also describe the Galois action on the isogeny class of the elliptic L-curve, in the classical case, or the abelian L-surface with quaternionic multiplication, in the general case.