Abelian varieties genuinely of ${\mathrm{GL}}_n$-type

Abstract

A simple abelian variety $A$ defined over a number field $k$ is called of ${\mathrm{GL}}_n$-type if there exists a number field of degree $2\dim (A)/n$ which is a subalgebra of ${\mathrm{End}}^0(A)$. We say that $A$ is genuinely of ${\mathrm{GL}}_n$-type if its base change $A_{\overline{k}}$ contains no isogeny factor of ${\mathrm{GL}}_m$-type for $m<n$. This generalizes the classical notion of abelian variety of ${\mathrm{GL}}_2$-type without potential complex multiplication introduced by Ribet. We develop a theory of building blocks, inner twists and nebentypes for these varieties. When the center of ${\mathrm{End}}^0(A)$ is totally real, Chi, Banaszak, Gajda, and Krasoń have attached to $A$ a compatible system of Galois representations of degree $n$ which is either symplectic or orthogonal. We extend their results under the weaker assumption that the center of ${\mathrm{End}}^0(A_{\overline{k}})$ be totally real. We conclude the article by showing an explicit family of abelian fourfolds genuinely of ${\mathrm{GL}}_4$-type. This involves the construction of a family of genus 2 curves defined over a quadratic field whose Jacobian has trivial endomorphism ring and is isogenous to its Galois conjugate.

Strengthened the criterion in Section 9, extending it to arbitrary dimension. In addition, several arguments throughout the paper have been clarified and simplified following the referee's suggestions