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

TL;DR

This paper generalizes the concept of abelian varieties of $ ext{GL}_2$-type to those genuinely of $ ext{GL}_n$-type, developing a theory of their building blocks, inner twists, and Galois representations, with explicit examples for $ ext{GL}_4$-type.

Contribution

It introduces the notion of genuinely $ ext{GL}_n$-type abelian varieties, extending previous concepts, and constructs explicit examples of such fourfolds.

Findings

Developed a theory of building blocks, inner twists, and nebentypes for these varieties.

Extended results on Galois representations to cases with totally real centers of endomorphism algebras.

Constructed explicit examples of abelian fourfolds genuinely of $ ext{GL}_4$-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 Kraso\'n 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.