Virtual Abelian varieties of $\mathrm{GL}_2$-type

TL;DR

This paper investigates Abelian varieties of $ ext{GL}_2$-type over number fields, classifying their isogeny classes, describing their Galois conjugates, and analyzing the geometry of associated moduli spaces, including explicit examples.

Contribution

It provides a classification of $ ext{GL}_2$-type Abelian varieties over number fields, links them to Shimura varieties, and explicitly determines the geometry of certain moduli spaces.

Findings

Identification of Abelian varieties via Galois conjugates and class group actions

Parametrization of these varieties by quotients of PEL Shimura varieties

Explicit determination of when moduli spaces are surfaces of general type or rational surfaces

Abstract

This paper studies a class of Abelian varieties that are of $\GL_2$-type and with isogenous classes defined over a number field $k$. We treat the cases when their endomorphism algebras are either (1) a totally real field $K$ or (2) a totally indefinite quaternion algebra over a totally real field $K$. Among the isogenous class of such an Abelian variety, we identify one whose Galois conjugates can be described in terms of actions of Atkin-Lehner operators and the class group of $K$. Thus we deduce that such Abelian varieties are parametrised by finite quotients of certain PEL Shimura varieties. These new families of moduli spaces are further analysed when they are of dimension $2$. We provide explicit numerical bounds for when they are surfaces of general type. In addition, for two particular examples, we show that they are both rational surfaces by computing the coordinates of inequivalent elliptic points and studying the intersections of Hirzebruch cycles with exceptional divisors.