Abelian varieties with quaternion and complex multiplication

TL;DR

This paper investigates abelian varieties with both quaternion and complex multiplication, providing explicit descriptions and degree estimates for isogenies to smaller-dimensional abelian varieties.

Contribution

It offers a detailed analysis of abelian varieties with combined quaternion and complex multiplication, including explicit constructions and isogeny degree bounds.

Findings

Explicit descriptions of abelian varieties with quaternion and complex multiplication

Estimates on the degree of isogenies to smaller-dimensional abelian varieties

Characterization of these varieties within the moduli space

Abstract

In this paper we study abelian varieties which correspond to CM points in the coarse moduli space of principally polarized abelian varieties with multiplication by a maximal order in a quaternion algebra over a totally real number field. These are abelian varieties of even dimension with quaternion and complex multiplication. We describe them explicitly via isogenies to products of abelian varieties of smaller dimension together with estimates on the degree of the isogeny.