Abelian surfaces with an automorphism and quaternionic multiplication

TL;DR

This paper constructs special families of Abelian surfaces with quaternionic multiplication and automorphisms, identifying their moduli spaces and relating them to Jacobians of genus two curves and real multiplication surfaces.

Contribution

It explicitly constructs families of Abelian surfaces with automorphisms and quaternionic multiplication, and describes their moduli spaces and relations to known curves.

Findings

Identified the Shimura curve parametrizing these Abelian surfaces.

Related these surfaces to Jacobians of genus two curves by Hashimoto and Murabayashi.

Described a Humbert surface parametrizing Abelian surfaces with real multiplication.

Abstract

We construct one dimensional families of Abelian surfaces with quaternionic multiplication which also have an automorphism of order three or four. Using Barth's description of the moduli space of (2,4)-polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space which parametrizes Abelian surfaces with real multiplication by Z[\sqrt{2}].