Genus two curves with quaternionic multiplication and modular jacobian

TL;DR

This paper presents a method to classify principal polarizations of modular abelian surfaces with quaternionic multiplication and provides an example of such a surface with an explicit curve whose Jacobian matches it.

Contribution

It introduces a new method for classifying principal polarizations of quaternionic modular abelian surfaces and demonstrates it with a concrete example.

Findings

Classified all isomorphism classes of principal polarizations for certain modular abelian surfaces.

Constructed an explicit example of a curve with Jacobian isogenous to a quaternionic multiplication surface.

Proved the Jacobian of the constructed curve has quaternionic multiplication.

Abstract

We describe a method to determine all the isomorphism classes of principal polarizations of the modular abelian surfaces $A_f$ with quaternionic multiplication attached to a normalized newform $f$ without complex multiplication. We include an example of $A_f$ with quaternionic multiplication for which we find numerically a curve $C$ whose Jacobian is $A_f$ up to numerical approximation, and we prove that it has quaternionic multiplication and is isogenous to $A_f$.