Parametrization of abelian $K$-surfaces with quaternionic multiplication

TL;DR

This paper establishes a parametrization of abelian K-surfaces with quaternionic multiplication using K-rational points on specific Shimura curves mod Atkin-Lehner involutions, linking algebraic geometry and number theory.

Contribution

It provides a new explicit description of abelian K-surfaces with quaternionic multiplication via Shimura curves and Atkin-Lehner involutions.

Findings

Parametrization of abelian K-surfaces with quaternionic multiplication

Connection between these surfaces and Shimura curves

Use of Atkin-Lehner involutions in the parametrization

Abstract

We prove that the abelian $K$-surfaces whose endomorphism algebra is a quaternion algebra are parametrized, up to isogeny, by the $K$-rational points of the quotient of certain Shimura curves by the group of their Atkin-Lehner involutions.