Abelian Surfaces over totally real fields are Potentially Modular

TL;DR

This paper proves that abelian surfaces over totally real fields are potentially modular, leading to important consequences for their zeta functions and extending modularity results to certain genus 2 curves.

Contribution

It establishes potential modularity for abelian surfaces over totally real fields, a significant advancement in understanding their arithmetic properties.

Findings

Potential modularity of abelian surfaces over totally real fields.

Meromorphic continuation and functional equations of Hasse--Weil zeta functions.

Modularity results for genus one curves over quadratic extensions.

Abstract

We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse--Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces A over Q with End_C(A)=Z. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.