# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1812.09269