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arXiv · 1812.09269

Abelian Surfaces over totally real fields are Potentially Modular

George Boxer·Frank Calegari·Toby Gee·Vincent Pilloni
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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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George Boxer, Frank Calegari, Toby Gee, Vincent Pilloni. 2018-12-21. Abelian Surfaces over totally real fields are Potentially Modular. https://arxiv.org/abs/1812.09269

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