Abelian Surfaces over totally real fields are Potentially Modular

2018/12/21 by Boxer, George, Calegari, Frank, Gee, Toby +1 · 3 citations
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1812.09269

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.

Cited by

• Potential Automorphy of K3 Surfaces with Large Picard Rank
• Modularity theorems for abelian surfaces
• The residual monodromy for the Dwork family in even characteristic and its applications to Galois representations

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