Abelian Surfaces over totally real fields are Potentially Modular
Number Theory
2021-11-30 v3
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.
Cite
@article{arxiv.1812.09269,
title = {Abelian Surfaces over totally real fields are Potentially Modular},
author = {George Boxer and Frank Calegari and Toby Gee and Vincent Pilloni},
journal= {arXiv preprint arXiv:1812.09269},
year = {2021}
}
Comments
Final version (fixing minor typos found in copyediting). 292 pages, to appear in Publ. Math. de l'IHES