Abelian Surfaces over totally real fields are potentially modular
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Author(s)
Boxer, George
Calegari, Frank
Gee, Toby
Pilloni, Vincent
Type
Journal Article
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 EndCA = Z. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.
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 EndCA = Z. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.
Date Issued
2021-11-29
Date Acceptance
2021-10-27
Citation
Publications mathématiques de l'IHÉS, 2021, 134, pp.153-501
ISSN
0073-8301
Publisher
Springer
Start Page
153
End Page
501
Journal / Book Title
Publications mathématiques de l'IHÉS
Volume
134
Copyright Statement
© 2021 The Author(s). This article is licensed under a Creative Commons Attribution 4.0 International
License, which permits use, sharing, adaptation, distribution and reproduction in any
medium or format, as long as you give appropriate credit to the original author(s) and
the source, provide a link to the Creative Commons licence, and indicate if changes were
made. The images or other third party material in this article are included in the article’s
Creative Commons licence, unless indicated otherwise in a credit line to the material. If
material is not included in the article’s Creative Commons licence and your intended use
is not permitted by statutory regulation or exceeds the permitted use, you will need to
obtain permission directly from the copyright holder. To view a copy of this licence, visit
http://creativecommons.org/licenses/by/4.0/.
License, which permits use, sharing, adaptation, distribution and reproduction in any
medium or format, as long as you give appropriate credit to the original author(s) and
the source, provide a link to the Creative Commons licence, and indicate if changes were
made. The images or other third party material in this article are included in the article’s
Creative Commons licence, unless indicated otherwise in a credit line to the material. If
material is not included in the article’s Creative Commons licence and your intended use
is not permitted by statutory regulation or exceeds the permitted use, you will need to
obtain permission directly from the copyright holder. To view a copy of this licence, visit
http://creativecommons.org/licenses/by/4.0/.
License URL
Sponsor
Commission of the European Communities
The Leverhulme Trust
Engineering & Physical Science Research Council (EPSRC)
The Royal Society
Identifier
https://link.springer.com/article/10.1007/s10240-021-00128-2
Grant Number
FP7-ERC-StG-2012-306326
LH.PZ.GEE.2012
EP/L025485/1
WM150076
Subjects
0101 Pure Mathematics
General Mathematics
Publication Status
Published
Date Publish Online
2021-11-29