Potential automorphy over CM fields

Pages 897-1113 from Volume 197 (2023), Issue 3 by Patrick B. Allen, Frank Calegari, Ana Caraiani, Toby Gee, David Helm, Bao V. Le Hung, James Newton, Peter Scholze, Richard Taylor, Jack A. Thorne

Abstract

Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and furthermore satisfy the Sato–Tate conjecture. As an application of a different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for $\mathrm {GL}_2(\mathbf {A}_F)$.

Keywords
,
Mathematical Subject Classification

Primary 2000: , , ,

DOI
https://doi.org/10.4007/annals.2023.197.3.2
MR
4564261
zbMATH
7668527
Milestones
Received: 24 December 2018
Revised: 14 June 2022
Accepted: 14 June 2022
Published online: 23 March 2023

Authors

Patrick B. Allen

McGill University, Montreal, Canada

Frank Calegari

University of Chicago, Chicago, IL, USA

Ana Caraiani

Mathematisches Institüt der Universität de Bonn, Bonn, Germany and Department of Mathematics, Imperial College London, London, UK

Toby Gee

Imperial College London, London, UK

David Helm

Imperial College London, London, UK

Bao V. Le Hung

Northwestern University, Evanston, IL, USA

James Newton

Mathematical Institute, University of Oxford, Oxford, UK

Peter Scholze

Mathematisches Institüt der Universität de Bonn, Bonn, Germany

Richard Taylor

Stanford University, Stanford CA, USA

Jack A. Thorne

Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK