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arXiv · 1812.09999

Potential automorphy over CM fields

Patrick B. Allen·Frank Calegari·Ana Caraiani·Toby Gee·David Helm·Bao V. Le Hung·James Newton·Peter Scholze·Richard Taylor·Jack A. Thorne
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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 $\mathrm{GL}_2(\mathbf{A}_F)$.

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Patrick B. Allen, Frank Calegari, Ana Caraiani, Toby Gee, David Helm, Bao V. Le Hung, James Newton, Peter Scholze, Richard Taylor, Jack A. Thorne. 2018-12-25. Potential automorphy over CM fields. https://arxiv.org/abs/1812.09999

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