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Potential automorphy over CM fields

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arxiv 1812.09999 v2 pith:XOZGXW3S submitted 2018-12-25 math.NT

Patrick B. Allen , Frank Calegari , Ana Caraiani , Toby Gee , David Helm , Bao V. Le Hung , James Newton , Peter Scholze
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Richard Taylor Jack A. Thorne
This is my paper · ORCID
classification math.NT
keywords conjectureproverepresentationsapplicationautomorphicautomorphyconditioncurves
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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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  1. On the generic part of the cohomology of non-compact unitary Shimura varieties

    math.NT 2019-09 accept novelty 8.0 of 10

    Generic mod ℓ cohomology of non-compact unitary Shimura varieties is concentrated above the middle degree, and fibers of the compactified Hodge-Tate period map are compactified Igusa varieties.

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