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Potential automorphy over CM fields

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Allen, PB 
Calegari, F 
Caraiani, A 
Gee, T 
Helm, D 


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 poten- tially 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 GL2(AF ).



Galois representations, automorphic forms

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Annals of Mathematics

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Annals of Mathematics
European Research Council (714405)
P.A. was supported in part by Simons Foundation Collaboration Grant for Mathemati- cians 527275, NSF Grant DMS-1902155, and by NSERC. F.C. was supported in part by NSF Grants DMS-1701703 and DMS-2001097. A.C. was supported in part by NSF Grant DMS-1501064, by a Royal Society University Research Fellowship, by ERC Starting Grant 804176 and by a Leverhulme Prize. T.G. was supported in part by a Leverhulme Prize, EPSRC grant EP/L025485/1, ERC Starting Grant 306326, and a Royal Society Wolfson Research Merit Award. B.L. was supported in part by NSF Grant DMS-1802037, NSF Grant DMS-1952678 and the Alfred P. Sloan Foundation. J.N. was supported by a UKRI Future Leaders Fellowship, grant MR/V021931/1. P.S. was supported in part by a DFG Leibniz Grant, and by the DFG under the Excellence Strategy EXC-2047/1-390685813. R.T. was supported by NSF Grant DMS-1902265 during the revision of this paper. J.T. was supported by a Clay Research Fellowship and ERC Starting Grant 714405.