/sci/ - Science & Math

https://arxiv.org/pdf/1812.09999.pdf
>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
>(Submitted on 25 Dec 2018)

>Let [math]F[/math] be a CM number field. We prove modularity lifting theorems for regular [math]n[/math]-dimensional Galois representations over [math]F[/math] without any self-duality condition. We deduce that all elliptic curves [math]E[/math] over [math]F[/math] 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 [math]GL_2(\mathbb{A}_F)[/math].