Potential automorphy over CM fields
Patrick B. Allen, Frank Calegari, Ana Caraiani, Toby Gee, David Helm, Bao 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 $\mathrm{GL}_2(\mathbb{A}_F)$.
Topics & Concepts
MathematicsConjectureGalois moduleModular formAutomorphic formRamanujan's sumElliptic curvePure mathematicsZero (linguistics)Duality (order theory)Field (mathematics)Algebraic number fieldModularity (biology)sortDiscrete mathematicsArithmeticGeneticsLinguisticsBiologyPhilosophyAdvanced Algebra and GeometryAlgebraic Geometry and Number TheoryAnalytic Number Theory Research