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Potential automorphy over CM fields. (English) Zbl 1521.11034

In this so-called “ten author paper”, the authors prove that the Sato-Tate conjecture holds for any elliptic curve over a CM field.
Let us give some background. F. Calegari and D. Geraghty [Invent. Math. 211, No. 1, 297–433 (2018; Zbl 1476.11078)] had an insight for how to extend the Taylor-Wiles method for modularity lifting to a very general setting. This depends on two conjectures about torsion classes in the cohomology of locally symmetric spaces. The first conjecture was that the Galois representations constructed by Scholze satisfy a strong form of local-global compatibility at all primes. The second was a vanishing conjecture for the mod-\(p\) cohomology of arithmetic groups localized at non-Eisenstein primes that mirrored the corresponding vanishing theorems for cohomology corresponding to tempered automorphic representations in characteristic zero.
In the present paper, the authors implement the Calegari-Geraghty method unconditionally. They prove the first unconditional modularity lifting theorems for \(n\)-dimensional regular Galois representations without any self-duality conditions. In fact, they prove many cases of the first of these conjectures. Their arguments crucially exploit work of Caraiani and Scholze on the cohomology of non-compact Shimura varieties. On the other hand, they do not resolve the second conjecture concerning the vanishing of mod-\(p\) cohomology. Instead, they sidestep this difficulty by a new technical innovation: a derived version of Ihara avoidance. In particular, they prove the local-global compatibility for \(l \neq p\) or \(l=p\) in both the ordinary and Fontaine-Laffaille case. They obtain quite general modularity lifting theorems in both the ordinary and Fontaine-Laffaille case for general \(n\)-dimensional representations over CM fields. As an application, they show that if \(E\) is an elliptic curve over a CM number field \(F\), then \(E\) and all the symmetric powers of \(E\) are potentially modular. Consequently, the Sato-Tate conjecture holds for \(E\). As an application of a different sort, they also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for \(\mathrm{GL}_2(\mathbb{A}_F)\).
Reviewer: Lei Yang (Beijing)

MSC:

11F80 Galois representations
11F55 Other groups and their modular and automorphic forms (several variables)
11G18 Arithmetic aspects of modular and Shimura varieties
14G05 Rational points
14G35 Modular and Shimura varieties

Citations:

Zbl 1476.11078
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References:

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