Potential automorphy over CM fields (1812.09999v2)

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 $\mathrm{GL}_2(\mathbf{A}_F)$.

Potential Automorphy Over CM Fields: A Detailed Overview

This essay provides an expert summary of the paper "Potential Automorphy Over CM Fields," which targets the profound theorems and conjectures related to modularity and automorphy of Galois representations. The paper explores the intricate relationship between arithmetic geometry and number theory, focusing on regular $n$-dimensional Galois representations over CM fields and their automorphic properties, without assuming any self-duality conditions.

Core Contributions

The paper establishes the first unconditional modularity lifting theorems for regular $n$-dimensional Galois representations over a CM field $F$. This advances prior results which were conditional on conjectures concerning local-global compatibilities and the vanishing of mod-$p$ cohomology for arithmetic groups. To bypass these conjectures, the authors introduce a derived version of "Ihara avoidance," expanding classical Ihara techniques to broader applications and remove dependence on vanishing results.

Technical Results and Claims

1. Modularity Lifting Theorems: Key results include theorems for regular $n$-dimensional representations proving potential modularity, and thereby confirming the Sato-Tate conjecture for elliptic curves over CM fields. This creates foundational groundwork for correlating representations and automorphy in scenarios devoid of self-duality.
2. Ramanujan Conjecture: The paper extends results to the Ramanujan conjecture, proving that weight zero cuspidal automorphic representations for $GL_2(A_F)$ are tempered, addressing situations where associated Galois representations are not present in the cohomology of algebraic varieties.
3. Local-Global Compatibility: For $n$-dimensional representations, the paper navigates through challenging conditions of local-global compatibility using innovative techniques and significant extensions of Caraiani-Scholze's work on Shimura varieties. The modularity proofs leverage these deepenings using base-change arguments and properties derived from Shimura varieties' boundary.

Implications and Future Developments

The implications of proving the modularity of representations are vast, impacting number theory's conjectures such as the Birch and Swinnerton-Dyer conjecture and general Langlands program. The modularity lifting theorems potentially influence constructions around elliptic curves' arithmetic properties, allowing deeper explorations into their congruence relations.

Concerning theoretical developments, these results encourage further research into the automorphy of higher symmetric powers and the exploration of representations where local-global compatibility might extend beyond prime-specific characteristics. The insights could eventually aid in formulating equivalences between Galois representations and automorphic forms for broader classes of $L$-functions.

Conclusion

The paper "Potential Automorphy Over CM Fields" is a critical advancement in arithmetic geometry, presenting robust solutions to modularity lifting problems. Its approach reframes the technical landscape on foundational number theory conjectures, potentially opening new avenues in complex shapes of understanding within the Langlands program. Further exploration into the consequences of these results may find enhanced links between elliptic curves and Galois representations, impacting both theoretical and computational paradigms within mathematics.