Modularity Lifting beyond the Taylor-Wiles Method (1207.4224v2)

Abstract: We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions -- one must be in a Shimura variety setting and the automorphic forms must be of regular weight at infinity. In this paper, we essentially show how to remove these restrictions. Our most general result is a modularity lifting theorem which, on the automorphic side applies to automorphic forms on the group GL(n) over a general number field; it is contingent on a conjecture which, in particular, predicts the existence of Galois representations associated to torsion classes in the cohomology of the associated locally symmetric space. We show that if this conjecture holds, then our main theorem implies the following: if E is an elliptic curve over an arbitrary number field, then E is potentially automorphic and satisfies the Sato--Tate conjecture. In addition, we also prove some unconditional results. For example, in the setting of GL(2) over Q, we identify certain minimal global deformation rings with the Hecke algebras acting on spaces of p-adic Katz modular forms of weight one. Such algebras may well contain p-torsion. Moreover, we also completely solved the problem (for p odd) of determining the multiplicity of an irreducible modular representation rhobar in the Jacobian J_1(N), where N is the minimal level such that rhobar arises in weight two.

Modularity Lifting beyond the Taylor--Wiles Method

This article provides a comprehensive treatment on extending the methodologies for modularity lifting theorems beyond the classic Taylor--Wiles method. Authored by Frank Calegari and David Geraghty, the paper introduces novel modularity lifting theorems applicable to $p$-adic Galois representations, especially in scenarios where traditional Taylor--Wiles techniques are not applicable. This paper removes several key constraints historically associated with the applicability of these methods, particularly the reliance on Shimura variety settings and regular weight restrictions.

Key Contributions

1. General Modularity Lifting Theorems: The authors formulate theorems that apply to automorphic forms on the group $GL(n)$ over arbitrary number fields. This includes a contingent conjecture that predicts the existence of Galois representations linked to torsion classes in the cohomology of associated locally symmetric spaces.
2. Potential Automorphy: If the conjecture holds true, the authors prove that elliptic curves over arbitrary number fields, such as $E$, are potentially automorphic and adhere to the Sato--Tate conjecture.
3. Unconditional Results: Apart from conditional results, some unconditional modularity lifting results are also proven. For example, for $GL(2)$ over $Q$, the paper identifies minimal global deformation rings with Hecke algebras acting on spaces of $p$-adic Katz modular forms of weight 1.
4. Addressing Cohomological Degree Restrictions: The authors successfully remove the need for automorphic forms to contribute solely to a single degree of cohomology. This advancement reflects the broader applicability of their theorems beyond traditional Shimura varieties.

Implications and Speculations

• Practical Implications: By generalizing the modularity lifting conditions, this research enables the paper and verification of broader classes of elliptic curves and Galois representations, opening doors for future work in mathematics concerning automorphy and related topics.
• Theoretical Insights: The divergence from traditional constraints suggests deeper connections in the mathematics of modular forms and Galois representations, potentially leading to an expansive framework unifying various complex structures.
• Future Developments: One might anticipate further developments in AI for automorphic forms computation, leveraging these broader modularity conditions to build algorithms that automate the verification of modularity in larger classes of objects.

Conclusion

Calegari and Geraghty present a visionary extension of modularity theorems, pushing the boundaries of applicability for Taylor--Wiles methods. While certain results remain contingent on conjectures, their paper crafts a pathway for substantial advancement in the interplay between Galois representations and automorphic forms, marking significant progress in the field of number theory.