Congruence modules and the Wiles-Lenstra-Diamond numerical criterion in higher codimensions

• 成果类型: Article
• 署名作者: Iyengar, Srikanth B.; Khare, Chandrashekhar B.; Manning, Jeffrey
• 署名单位: Utah System of Higher Education; University of Utah; University of California System; University of California Los Angeles
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01292-y
• 发表日期: 2024
• 页码: 769-864
• 关键词: breuil-mezard conjecture; forms; rings; automorphy; ALGEBRAS; CURVES

摘要

We define a congruence module Psi(A)(M) associated to a surjective O\-algebra morphism lambda:A -> O, with O a discrete valuation ring, A a complete noetherian local O-algebra regular at p, the kernel of lambda, and M a finitely generated A-module. We establish a numerical criterion for M to have a free direct summand over A of positive rank. It is in terms of the lengths of Psi(A)(M) and the torsion part of p/p(2). It generalizes results of Wiles, Lenstra, and Diamond, that deal with the case when the codimension of p is zero. Number theoretic applications include integral (non-minimal) R=T theorems in situations of positive defect conditional on certain standard conjectures. Here R is a deformation ring parametrizing certain Galois representations and T is a Hecke algebra. An example of a positive defect situation is that of proving modularity lifting for 2-dimensional l-adic Galois representations over an imaginary quadratic field. The proofs combine our commutative algebra results with a generalization due to Calegari and Geraghty of the patching method of Wiles and Taylor-Wiles and level raising arguments that go back to Ribet. The results provide new evidence in favor of the intriguing, and as yet fledgling, torsion analog of the classical Langlands correspondence. We also prove unconditional integral R=T results for Hecke algebras T acting on weight one cohomology of Shimura curves over Q. This leads to a torsion Jacquet-Langlands correspondence comparing integral Hecke algebras acting on weight one cohomology of Shimura curves and modular curves. In this case the cohomology has abundant torsion and so our correspondence cannot be deduced by means of the classical Jacquet-Langlands correspondence.