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arXiv · 2108.09729

Wiles defect of Hecke algebras via local-global arguments

Gebhard Boeckle·Chandrashekhar B. Khare·Jeffrey Manning
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Abstract

We continue our study of the Wiles defect of deformation rings $R$ and Hecke rings $T$ (at a newform $f$) acting on the cohomology of Shimura curves. The Wiles defect at an augmentation $λ_f:T \to O$ measures the failure of $R,T$ to be complete intersections locally at $λ_f$. In situations we study here the Taylor-Wiles-Kisin patching method gives an isomorphism $R=T$ without the rings being complete intersections. Using novel arguments in commutative algebra and patching, we generalize significantly and give different proofs of our earlier results that compute the Wiles defect at $λ_f: R=T \to O$, and explain in an a priori manner why the answer is a sum of local defects. As a curious application of our work we give a new and more robust approach to the result of Ribet--Takahashi that computes change of degrees of optimal parametrizations of elliptic curves by Shimura curves as we vary the Shimura curve.

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Gebhard Boeckle, Chandrashekhar B. Khare, Jeffrey Manning. 2021-08-24. Wiles defect of Hecke algebras via local-global arguments. https://arxiv.org/abs/2108.09729

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