Wiles defect of Hecke algebras via local-global arguments

TL;DR

This paper investigates the Wiles defect of deformation and Hecke rings associated with modular forms, using advanced algebraic and patching techniques to understand their structure and implications for elliptic curve parametrizations.

Contribution

It generalizes previous results on the Wiles defect, providing new proofs and a priori explanations for its local decomposition, and introduces novel algebraic methods in the context of Shimura curves.

Findings

Computed the Wiles defect at specific augmentation maps.

Established that the Taylor-Wiles-Kisin patching method can yield isomorphisms without rings being complete intersections.

Provided a new approach to the change of degrees in elliptic curve parametrizations.

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 $\lambda_f:T \to O$ measures the failure of $R,T$ to be complete intersections locally at $\lambda_f$. 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 $\lambda_f: R=T \to O$, 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.