Wiles defect for Hecke algebras that are not complete intersections

TL;DR

This paper investigates the failure of Wiles' numerical criterion for certain Hecke algebras associated with Shimura curves, especially when these algebras are not complete intersections, by introducing the concept of Wiles defect.

Contribution

It introduces the Wiles defect as a measure of failure of the numerical criterion in non-complete intersection cases and computes it explicitly using local Galois representation data.

Findings

Hecke algebras can fail to be Gorenstein when not complete intersections.

The Wiles defect quantifies the deviation from the numerical criterion.

The defect is computed in terms of local behavior at primes dividing the discriminant.

Abstract

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings $R\to T$ to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod $p$ Galois representation at non-minimal level were isomorphic and complete intersections, provided the same was true at minimal level. In this paper we study Hecke algebras acting on cohomology of Shimura curves arising from maximal orders in indefinite quaternion algebras over the rationals localized at a semistable irreducible mod $p$ Galois representation $\overline \rho$. If $\overline \rho$ is scalar at some primes dividing the discriminant of the quaternion algebra, then Hecke algebra is still isomorphic to the deformation ring, but is not a complete intersection, or even Gorenstein, so the Wiles numerical criterion cannot apply. We consider a weight 2 newform $f$ which contributes to the cohomology of the Shimura curve and gives rise to an augmentation $\lambda_f$ of the Hecke algebra. We quantify the failure of the Wiles numerical criterion at $\lambda_f$ by computing the associated {\it Wiles defect} purely in terms of the local behavior at primes dividing the discriminant of the global Galois representation $\rho_f$ which $f$ gives rise to by the Eichler--Shimura construction. One of the main tools used in the proof is Taylor--Wiles--Kisin patching.