Wiles defect for Hecke algebras that are not complete intersections - Böckle, Gebhard; Khare, Chandrashekhar; Manning, Jeffrey (Compositio mathematica, 2021)
In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings R→TR→TR→ T to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod ppp Galois representation at non-minimal level are isomorphic and complete intersections, provided the same is 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 ppp Galois representation ˉρρ¯\bar {ρ }. If ˉρρ¯\bar {ρ } is scalar at some primes dividing the discriminant of the quaternion algebra, then the 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 fff which contributes to the cohomology of the Shimura curve and gives rise to an augmentation λfλfλ _f of the Hecke algebra. We quantify the failure of the Wiles numerical criterion at λfλfλ _f by computing the associated Wiles defect purely in terms of the local behavior at primes dividing the discriminant of the global Galois representation ρfρfρ _f which fff gives rise to by the Eichler-Shimura construction. One of the main tools used in the proof is Taylor-Wiles-Kisin patching.
Gesehen am 29.10.2021