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The Wiles defect for Hecke algebras via local-global arguments

Posted in
Jeff Manning
Wed, 27/10/2021 - 14:30 - 15:30
Parent event: 
Number theory lunch seminar

Zoom ID: 919 6497 4060
For password please contact Pieter Moree (moree@mpim...).


In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings R->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 addition to proving modularity theorems, this numerical criterion also implies a connection between the order of a certain Selmer group and a special value of an L-function.

In this talk I will consider the case of a Hecke algebra acting on the cohomology of a Shimura curve associated to a quaternion algebra. In this case, one has an analogous map of ring R->T which is known to be an isomorphism, but in many cases the rings R and T fail to be complete intersections. This means that Wiles' numerical criterion will fail to hold. I will describe a method for precisely computing the extent to which the numerical criterion fails (i.e. the 'Wiles defect"). In particular, I will show that it can be computed entirely from the structure of certain local Galois deformation rings at the primes dividing the discriminant of the quaternion algebra, and thus depends only on local information.

This is joint work with Gebhard Bockle and Chandrashekhar Khare.

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