Wiles defect for Hecke algebras that are not complete intersections

Wiles defect for Hecke algebras that are not complete intersections

Chandrashekhar Khare, University of California, Los Angeles

Please note that this seminar will take place online via Zoom. You can connect to this seminar via the following link:



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. 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 a Shimura curve associated to a quaternion algebra. In this case, one has an analogous map of rings 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's 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") at a newform f which gives rise to an augmentation T -> Z_p. The defect turns out to be determined entirely by local information at the primes q dividing the discriminant of the quaternion algebra at which the mod p representation arising from f is ``trivial''. (For instance if f corresponds to a semistable elliptic curve, then the local defect at q is related to the ``tame regulator'' of the Tate period of the elliptic curve at q.)

This is joint work with Gebhard Boeckle and Jeffrey Manning.