Wiles defect for Hecke algebras that are not complete intersections

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  of the newform f 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.