Wiles’ proof of Fermat’s last theorem boils down to proving the existence of a ringisomorphism R -> T, where R is a Galois deformation ring and T is a Hecke algebra acting on a space of cusp forms. This relies on a numerical criterion for such a map to be an isomorphism of complete intersections.

In [3] and [4], the authors study contexts where R and T are not complete intersections,thus the Wiles numerical criterion cannot hold. They quantify the failure of the numerical criterion by computing the associated Wiles defect in terms of the local behavior of a global Galois representation ρ_f associated to a modular form f. We use the methods of [4] to compute the Wiles defect in the case where we demand that the given modular representation ρf is of principal series type at a fixed set of primes