Joint IAS/Princeton University Number Theory Seminar

Wiles’s proof of the modularity of (semistable) elliptic curves over the rationals and Fermat’s Last Theorem relied on his invention of a modularity lifting method. There were two strands to the method:

(i) A numerical criterion to for a map of rings to be an isomorphism between complete intersections that are finite flat over Z_p in Wiles’s paper on FLT, subsequently generalized by Fred Diamond.

(ii) Patching (in his paper with Taylor) The patching method has been vastly generalized; in particular Calegari-Geraghty found a way to generalize it in principle to prove (potential) modularity of elliptic curves over imaginary quadratic fields (a situation of ``positive defect’’).

Their method has been made unconditional to prove modularity lifting results over CM fields in the ten author paper. The numerical criterion has yet to be generalized to positive defect.

In joint work with Srikanth Iyengar and Jeff Manning we give a development of the Wiles-Diamond numerical criterion to situations of positive defect (for example to proving modularity results for torsion Galois representations over imaginary quadratic fields). This in principle allows one to prove integral R=T theorems (in minimal and non-minimal situations), for which just the use of patching seems inadequate. One interest of proving such integral versions of modularity lifting is that in these situations, the Betti cohomology groups of 3-dimensional Bianchi manifolds (the analog of the modular curves over imaginary quadratic fields) have a lot of torsion. Our strategy consists of proving a higher dimensional version of the numerical criterion of Wiles-Diamond and applying it to prove integral R=T theorems (in the non-minimal case) after patching.