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Wiles’s proof of Ìýthe modularity of (semistable) elliptic curves over the rationals and Fermat’s Last Theorem relied on his invention of Ìýa modularity lifting method.

There were two strands to the method:

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

(ii) Patching (in his paper with Taylor)

The patching method Ìýhas been vastly generalized; in particular Calegari-Geraghty Ìýfound a way to generalize it in principle to prove (potential) modularity of elliptic curves over imaginary quadratic fields (a situation of ``positive defect’’).Ìý Their method has been made unconditional to prove Ìýmodularity lifting results Ìýover CM fields in the ten author paper. The numerical criterion has yet to be be generalized to positive defect.

In joint work with Srikanth Iyengar and Jeff Manning we give Ìýa development of the Wiles-Diamond numerical criterion to situations of positive defect (for example to proving modularity results for torsion Galois representations Ìýover imaginary quadratic fields). This in principle allows one to prove integral R=T theorems (in minimal and non-minimal situations), Ìýfor which just the use of patching seems inadequate. One interest of proving such integral versions of modularity lifting is that in these situations, the ÌýBetti cohomology groups Ìýof Ìý3-dimensional Bianchi manifolds Ìý(the analog of the modular curves over imaginary quadratic fields) Ìýhave a lot of torsion. Our strategy consists of Ìýproving a higher dimensional version of the numerical criterion of Wiles-Diamond and applying it Ìýto prove integral R=T theorems (in the non-minimal case) after patching.