Hybrid talk.

Contact for zoom details: Pieter Moree (moree@mpim-bonn.mpg.de)

Wiles' modularity lifting theorem was the central argument in

his proof of modularity of (semistable) elliptic curves over Q, and hence

of Fermat's Last Theorem. His proof relied on two key components: his

"patching" argument (developed in collaboration with Taylor) and his

numerical isomorphism criterion.

In the time since Wiles' proof, the patching argument has been generalized

extensively to prove a wide variety of modularity lifting results. In

particular Calegari and Geraghty have found a way to generalize it to prove

potential modularity of elliptic curves over imaginary quadratic fields

(contingent on some standard conjectures). The numerical criterion on the

other hand has proved far more difficult to generalize, although in

situations where it can be used it can prove stronger results than what can

be proven purely via patching.

In this talk I will present joint work with Srikanth Iyengar and

Chandrashekhar Khare which proves a generalization of the numerical

criterion to the context considered by Calegari and Geraghty (and

contingent on the same conjectures). This allows us to prove integral "R=T"

theorems at non-minimal levels over imaginary quadratic fields, which are

inaccessible by Calegari and Geraghty's method. The results provide new

evidence in favor of a torsion analog of the classical Langlands

correspondence.

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