The Wiles-Lenstra-Diamond numerical criterion over imaginary quadratic fields

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.