On the Wiles--Lenstra--Diamond numerical criterion for freeness of modules over complete intersections

Abstract: I will talk about recent work with Srikanth Iyengar and Jeff Manning on a higher codimension version of the Wiles--Lenstra--Diamond numerical criterion (the original version is in  codimension $0$). The original version played a key role in Wiles’s  work on  the modularity of semistable elliptic curves over the rationals.

I will sketch some (conditional)  applications of the commutative algebra we develop to proving integral $R=T$ theorems in positive defect (as arise when considering $2$-dimensional Galois representations over imaginary quadratic fields, a defect one situation), and other questions/perspectives  the work leads to.  There is an  unconditional application to proving an analog of the Jacquet--Langlands correspondence for Hecke algebras acting on the cohomology of Shimura curves  with coefficients in weight one sheaves. As these Hecke algebras typically have a lot of torsion, such results cannot be deduced from the classical Jacquet--Langlands correspondence for classical  weight one forms.