A numerical criterion in commutative algebra played a decisive role in Wiles’s proof of Fermat’s Last Theorem. In joint work with Srikanth Iyengar and Jeffrey Manning, we extend it to higher codimension. Exploring the number-theoretic meaning of the key new definition we make—congruence modules in higher codimension—has led to further arithmetic applications.

In ongoing work with Fred Diamond, Iyengar and Manning, we use local congruence ideals to prove new cases of the Bloch–Kato conjecture for special values of adjoint L-functions of newforms f. A central insight of the work is that local congruence ideals are counterparts of  the classical global congruence ideals attached to f, paralleling the relation between local and global Galois cohomology for the adjoint motive of f.