Abstracts

Congruences in number theory, through commutative algebra

Srikanth Iyengar, University of Utah, Salt Lake City, USA

Congruences—whether between integers, functions, or ideals—appear throughout mathematics. This talk is inspired by the theory of congruences associated with modular forms, which plays a critical role in algebraic number theory and, notably, Wiles’ proof of Fermat’s Last Theorem.

The focus will be on the ``congruence ideal," an algebraic tool used to capture these relationships and a key ingredient in Wiles’ numerical criterion for detecting ring isomorphisms. Recently, Chandrashekhar Khare, Jeff Manning, and I extended Wiles' work by introducing congruence ideals in higher codimension and generalizing the numerical criterion to establish new modularity theorems. Beyond their original purpose, these ideals are finding surprising new applications. I will survey these developments, focusing on their commutative algebraic aspects. The talk should be accessible to anyone familiar with basic algebra. It is based on the following preprint: https://arxiv.org/abs/2510.05418