On Hecke algebras at exceptional Eisenstein primes

Mazur’s seminal work on the Eisenstein ideal initiated the study of Hecke algebras acting on cusp forms of weight $2$ that are congruent modulo $p$ to an Eisenstein series. This theory has since been developed by Merel, Calegari--Emerton, Lecouturier, Wake--Wang-Erickson, Deo, and others, particularly in weights $k\geq 2$, using geometric and Galois-theoretic methods. The weight $1$ case, especially at exceptional Eisenstein primes, remains comparatively less understood. In this talk, we investigate this setting and give a criterion for the relevant Hecke algebra to be Gorenstein. Our approach relies on modularity theorems, Iwasawa theory, and $p$-adic $L$-functions.