On Hecke algebras at exceptional Eisenstein primes

We study the structure over $\mathbb{Z}_p$ of the Hecke algebra acting on ordinary cusp forms in the Eisenstein, non-$p$-distinguished case. Our results and methods draw on four complementary approaches: deformation theory, through $R=\mathbf{T}$ theorems; Iwasawa theory, through connections to Sharifi's conjecture; $p$-adic $L$-functions, through an exceptional zero formula; and computational experiments using Magma. We also obtain new results regarding the Coleman--Greenberg conjecture.