On Hecke algebras at exceptional Eisenstein primes

We study the structure over Zₚ 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=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.