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arXiv · 2610.35559

On Hecke algebras at exceptional Eisenstein primes

Alexandre Maksoud
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Abstract

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.

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Alexandre Maksoud. 2026-10-26. On Hecke algebras at exceptional Eisenstein primes. https://arxiv.org/abs/2610.35559

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