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arXiv · 2608.20125

Modularity theorems for Eisenstein congruences in prime-power level

Jaclyn Lang·Katharina Müller·Bharathwaj Palvannan
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Abstract

Let $p, N \geq 5$ be primes such that $N \equiv 1 \bmod p$. We prove modularity theorems at levels $N$ and $N^2$, showing that suitable Eisenstein localizations of the weight-$2$ $p$-adic Hecke algebra at these levels are isomorphic to certain natural quotients of a universal pseudodeformation ring. This universal ring parametrizes pseudorepresentations that are residually Eisenstein, unramified outside $Np$ and finite-flat at $p$, and satisfy appropriate conditions at $N$ depending on the level.

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Jaclyn Lang, Katharina Müller, Bharathwaj Palvannan. 2026-08-20. Modularity theorems for Eisenstein congruences in prime-power level. https://arxiv.org/abs/2608.20125

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