#### Vol. 11, No. 1, 2017

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Structure of Hecke algebras of modular forms modulo $p$

### Shaunak V. Deo

Vol. 11 (2017), No. 1, 1–38
##### Abstract

Generalizing the recent results of Bellaïche and Khare for the level-$1$ case, we study the structure of the local components of the shallow Hecke algebras (i.e., Hecke algebras without ${U}_{\phantom{\rule{0.3em}{0ex}}p}$ and ${U}_{\ell }$ for all primes $\ell$ dividing the level $N$) acting on the space of modular forms modulo $p$ for ${\Gamma }_{0}\left(N\right)$ and ${\Gamma }_{1}\left(N\right)$. We relate them to pseudodeformation rings and prove that in many cases, the local components are regular complete local algebras of dimension $2$.

##### Keywords
Modular forms modulo $p$, Hecke algebras, deformations of Galois representations
##### Mathematical Subject Classification 2010
Primary: 11F80
Secondary: 11F25, 11F33