Structure of Hecke algebras of modular forms modulo $p$

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 $U_p$ 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 $2$.