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
Milestones
Received: 31 July 2015
Revised: 5 August 2016
Accepted: 17 November 2016
Published: 23 January 2017
Authors
 Shaunak V. Deo Department of Mathematics MS 050, Brandeis University 415 South Street Waltham, MA 02453 United States Mathematics Research Unit Université du Luxembourg Faculté des Sciences, de la Technologie et de la Communication 6, rue Richard Coudenhove-Kalergi L-1359 Luxembourg Luxembourg