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

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 Up and U for all primes dividing the level N) acting on the space of modular forms modulo p for Γ0(N) and Γ1(N). We relate them to pseudodeformation rings and prove that in many cases, the local components are regular complete local algebras of dimension 2.

Modular forms modulo $p$, Hecke algebras, deformations of Galois representations
Mathematical Subject Classification 2010
Primary: 11F80
Secondary: 11F25, 11F33
Received: 31 July 2015
Revised: 5 August 2016
Accepted: 17 November 2016
Published: 23 January 2017
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