Structure of Hecke algebras of modular forms modulo \(p\)
type: Item
instance of: scholarly article
title: Structure of Hecke algebras of modular forms modulo \(p\)
zbMATH Open document ID: 1361.11038
author: Shaunak V. Deo
published in: Algebra \& Number Theory
publication date: 2017-01-31T00:00:00Z
review text: Let \(p\geq 5\) be a prime number, \(K\) a finite extension field of \(\mathbb Q_p\) with ring of integers \(\mathcal O\) and uniformizing element \(\pi.\) Let \(N\) be a natural number and \(\Gamma\) be one of the congruence groups \(\Gamma_0(N)\) or \(\Gamma_1(N)\) inside \(\text{SL}(2,\mathbb Z).\) Denote by \({\mathcal S}_k\) the module of weight \(k\) cuspidal forms for \(\Gamma\) with \(q\)-expansion in \({\mathcal O}q.\) The author studies local components of the shallow Hecke algebras (i.e., avoiding \(U_\ell\) for primes \(\ell\) dividing \(pN\)) acting on \(\sum_{i\leq k} {\mathcal S}_i\) mod \(\pi.\) As in the case \(N=1\) treated by \textit{J. Bellaïche} and \textit{C. Khare} [Compos. Math. 151, No. 3, 397--415 (2015; Zbl 1328.11046)], it turns out that the local components of the Hecke algebras are fairly often (stated more precisely in the paper) regular complete local algebras of dimension 2. Divided congruence modules and deformation spaces for pseudo-representations play a vital role in the arguments.
reviewed by: Stefan Kühnlein
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11F25
Mathematics Subject Classification ID: 11F33
zbMATH DE Number: 6679111
zbMATH Keywords: modular forms modulo \(p\)
zbMATH Keywords: Hecke algebras
zbMATH Keywords: deformations of Galois representations
MaRDI profile type: Publication
full work available at URL: ant.2017.11.1
OpenAlex ID: W2580302859
DOI: 10.2140/ANT.2017.11.1
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description: scientific article; zbMATH DE number 6679111