Representation theory and the cuspidal group of \(X(p)\)
type: Item
instance of: scholarly article
title: Representation theory and the cuspidal group of \(X(p)\)
zbMATH Open document ID: 0629.14021
DOI: 10.1215/S0012-7094-87-05406-8
published in: Duke Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: The author uses the characteristic p representation theory of \(\mathrm{GL}_ 2(\mathbb Z/p\mathbb Z)\) to study the structure of the \(p\)-part of the cuspidal divisor class group \(C\) of the principal modular curve \(X(p)\) as a module over the Galois group of \(X(p)\). In the process he recovers several results of \textit{D. S. Kubert} and \textit{S. Lang} [``Modular units.'' New York etc.: Springer (1981; Zbl 0492.12002)]. In particular, he sees that the existence of the special group is required by the non-semisimplicity of the principal series representations of \(\mathrm{GL}_ 2(\mathbb Z/p\mathbb Z)\) in characteristic \(p\). In the same vein, the fact that certain modular representations are not semisimple is used to show that, when \(p\geq 5\), the quotient \(C/pC\) is of dimension \(\geq (p-5)(p-1)/4\) with equality if and only if \(p\) is a regular prime.
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 14G25
zbMATH DE Number: 4023464
zbMATH Keywords: characteristic p representation
zbMATH Keywords: principal modular curve
author: Benedict H. Gross
reviewed by: Sheldon Kamienny
MaRDI profile type: Publication
cites work: Grundlagen einer Theorie der Integralgruppen und der Integralperioden bei den Normalteilern der Modulgruppe
cites work: Q3953865
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cites work: Q4126536
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full work available at URL: s0012-7094-87-05406-8
OpenAlex ID: W2014852431
description: scientific article; zbMATH DE number 4023464