Metacyclic groups and modular forms
type: Item
instance of: scholarly article
title: Metacyclic groups and modular forms
zbMATH Open document ID: 0972.11031
DOI: 10.1007/BF02686239
published in: Mathematical Notes
publication date: 2001-07-12T00:00:00Z
review text: In several papers the author investigated the relationship between representations of finite groups and multiplicative eta products [J. Théor. Nombres Bordx. 11, 247-262 (1999; Zbl 0954.11014); Mat. Zametki 52, 25-31 (1992; Zbl 0771.11022) among others]. Now she deals with metacyclic groups. Let \(\Phi\) be a unimodular representation of a finite group \(G\). For \(g\in G\), let \(P_g(x)= \prod_k (x^{a_k}-1)^{t_k}\) with \(a_k\in \mathbb{N}\), \(t_k\in \mathbb{Z}\) be the characteristic polynomial of \(\Phi(g)\). Then the eta product \(\eta_g(z)= \prod_k \eta^{t_k} (a_kz)\) is associated with \(g\), where \(\eta(z)\) is the Dedekind eta function. A finite group \(G\) is called metacyclic if there is a normal subgroup \(N\) of \(G\) such that \(G/N\) is also cyclic. The author determines all metacyclic (and all cyclic) groups \(G\) and the corresponding eta products such that a faithful unimodular representation of \(G\) with associated multiplicative eta products exists. As is well-known, there are exactly 28 integral weight and two half-integral weight multiplicative eta products, all of which have \(t_k\in \mathbb{N}\).
reviewed by: Günter Köhler
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 11F22
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20F99
zbMATH DE Number: 1517695
zbMATH Keywords: cyclic groups
zbMATH Keywords: representations of finite groups
zbMATH Keywords: multiplicative eta products
zbMATH Keywords: metacyclic groups
zbMATH Keywords: Dedekind eta function
zbMATH Keywords: unimodular representation
author: Galina V. Voskresenskaya
MaRDI profile type: Publication
cites work: Q3362758
cites work: Cusp forms and finite subgroups in \(SL(5, \mathbb{C})\)
cites work: Modular forms and regular representations of groups of order 24
cites work: Modular forms and representations of the dihedral group
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description: scientific article; zbMATH DE number 1517695