Metacyclic groups and modular forms
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 \(t_k\in \mathbb{N}\).
scientific article; zbMATH DE number 1517695