Theta series on the theta group
type: Item
instance of: scholarly article
title: Theta series on the theta group
zbMATH Open document ID: 0652.10020
DOI: 10.1007/BF02941367
author: Günter Köhler
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1988-01-01T00:00:00Z
review text: This paper is a continuation of a similar study for the Hecke groups \(G(\sqrt{2})\) and \(G(\sqrt{3})\) [Math. Z. 197, 69--96 (1988; Zbl 0632.10024)]. It contains a complete description of all modular forms f on subgroups of the theta group \(\Gamma_{\tau}\) which satisfy: (1) \(f\) is a sum of modular forms of integral weight and different multiplier systems on \(\Gamma_{\tau}\); (2) \(f\) is of CM-type, i.e., \(f\) is a theta series with Hecke character of an imaginary quadratic number field. The only number fields which occur are \({\mathbb Q}(\sqrt{-d})\) with \(d\in \{1,2,3,6\}\). The Dirichlet series associated to these \(f\) have Euler products; this follows in an elementary way without using the theory of Hecke operators. For small weights, the functions \(f\) are identified with combinations of the eta-function and Eisenstein series. At the end of the paper there is a list of all Hecke eigenforms of integral weights \(k\leq 6\) which satisfy (1). The first cusp eigenforms not satisfying (2) occur for weight \(k=4\).
reviewed by: Günter Köhler
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F66
zbMATH DE Number: 4063192
zbMATH Keywords: modular forms
zbMATH Keywords: theta group
zbMATH Keywords: integral weight
zbMATH Keywords: multiplier systems
zbMATH Keywords: CM-type
zbMATH Keywords: theta series
zbMATH Keywords: Hecke character
zbMATH Keywords: Dirichlet series
zbMATH Keywords: Euler products
zbMATH Keywords: Hecke eigenforms of integral weights
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4063192