Theta series on the Hecke groups \(G(\sqrt{2})\) and \(G(\sqrt{3})\)
type: Item
instance of: scholarly article
title: Theta series on the Hecke groups \(G(\sqrt{2})\) and \(G(\sqrt{3})\)
zbMATH Open document ID: 0632.10024
DOI: 10.1007/BF01161631
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183719
review text: For \(q=2,3\), the Hecke group \(G(\sqrt{q})\) is the group which is generated by the transformations \(z\mapsto z+\sqrt{q}\) and \(z\mapsto - 1/z\) of the upper half plane. The object of this paper are those modular forms of integral weight and an arbitrary multiplier system on \(G(\sqrt{q})\) which can be represented as a theta series with a grössencharacter (Hecke character) attached to some imaginary quadratic number field. Modular forms of this kind have been called of CM-type [\textit{K. A. Ribet}, Lect. Notes Math. 601, 17--52 (1977; Zbl 0363.10015)]. One of the reasons why they are interesting is that they can be recognized to be Hecke eigenforms without using Hecke operators: From elementary arithmetic in imaginary quadratic number fields it follows that the associated Dirichlet series have Euler product expansions. There are several infinite sequences of modular forms which are theta series with characters attached to \({\mathbb Q}(\sqrt{-3})\) for \(G(\sqrt{3})\), and to \({\mathbb Q}(\sqrt{-1})\), \({\mathbb Q}(\sqrt{-2})\) for \(G(\sqrt{2}).\) The paper contains many identities which express these theta series, for small weights, as combinations of Eisenstein series and the Dedekind eta-function \(\eta\). The theta series representations yield congruences and estimates for the Fourier coefficients of the modular forms. The estimates are slightly sharper than those which hold generally for cusp eigenforms (Ramanujan-Petersson-Deligne). Moreover, properties of the characters imply that the Fourier coefficients of otherwise unrelated modular forms coincide at certain infinite sets of primes. Results of this kind (congruences, estimates, coincidence of Fourier coefficients) are also given for the forms \(\eta^ 2\), \(\eta^ 4\), \(\eta^ 6\), \(\eta^ 8\) on the modular group. Theta series with grössencharacters on the modular group have been investigated by \textit{B. Schoeneberg} [Math. Ann. 126, 177--184 (1953; Zbl 0053.05403)], and they have been reconside…
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F06
zbMATH DE Number: 4027572
zbMATH Keywords: Hecke group
zbMATH Keywords: modular forms of integral weight
zbMATH Keywords: multiplier system
zbMATH Keywords: theta series with a grössencharacter
zbMATH Keywords: Hecke character
zbMATH Keywords: imaginary quadratic number field
zbMATH Keywords: CM-type
zbMATH Keywords: Hecke eigenforms
zbMATH Keywords: identities
zbMATH Keywords: Eisenstein series
zbMATH Keywords: Dedekind eta-function
zbMATH Keywords: congruences
zbMATH Keywords: estimates for the Fourier coefficients
zbMATH Keywords: coincidence of Fourier coefficients
author: Günter Köhler
reviewed by: Günter Köhler
MaRDI profile type: Publication
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OpenAlex ID: W1970700072
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description: scientific article; zbMATH DE number 4027572