On indefinite modular forms of weight one
type: Item
instance of: scholarly article
title: On indefinite modular forms of weight one
zbMATH Open document ID: 0587.12005
author: Toyokazu Hiramatsu
author: Noburo Ishii
author: Yoshio Mimura
published in: Journal of the Mathematical Society of Japan
publication date: 1986-01-01T00:00:00Z
review text: Let us suppose that the ring class field \(N_ f\) modulo f (f\(\in {\mathbb{N}})\) of an imaginary quadratic field \(\Sigma\) is a dihedral extension over \({\mathbb{Q}}\) with Galois group \(D_ 4\). Let K be the unique real quadratic subfield of \(N_ f\). From a result of Shintani it follows that the L-series associated to \(\Sigma\) and to K (with special ring (ray) class characters \(\chi\) ' and \(\chi)\) coincide: \[ (1)\quad L_{\Sigma}(s,\chi ')=L_ K(s,\chi). \] The Mellin transform of the above L-series now gives an identity between the corresponding positive definite and indefinite theta series. This relation has been given explicitely in several numerical examples. At the end of the paper the higher reciprocity law on extensions of imaginary quadratic fields with Galoisgroup \(D_ 4\) [cf. \textit{C. J. Moreno}, J. Number Theory 12, 57-70 (1980; Zbl 0426.10024), and the reviewer, J. Reine Angew. Math. 361, 11- 22 (1985; Zbl 0562.12014)] is transfered to the case of real quadratic fields using equation (1).
Mathematics Subject Classification ID: 11R39
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 3941642
zbMATH Keywords: modular forms in one variable
zbMATH Keywords: nonabelian class field theory
zbMATH Keywords: ring class field
zbMATH Keywords: dihedral extension
zbMATH Keywords: L-series
zbMATH Keywords: theta series
zbMATH Keywords: higher reciprocity law
zbMATH Keywords: imaginary quadratic fields
reviewed by: Jannis A. Antoniadis
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03810067
OpenAlex ID: W1963494756
DOI: 10.2969/JMSJ/03810067
description: scientific article; zbMATH DE number 3941642