On the Shintani zeta function for the space of binary quadratic forms
type: Item
instance of: scholarly article
title: On the Shintani zeta function for the space of binary quadratic forms
zbMATH Open document ID: 0757.11027
DOI: 10.1007/BF01444626
author: Akihiko Yukie
published in: Mathematische Annalen
publication date: 1992-09-26T00:00:00Z
full work available at URL: https://eudml.org/doc/164919
review text: The purpose of this paper is to define an adjusted zeta function \(Z_{ad}(\Phi,\omega,s)\) of the space of quadratic forms and calculate the principal part of the poles. Shintani's zeta function is a Dirichlet series whose coefficients are the number of equivalence classes of integral quadratic forms. We can consider the adelization of Shintani's zeta function. It is denoted by \(Z(\Phi,\omega,s)\). It is a distribution with a meromorphic parameter \(s\) on the adelic space of quadratic forms. Here, \(\Phi\) is a test function of the distribution and \((\omega,s)\) are parameters of the zeta function. By reducing it to the adjusted one, \(Z_{ad}(\Phi,\omega,s)\), it becomes a meromorphic function with finite poles. The coefficients of the principal parts of the poles are also distributions. The author of this paper gives an explicit form of the principal part of \(Z_{ad}(\Phi,\omega,s)\).
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11E45
Mathematics Subject Classification ID: 14M17
zbMATH DE Number: 56287
zbMATH Keywords: prehomogeneous vector space
zbMATH Keywords: adjusted zeta function
zbMATH Keywords: quadratic forms
zbMATH Keywords: principal part of the poles
zbMATH Keywords: adelization of Shintani's zeta function
zbMATH Keywords: test function
zbMATH Keywords: distributions
reviewed by: Masakazu Muro
MaRDI profile type: Publication
cites work: Some finiteness properties of adele groups over number fields
cites work: Q3139651
cites work: The adelic zeta function associated to the space of binary cubic forms. II: Local theory.
cites work: Density of discriminants of cubic extensions.
cites work: Eisenstein series of \(1\over 2\)-integral weight and the mean value of real Dirichlet L-series
cites work: On the functional equations satisfied by Eisenstein series
cites work: Q5622438
cites work: On zeta functions associated with prehomogeneous vector spaces
cites work: On Dirichlet series whose coefficients are class numbers of integral binary cubic forms
cites work: Q4071830
cites work: The average measure of quadratic forms with given determinant and signature
cites work: Adèles and algebraic groups. (Appendix 1: The case of the group \(G_2\), by M. Demazure. Appendix 2: A short survey of subsequent research on Tamagawa numbers, by T. Ono)
cites work: Q4090435
cites work: Sur certains groupes d'opérateurs unitaires
cites work: Sur la formule de Siegel dans la théorie des groupes classiques
cites work: Q4282628
description: scientific article; zbMATH DE number 56287