Zeta functions and genus of quadratic forms
type: Item
instance of: scholarly article
title: Zeta functions and genus of quadratic forms
zbMATH Open document ID: 0702.11022
author: Pilar Bayer
author: Enric Nart
published in: L'Enseignement Mathématique. 2e Série
publication date: 1989-01-01T00:00:00Z
review text: In his prize winning paper for the Paris Academy ``Grundlagen für die Theorie der quadratischen Formen mit ganzzahligen Koeffizienten'' (Ges. Abh. Bd. 1, 3-144) \textit{H. Minkowski} proved (in an appendix) that two nondegenerate integral quadratic forms of rank k, having the same discriminant and the same congruential representation numbers (and the same 2-type if \(k\geq 5)\) belong to the same genus. The present paper starts out with a reformulation of Minkowski's proof using modern terminology and removing the restriction on the discriminant. It is then shown how to translate this statement into the language of local (Section 2), adelic (Section 4) and global (Section 5) representation masses. As a consequence it is shown that the representation masses of the form f as well as those of the genus of f determine the genus of f with the same restricting conditions as above.
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 11E45
zbMATH DE Number: 4150275
zbMATH Keywords: Gauss sums
zbMATH Keywords: integral quadratic forms
zbMATH Keywords: representation numbers
zbMATH Keywords: representation masses
zbMATH Keywords: genus
reviewed by: Rainer Schulze-Pillot
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4150275