A generalized Igusa local zeta function and local densities of quadratic forms
type: Item
instance of: scholarly article
title: A generalized Igusa local zeta function and local densities of quadratic forms
zbMATH Open document ID: 0796.11015
author: Hidenori Katsurada
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1994-09-06T00:00:00Z
review text: The paper proves that a certain formal power series attached to the local densities of quadratic forms can be realized as a coefficient of the Laurent expansion of a generalization of Igusa's local zeta function. Let \(A\) and \(B\) be non-degenerate symmetric matrices of degrees \(m\) and \(n\), respectively with entries of \(p\)-adic integers \(\mathbb{Z}_ p\). We define a local density attached to the symmetric matrices \(A\), \(B\) by \[ \alpha_ p(B,A)= \lim_{e\to\infty} p^{(-mn+ (n(n+1)/2))e} \# \{\overline{X}\in M_{mn} (\mathbb{Z}_ p)/p^ e M_{mn}(\mathbb{Z}_ p);\;A[X]\equiv B\bmod q\cdot p^ e\}, \] (\(A[X]:={}^ tXAX\)). Then we define a formal power series \(P(B,A; x_ 1,\dots,x_ n)\) by \[ P(B,A; x_ 1,\dots, x_ n)+ \sum_{r_ 1,\dots,r_ n=0}^ \infty \alpha_ p(B [\text{diag} (p^{r_ 1},\dots, p^{r_ n})], A)x_ 1^{r_ 1}\cdots x_ n^{r_ n}. \] The main purpose of the present paper is to show that the series \(P(B,A; x_ 1,\dots, x_ n)\) appears as a coefficient of the Laurent expansion of a generalized Igusa's local zeta function.
Mathematics Subject Classification ID: 11E45
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11E08
zbMATH DE Number: 495479
zbMATH Keywords: formal power series
zbMATH Keywords: local densities
zbMATH Keywords: quadratic forms
zbMATH Keywords: Laurent expansion
zbMATH Keywords: generalization of Igusa's local zeta function
reviewed by: Masakazu Muro
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227337
OpenAlex ID: W1991449160
cites work: Q3765850
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DOI: 10.2748/TMJ/1178227337
description: scientific article; zbMATH DE number 495479