Rationality of a certain formal power series attached to local densities of quadratic forms
type: Item
instance of: scholarly article
title: Rationality of a certain formal power series attached to local densities of quadratic forms
zbMATH Open document ID: 0818.11020
DOI: 10.1007/BF02567692
author: Hidenori Katsurada
published in: Manuscripta Mathematica
publication date: 1994-09-06T00:00:00Z
full work available at URL: https://eudml.org/doc/155922
review text: Let \(p\) be a prime number. Let \(A\) and \(B\) be non-degenerate symmetric matrices of degree \(m\) and \(n\) with \(m\geq n\geq 1\), respectively, with entries in the ring \(\mathbb{Z}_ p\) of \(p\)-adic integers. We define a local density \(\alpha_ p (B, A)\) by \[ \alpha_ p (B,A)= \lim_{e\to \infty} p^{(-mn+ \langle n\rangle )e} \#\{ \overline {X}\in M_{mn} (\mathbb{Z}_ p)/ p^ e M_{mn} (\mathbb{Z}_ p);\;{}^ t XAX\equiv B\bmod p^ e\}, \] where \(\langle n\rangle= n(n+ 1)/2\) and \(M_{mn} (\mathbb{Z}_ p)\) denotes the set of \((m,n)\)-matrices with entries in \(\mathbb{Z}_ p\). For the matrices \(A\) and \(B\) stated above, and an \(l\)-tuple \(\vec n:= (n_ 1,\dots, n_ l)\) of positive integers \(n_ 1,\dots, n_ l\) such that \(n_ 1+ \cdots+ n_ l=n\), we define a formal power series \[ P(B,A, \vec n; x_ 1,\dots, x_ l)= \sum_{r_ 1,\dots, r_ l=0}^ \infty \alpha_ p \bigl(B [\text{diag} (p^{r_ 1} E_{n_ 1},\dots, p^{r_ l} E_{n_ l}) ], A\bigr) x_ 1^{r_ 1}\dots x_ l^{r_ l}, \] where for an \((m,n)\)-matrix \(C\) and an \((m,m)\)-matrix \(D\) we write \(D[ C]:= t{}^ t CDC\), and for square matrices \(A_ 1,\dots, A_ r\). In this paper, the author proves the rationality of the function for arbitrary symmetric matrices in his original way.
Mathematics Subject Classification ID: 11E45
Mathematics Subject Classification ID: 11E95
Mathematics Subject Classification ID: 11S40
zbMATH DE Number: 599653
zbMATH Keywords: quadratic form
zbMATH Keywords: Igusa's zeta function
zbMATH Keywords: symmetric matrices
zbMATH Keywords: local density
zbMATH Keywords: formal power series
zbMATH Keywords: rationality
reviewed by: Masakazu Muro
MaRDI profile type: Publication
cites work: Q3765850
cites work: The rationality of the Poincaré series associated to the p-adic points on a variety
cites work: Rationality criteria and application to the generating series of a system of equations with coefficients in a local field
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: A generalized Igusa local zeta function and local densities of quadratic forms
cites work: Local densities of quadratic forms and Fourier coefficients of Eisenstein series
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description: scientific article; zbMATH DE number 599653