On power series attached to local densities
type: Item
instance of: scholarly article
title: On power series attached to local densities
zbMATH Open document ID: 0857.11018
author: Hidenori Katsurada
published in: Proceedings of the Japan Academy. Series A
publication date: 1997-03-11T00:00:00Z
review text: Let \(p\) be an odd prime. Let \(S \in M(s; \mathbb{Z}_p)\), \(T\in M(t; \mathbb{Z}_p)\) be nondegenerate symmetric matrices with entries in the \(p\)-adic integers \(\mathbb{Z}_p\). Set \[ A_e(T,S) = \bigl\{\overline X \in M(s,t; \mathbb{Z}_p)/p^eM(s,t; \mathbb{Z}_p) \mid{^tXSX}-T \in p^e M(s,t; \mathbb{Z}_p) \bigr\}. \] Then the local density \(\alpha_p(T,S)\) is defined by \[ \alpha_p (T,S) = \lim_{e\to \infty} p^{(-2st + t(t+1)) e/2}\cdot\# A_e(T,S). \] With it we attach the following power series \[ P(T,S;x_1,\dots, x_t): = \sum^\infty_{r_1, \dots, r_t=0} \alpha_p \bigl(\langle p^{r_1}, \dots, p^{r_t}\rangle \cdot T \cdot \langle p^{r_1}, \dots, p^{r_t}\rangle, S\bigr) \cdot x^{r_1}_1 \cdots x_t^{r_t}. \] In [J. Number Theory 51, 169-209 (1995; Zbl 0822.11026) and Proc. Japan Acad., Ser. A 70, 208-211 (1994; Zbl 0822.11027)] the author studied these power series. He proved that \(P(T,S;x_1,\dots,x_t)\) is a rational function of \(x_1, \dots, x_t\). He further gave an explicit form for its denominator in the case of a diagonal matrix \(T\). Now he gives more precise results if in addition \(S\) is a unimodular matrix of degree \(\geq 2 \cdot \deg T\). The proof requires a careful analysis of the proof of the main theorem in the first article by the author mentioned here.
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 11E95
zbMATH DE Number: 897685
zbMATH Keywords: symmetric matrices
zbMATH Keywords: local density
zbMATH Keywords: power series
zbMATH Keywords: unimodular matrix
reviewed by: Martin Epkenhans
MaRDI profile type: Publication
full work available at URL: pjaa.71.187
OpenAlex ID: W2051997772
cites work: Q3765850
cites work: A generalized Igusa local zeta function and local densities of quadratic forms
cites work: Rationality of a certain formal power series attached to local densities of quadratic forms
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cites work: A note on local densities of quadratic forms
cites work: Local densities of quadratic forms and Fourier coefficients of Eisenstein series
DOI: 10.3792/PJAA.71.187
description: scientific article; zbMATH DE number 897685