A recursion formula for local densities
type: Item
instance of: scholarly article
title: A recursion formula for local densities
zbMATH Open document ID: 0872.11021
author: Hidenori Katsurada
author: Masaki Hisasue
published in: Journal of Number Theory
publication date: 1997-06-23T00:00:00Z
review text: Let \(S\) and \(T\) be non-degenerate matrices of degree \(s\) and \(t\) \((s\geq t\geq 1)\) with entries in the ring \(\mathbb{Z}_p\) of \(p\)-adic integers. Then the local density \(\alpha_p(T,S)\) is defined as follows: Set \[ A_e(T,S)= \{X\in M(s,t;\mathbb{Z}_p/_{p^e}\mathbb{Z}_p);{^t XSX}\equiv T\text{ mod }p^e\} \] and \(\alpha_p(T,S)= \lim_{e\to\infty}p^{-est+et(t- 1)/2}\# A_e(T,S)\). The paper under review contains some useful recursion formulas for local densities \(\alpha_p(T,S)\) in the case of a unimodular matrix \(X\). These formulas determine a denominator of the formal power series \[ R((B_1,\dots, B_s),A;x_1,\dots,x_s)= \sum_{e_1\geq\cdots\geq e_s\geq 0}\alpha_p (p^{e_1}B_1\perp\cdots\perp p^{e_s}B_s,A) x^{e_1}_1\cdots x^{e_s}_s, \] which is known to the rational. Here \(B_1,\dots,B_s\), \(A\) are non-degenerate symmetric matrices and \(A\) is unimodular.
reviewed by: Martin Epkenhans
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 11E95
zbMATH DE Number: 1025608
zbMATH Keywords: local density
zbMATH Keywords: recursion formulas
zbMATH Keywords: unimodular matrix
MaRDI profile type: Publication
full work available at URL: jnth.1997.2127
OpenAlex ID: W2041758319
cites work: Q3765850
cites work: A certain formal power series of several variables attached to local densities of quadratic forms. I
cites work: A certain formal power series attached to local densities of quadratic forms. II
cites work: Dirichlet series in the theory of Siegel modular forms
cites work: Local densities of quadratic forms and Fourier coefficients of Eisenstein series
cites work: Q4271318
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DOI: 10.1006/JNTH.1997.2127
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description: scientific article; zbMATH DE number 1025608