Almost strong approximations for definite quadratic spaces
type: Item
instance of: scholarly article
title: Almost strong approximations for definite quadratic spaces
zbMATH Open document ID: 0894.11013
author: J. S. Hsia
author: M. Jöchner
published in: Inventiones Mathematicae
publication date: 1998-09-08T00:00:00Z
review text: The strong approximation theorem for the spin group of an indefinite quadratic space of dimension at least 3 [see, e.g., 104:4 of \textit{O. T. O'Meara}'s book [Introduction to quadratic forms, Grundlehren Math. Wiss. 117. Springer-Verlag, New York (1973; Zbl 0259.10018)], which was proven in the work of Eichler and Kneser in the 1950's, has since that time played a key role in the proofs of fundamental results on the representation and equivalence of indefinite integral quadratic lattices. The purpose of the present paper is to prove ``almost strong approximation theorems'' (one for the spinorial kernel \(O_m'\) and one for the rotation group \(O_m^+\)) for definite quadratic spaces. The statement of the theorem in the case of \(O_m'\) is as follows. Let \(V\) be a regular quadratic space over \(\mathbb{Q}\) of dimension \(m\geq 3\). Then, for any finite set \(T\) of primes, any set \(\{f_p\in O_m' (V_p)\}_{p\in T}\), and any \(\varepsilon> 0\), there exists an extended finite set \(T_e\supset T\) of primes such that for any prime \(q\not\in T_e\) there is a \(\sigma(q)\in O_m' (V)\) satisfying: (i) \(\|\sigma(q)-f_p\|_p< \varepsilon\) for each \(p\in T\), (ii) \(\|\sigma(q)\|_p= 1\) for all \(p\not\in T\cup\{q\}\), and (iii) \(\|\sigma(q)\|_q= q^2\). (The authors show by an example that the exponent 2 appearing in condition (iii) is the best possible; i.e., that this exponent cannot in general be reduced to 1.) The statement of the theorem for the rotation group \(O_m^+\) is somewhat more involved and will not be given here. The proof is correspondingly more difficult, but the applications in some instances yield stronger results than those obtained from the theorem for \(O_m'\). The proofs of both theorems depend upon a refinement and extension of previous results on the primitive representation by a positive definite quaternary quadratic \(\mathbb{Z}\)-lattice of all sufficiently large integers which are primitively represented by its genus. For the present purp…
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 11E20
Mathematics Subject Classification ID: 11E88
zbMATH DE Number: 1054506
zbMATH Keywords: almost strong approximation theorems
zbMATH Keywords: spinorial kernel
zbMATH Keywords: rotation group
zbMATH Keywords: definite quadratic spaces
zbMATH Keywords: primitive representation by a positive definite quaternary quadratic \(\mathbb{Z}\)-lattice
zbMATH Keywords: local representations
zbMATH Keywords: asymptotic representation of forms
zbMATH Keywords: primitive representation of integers by positive ternary quadratic forms
reviewed by: Andrew G. Earnest
MaRDI profile type: Publication
full work available at URL: s002220050169
OpenAlex ID: W1998873958
DOI: 10.1007/S002220050169
description: scientific article; zbMATH DE number 1054506