Strong approximation of quadrics and representations of quadratic forms
type: Item
instance of: scholarly article
title: Strong approximation of quadrics and representations of quadratic forms
zbMATH Open document ID: 1206.11044
author: Constantin N. Beli
author: Wai Kiu Chan
published in: Journal of Number Theory
publication date: 2008-12-08T00:00:00Z
review text: Let \(F\) be a number field, \(S\) a finite set of primes of \(F\) containing all the archimedean primes of \(F\), and \(\mathfrak o\) the ring of \(S\)-integers of \(F\). Let \(V\) be a nondegenerate quadratic space over \(F\) of dimension \(m\) such that \(V_{\mathfrak p}\) is isotropic for some prime \(\mathfrak p \in S\), and let \(U\), \(M\) and \(N\) be a nondegenerate subspace of \(V\) of dimension \(n\), an \(\mathfrak o\)-lattice on \(V\), and an \(\mathfrak o\)-lattice on \(U\) that is represented everywhere locally by \(M\), respectively. The main result of this paper gives necessary and sufficient conditions, when \(m-n \geq 2\), for the existence of a representation of \(N\) by \(M\) which approximates prescribed representations given on the completions with respect to a finite set of primes disjoint from \(S\). To state the result more precisely, let \(T\) be a finite set of primes disjoint from \(S\). Then \(M\) is said to represent \(N\) with approximation at \(T\) if, for any given integer \(k \geq 1\) and any family of representations \(\phi_{\mathfrak p}:N_{\mathfrak p}\rightarrow M_{\mathfrak p}\) for \(\mathfrak p \in T\), there exists a representation \(f:N\rightarrow M\) such that \(f \equiv \phi_{\mathfrak p} \text{ mod } \mathfrak p^kM_{\mathfrak p}\) for all \(\mathfrak p \in T\). Under the assumptions stated above, it is proven that if \(W\) is isotropic or \(m-n\geq 3\), then \(M\) represents \(N\) with approximation at any finite set of primes disjoint from \(S\). When \(W\) is anisotropic and \(m-n = 2\), the conditions under which \(M\) represents \(N\) with approximation at any finite set of primes disjoint from \(S\) are more subtle, and the result depends on the theory of spinor exceptional representations. This result on representation with approximation is applied to provide necessary and sufficient conditions stated in terms of the arithmetic of the underlying quadratic spaces for the variety \(Y\) of representations of \(U\) by …
reviewed by: Andrew G. Earnest
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 20G30
zbMATH DE Number: 5379237
zbMATH Keywords: representation of quadratic lattices
zbMATH Keywords: representation with approximation property
zbMATH Keywords: strong approximation of quadrics
MaRDI profile type: Publication
full work available at URL: j.jnt.2007.07.010
OpenAlex ID: W1987894004
cites work: Bounded generation of \(S\)-arithmetic subgroups of isotropic orthogonal groups over number fields.
cites work: Representations of indefinite quadratic forms
cites work: Q5675787
DOI: 10.1016/J.JNT.2007.07.010
description: scientific article; zbMATH DE number 5379237