Representation of positive definite quadratic forms with congruence and primitive conditions. II
type: Item
instance of: scholarly article
title: Representation of positive definite quadratic forms with congruence and primitive conditions. II
zbMATH Open document ID: 0842.11013
author: M. Jöchner
published in: Journal of Number Theory
publication date: 1996-02-21T00:00:00Z
review text: Let \(\text{SR}(m,n)\) be the strong representation property: \(M\) is a positive definite \(\mathbb Z\)-module of \(\text{rank}\leq m\), \(s\in \mathbb N\), \(T\) is a finite set of primes containing 2 and all primes \(p\) for which \(M_p\) is not unimodular. Then there is a constant \(C\) depending on \(m\), \(n\), \(M\), \(T\), \(s\) such that if \(N\) is a quadratic \(\mathbb Z\)-module of rank \(n\) which is locally represented by \(M\), and if \(f_p: N_p\to M_p\) are the representations for \(p\in T\), and if \(C< \min (N)\) then there exists a representation \(f: N\to M\) so that \(f\equiv f_p\bmod p^s M_p\) for all \(p\in T\) and \(f(N_p)\) is primitive in \(M_p\) for all \(p\not\in T\). In [J. Number Theory 48, 88--101 (1994; Zbl 0810.11020)] the author and \textit{Y. Kitaoka} showed that \(\text{SR}(2n+3, n)\) holds true for \(n\geq 1\) if in addition \(\text{ord}_p (\text{Vol} (N))\) is bounded at any fixed prime \(q\not\in T\). The paper under review gives an algebraic proof of \(\text{SR}(7,2)\). It is not known whether \(\text{SR}(2n+ 3,n)\) holds unconditionally for all \(n\geq 1\).
Mathematics Subject Classification ID: 11E12
zbMATH DE Number: 714115
zbMATH Keywords: representation of positive definite quadratic forms
zbMATH Keywords: strong representation property
reviewed by: Martin Epkenhans
MaRDI profile type: Publication
full work available at URL: jnth.1995.1009
OpenAlex ID: W4212784332
DOI: 10.1006/JNTH.1995.1009
Recommended article: Q3003991
Recommended article: Representations of positive definite quadratic forms with congruence and primitive conditions
Recommended article: Über die Klassenzahl eines relativ-zyklischen Zhlkörpers vom Primzahlgrade
Recommended article: Über die Bedeutung der Fundamentalgleichung in der Theorie der algebraischen Körper.
Recommended article: Verzweigungstheorie in allgemeinen Ordnungen algebraischer Zahlkörper.
Recommended article: Representations of positive definite Hermitian forms with approximation and primitive properties
Recommended article: The number of embeddings of hermitian lattices
Recommended article: Primitive roots of ideals in algebraic number-fields.
Recommended article: Q4399968
Recommended article: The Noether numbers for cyclic groups of prime order
description: scientific article; zbMATH DE number 714115