Local criteria for universal and primitively universal quadratic forms
type: Item
instance of: scholarly article
title: Local criteria for universal and primitively universal quadratic forms
zbMATH Open document ID: 1475.11055
author: Andrew G. Earnest
author: B. L. K. Gunawardana
published in: Journal of Number Theory
publication date: 2021-06-07T00:00:00Z
review text: An integral quadratic lattice \(L\) over the \(p\)-adic integers \(\mathbb{Z}_p\) (for some prime \(p\)) is a module over \(\mathbb{Z}_p\) of full rank inside an underlying finite-dimensional \(\mathbb{Q}_p\)-vector space \(V\) equipped with a nondegenerate quadratic form \(q\) such that \(q(L)\subseteq\mathbb{Z}_p\). If equality holds, \(L\) is said to be universal, and if to each \(a\in\mathbb{Z}_p\setminus \{ 0\}\) there exists a primitive vector \(v\in L\) such that \(q(v)=a\), then \(L\) is said to be primitively universal. In this setting, the latter can only occur if \(L\) is isotropic, i.e. there exists some \(x\in L\setminus\{ 0\}\) with \(q(x)=0\). The main purpose of this paper is to give completely general criteria for a \(\mathbb{Z}_p\)-lattice to be universal resp. primitively universal. This paper builds on a previous article by the present authors [Ramanujan J. 55, No. 3, 1145--1163 (2021; Zbl 1478.11053)] where much of the foundational work was done, and it completes earlier efforts by \textit{N. Budarina} [Lith. Math. J. 50, No. 2, 140--163 (2010; Zbl 1247.11047)] and, much earlier, \textit{G. Pall} [Am. J. Math. 68, 47--58 (1946; Zbl 0060.11003)] where only partial results in the \(2\)-adic case had been obtained or given with full proofs. A noteworthy feature of the present paper is that it only relies on nothing more than the classical theory of local lattices as found in the books by \textit{O. T. O'Meara} [Introduction to quadratic forms. Berlin: Springer (2000; Zbl 1034.11003)] or \textit{L. J. Gerstein} [Basic quadratic forms. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1147.11002)]. The criteria for (primitive) universality are given in terms of the Jordan splitting of the lattice. Recall that a positive definite integral \(\mathbb{Z}\)-lattice is said to be (almost) primitively universal if it represents primitively (almost) all positive integers. It is known that a positive definite integral \(\mathbb{Z}\)-lattice of ra…
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11E20
Mathematics Subject Classification ID: 11E25
zbMATH DE Number: 7355825
zbMATH Keywords: quadratic form
zbMATH Keywords: quadratic lattice
zbMATH Keywords: universal quadratic lattice
zbMATH Keywords: primitively universal
zbMATH Keywords: locally primitively universal
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: j.jnt.2021.02.005
OpenAlex ID: W3136812880
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DOI: 10.1016/J.JNT.2021.02.005
description: scientific article; zbMATH DE number 7355825