Local criteria for universal and primitively universal quadratic forms
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 rank at least \(4\) is almost primitively universal iff it is locally primitively universal at all primes \(p\). As an application of the criteria obtained for primitive universality of \(\mathbb{Z}_p\)-lattices, the authors complete the classification (that had been begun by Budarina) of the almost primitively universal integral quadratic forms among the universal classically integral quaternary quadratic forms.
scientific article; zbMATH DE number 7355825