On primitively 2-universal quadratic forms (Q2892178)

A positive definite integral quadratic form or corresponding quadratic \(\mathbb Z\)-lattice is said to be (primitively) \(m\)-universal if it (primitively) represents all positive definite integral quadratic forms of rank \(m\). The well-known Conway-Schneeberger \(15\)-Theorem gives an effective criterion to determine whether a positive definite classically integral quadratic form is \(1\)-universal. Various generalizations of this result have been investigated. In particular, an analogous \(2\)-universality criterion has been obtained by \textit{B. M. Kim, M.-H. Kim} and \textit{B.-K. Oh} [Contemp. Math. 249, 51--62 (1999; Zbl 0955.11011)], and a partial result for primitive \(1\)-universality appears in a previous paper by the present author [Lith. Math. J. 50, No. 2, 140--163 (2010; Zbl 1247.11047)].NEWLINENEWLINEThe paper under review contains a detailed study of the binary quadratic forms that are primitively represented by a quadratic form \(Q\) of rank at least \(5\) over each of the rings \(\mathbb Z_p\) of \(p\)-adic integers, under restrictions on the structure of \(Q\). These local results lead to a criterion for a positive definite, classically integral quadratic form of rank at least \(5\) having odd square-free determinant to be locally primitively \(2\)-universal. A form of class number \(1\) satisfying this criterion is therefore primitively \(2\)-universal.