Primitively 2-universal senary integral quadratic forms (Q6585295)

All quadratic forms considered in this paper are positive definite and integral, in the sense that the associated symmetric matrix of coefficients has integral entries. For a positive integer \(m\), such a quadratic form is said to be (primitively) \(m\)-universal if it (primitively) represents all positive definite integral quadratic forms of rank \(m\). The smallest rank of a 1-universal quadratic form is 4. There is a long history of work on the problem of determining all equivalence classes of quaternary 1-universal quadratic forms, culminating in the 15-Theorem of J.H. Conway and W.A. Schneeberger. The problem of determining the primitively 1-universal quaternary forms was first considered by \textit{N. Budarina} [Lith. Math. J. 50, 143--160 (2010; Zbl 1247.11047)]. It was recently shown by \textit{J. Ju} et al. [J. Number Theory 242, 181--207 (2023; Zbl 1508.11045)] that there are exactly 107 equivalence classes of such forms.\N\NMoving up to \(m=2\), the smallest rank of a 2-universal quadratic form is 5, and it is known that there are exactly 11 equivalence classes of 2-universal quinary forms, as proved by \textit{B. M. Kim} et al. [Contemp. Math. 249, 51--62 (1999; Zbl 0955.11011)]. In the paper under review, the authors prove that the minimal rank of a primitively 2-universal quadratic form is 6, and that there are exactly 201 equivalence classes of primitive 2-universal quadratic forms of rank 6. A list of all potential candidates for forms of this type is produced using estimates on their successive minima and analyzing their possible leading sections. The authors then provide case-by-case verifications that all of the lattices on the list of candidates are in fact primitively 2-universal.