Local criteria for universal and primitively universal quadratic forms

Primitively universal quaternary quadratic forms

“…As mentioned in the introduction, Earnest and Gunawardana proved in [4] that there are exactly 152 equivalence classes of primitively almost universal quaternary quadratic forms that are universal. The primitively universalities of 37 quaternary quadratic forms listed in Table 1 were previously proved in [2], [4], and [5]. Among 37 quaternary quadratic forms, there are exactly 10 non-diagonal quadratic forms, all of whom have class number one.…”

“…In 2015, Earnest and his collaborators proved in [5] that there are exactly 27 equivalence classes of primitively universal quaternary diagonal quadratic forms, including 3 classes whose primitively universalities were proved by Budarina in [2]. The classification of all primitively universal quadratic forms over Z p for some prime p including the case when p " 2 was completed by Earnest and Gunawardana in [4]. They also proved that among 204 equivalence classes of universal quaternary quadratic forms, there are exactly 152 equivalence classes of primitively almost universal quaternary quadratic forms.…”

Exceptional sets for spinor regular ternary quadratic forms

“…This rules out integers of the type 8 + 16ℓ. Finally, the lattice 1, 1, 3 is isotropic over Z 2 and is therefore Z 2 -universal by [6,Proposition 4.1]. Hence, all positive integers divisible by 16 are represented by L 2 .…”

“…Let V denote the underlying quadratic space. By a computation of Hasse symbols, it follows that V 3 is anisotropic; hence α → V 3 for any α ∈ −dV = −3 Q2 3 (see, e.g., [6,Lemma 2.2]). As 9 k (6 + 9ℓ) = 3 2k • 3(2 + 3ℓ) ∈ −3 Q2…”

On $k$-universal quadratic lattices over unramified dyadic local fields

“…His method builds upon the general theory of bases of norm generators (BONGs), which he developed in his thesis [Bel01] (see also [Bel06], [Bel10], [Bel19]). Using only the more standard theory as presented in [O'M00], Earnest and Gunawardana also complete a classification of universal forms over the ring Z p of p-adic integers for any prime p ( [EG21]).…”