Representations of quadratic lattices over dyadic local fields

Abstract

1.11 If R i (L) = R i and α i (L) = α i we denote

On indefinite $k$-universal integral quadratic forms over number fields

“…By introducing the concept of bases of norm generators (BONGs in short), Beli has recently developed an integral representation theory over general dyadic local fields (cf. [2] and [3]). He classified 1-universal lattices over general dyadic fields in [4] by his theory.…”

Pythagoras number of quartic orders containing $\sqrt{2}$

“…The main result in this section, Theorem 2.5, is a criterion for the representability of a binary lattice as a sum of linear forms over a general dyadic local field. This result will be deduced from a general representation theorem in the theory of bases of norm generators, as developed by Beli in a series of papers [Bel01,Bel03,Bel06,Bel10,Bel19].…”

“…This is made possible by the theory of bases of norm generators developed by Beli (see e.g. [Bel01] and [Bel19]). We feel that this method, though important and powerful, has not been widely used in the literature.…”

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

“…For k = 1, Beli's work [Bel20] complements the analysis over dyadic fields in [XZ22, § 2], and gives necessary and sufficient conditions for an integral quadratic form over a general dyadic field to be universal. 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]).…”