Representations of integral quadratic forms over dyadic local fields

Abstract

Abstract. In this paper we give necessary and sufficient conditions for the representations of quadratic lattices over arbitrary dyadic fields. Our result is given in terms of Bases of Norm Generators (BONGs, for short). However, they can be translated in terms of the more traditional Jordan decompositions.

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].…”

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.…”

“…When p 1 ≡ ±1 (mod 8), the dyadic prime of F splits completely in F ( √ p * 1 )/F by [24,63:1.Local Square Theorem]. In this case, there is a classic ternary 1-LNG O F -lattice by Proposition 6.2 (2). When p 1 ≡ ±3 (mod 8), the dyadic prime of F is inert in F ( √ p * 1 )/F , so that F has no classic ternary 1-LNG lattice by Proposition 6.1.…”

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]).…”