Representations of quadratic lattices over dyadic local fields

TL;DR

This paper establishes necessary and sufficient conditions for quadratic lattices over dyadic local fields to represent each other, introducing BONGs as a new descriptive tool that simplifies the complex theory.

Contribution

It provides a novel characterization of quadratic lattice representations over dyadic local fields using BONGs, complementing traditional Jordan splitting methods.

Findings

Conditions for lattice representation are fully characterized.

BONGs offer a more compact description of quadratic lattices.

Results facilitate understanding of quadratic forms over dyadic fields.

Abstract

Given two quadratic lattices $M$ and $N$ over a dyadic local field $F$, i.e. a finite extension of ${\mathbb Q}$, we give necessary and sufficient conditions such that $M$ represents $N$. Previous results on this problem were obtained by O.T. O'Meara in 1958, who solved this problem in the case when the base field $F$ is 2-adic, i.e. a non-ramified extension of ${\mathbb Q}_2$, and by C. Riehm in 1964, who solved the case when $M$ is unimodular. Some notable work on quadratic forms over dyadic local fields, in particular, on representations has also been done by Fei Xu. Our result is given in terms of BONGs (bases of norm generators), which is a new way to describe quadratic lattices over dyadic local fields, complimentary to the more traditional Jordan splittings. However, they can be easily translated in the language of Jordan splittings. The use of BONGs is not meant to completely replace the Jordan splitting. Neither they make the proofs shorter. The study of quadratic lattices over dyadic local fields is well known to be very complicated and technical. However, by using BONGs the results can be written in a more compact form and it is easier to make educated guesses on general results by extrapolating results in small dimensions.