Universal integral quadratic forms over dyadic local fields

TL;DR

This paper classifies universal quadratic forms over dyadic local fields, extending previous work from non-dyadic fields and providing a comprehensive solution for all cases using BONGs and Jordan splittings.

Contribution

It solves the classification of universal quadratic forms over dyadic local fields in the general case, using BONGs and translating results into Jordan splittings.

Findings

Complete classification of universal quadratic forms over dyadic local fields.

Reduction of n-universality problem to cases with n ≤ 4.

Necessary conditions for n-universality when n ≥ 3 and n is odd.

Abstract

A quadratic form over a non-archimedian local field of characteristic zero $F$ is called universal if it is integral and it represents all non-zero integers of $F$. Xu Fei and Zhang Yang determined all universal quadratic forms in the case when $F$ is non-dyadic. In the more complicated dyadic case, when $F$ is a finite extension of $\mathbb Q_2$, they solved the same problem only in the ternary case. In our paper we solve this problem in the general case. Our result is given in terms of BONGs (bases of norm generators) but in section 3 of the paper we translate (without a proof) our result in terms of the more traditional Jordan splittings. In the last section we give some results on $n$-universality. We show that it can be reduced to the cases $n\leq 4$ and we give explicit necessary conditions for $n$-universality in the case when $n\geq 3$, $n$ odd. (A quadratic form is called $n$-maximal if it is integral and it represents all non-degenerate integral quadratic forms of rank $n$.)