On Kitaoka's conjecture and lifting problem for universal quadratic forms

TL;DR

This paper proves finiteness results for universal quadratic forms over totally real fields, constructs universal forms with bounded rank, and confirms a conjecture about the finiteness of fields with universal ternary forms.

Contribution

It establishes finiteness of extensions with universal forms, constructs a universal form with rank bounds, and proves a version of Kitaoka's conjecture for ternary forms.

Findings

Finitely many extensions have universal forms with a short basis.

Constructed universal forms with rank bounded by D(log D)^{d-1}.

Proved finiteness of fields with universal ternary quadratic forms.

Abstract

For a totally positive definite quadratic form over the ring of integers of a totally real number field $K$, we show that there are only finitely many totally real field extensions of $K$ of a fixed degree over which the form is universal (namely, those that have a short basis in a suitable sense). Along the way we give a general construction of a universal form of rank bounded by $D(\log D)^{d-1}$, where $d$ is the degree of $K$ over $\mathbb Q$ and $D$ is its discriminant. Furthermore, for any fixed degree we prove (weak) Kitaoka's conjecture that there are only finitely many totally real number fields with a universal ternary quadratic form.