Non-universality of ternary quadratic forms over fields containing $\sqrt2$

TL;DR

This work resolves Kitaoka's conjecture for all quartic totally real fields by showing there is no universal ternary classical quadratic form over such fields. The authors develop a comprehensive framework using Preliminaries on lattices and quadratic forms, lifting from $\mathbb{Q}(\sqrt{2})$, indecomposable elements, and cyclotomic subfields to rule out universality in quartic fields. They classify lattices into LIRE and non-LIRE types and exhaustively treat each case, including exceptional fields, to demonstrate nonuniversality. The results strongly support a refined conjecture that only $\mathbb{Q}(\sqrt{2}),\mathbb{Q}(\sqrt{3}),\mathbb{Q}(\sqrt{5})$ admit universal ternary classical lattices, with evidence extended to higher degrees and connections to universality criterion sets.

Abstract

We prove Kitaoka's conjecture for all totally real number fields of degree 4 -- namely, there is no positive definite classical quadratic form in three variables which is universal. To achieve this, we study the fields (often without restricting the degree) where 2 is a square, because in this arguably most difficult case, the recent results connecting Kitaoka's conjecture to sums of integral squares do not apply. We also prove some other properties of ternary quadratic forms over fields containing $\sqrt2$, for example in relation to the lifting problems for universal quadratic forms and for criterion sets.

Theorems & Definitions (95)

• Theorem 1.1
• Conjecture 1.2
• Theorem 1.3: cf. Theorem \ref{['th:QFNotUniversal']}
• Lemma 2.1: KY-lifting and KY-EvenBetter
• Lemma 2.2
• Remark
• Lemma 2.3: Kala-survey
• Lemma 2.4
• proof
• Lemma 2.5