Kitaoka's Conjecture and sums of squares

TL;DR

This work resolves Kitaoka's Conjecture for all totally real fields of odd discriminant by linking the existence of a universal ternary classical quadratic form to four-squares representations of elements in $2\mathcal{O}_{K}^{+}$. Using a lattice-theoretic framework and a unit-group dichotomy, the authors show that if $\sqrt{2}\notin K$ or the unit group has at least two nonsquare classes, then every element of $2\mathcal{O}_{K}^{+}$ is a sum of four squares, which forces strong structural constraints and rules out quartic counterexamples. They prove that $\sqrt2\notin K$ in the presence of a universal ternary form when $|\mathcal{U}_{K}^{+}/\mathcal{U}_{K}^{2}|=2$, and analyze both diagonal and non-diagonal ternary forms to establish universal-sums-of-squares representations via $\langle1,1,2,2\rangle$. The degree-four case is completely settled by reducing to two potential candidates, both of which are ruled out by known results, thereby supporting the conjecture in the odd-degree-discriminant setting and highlighting a clear pathway toward a full resolution of Kitaoka's conjecture in general.

Abstract

We connect the existence of a ternary classical universal quadratic form over a totally real number field $K$ with the property that all totally positive multiples of 2 are sums of squares (if $K$ does not contain $\sqrt 2$ or contains a nonsquare totally positive unit). In particular, we get that Kitaoka's Conjecture holds for all fields of odd discriminant.

Theorems & Definitions (66)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: KY-lifting
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4: cf. Kala-survey
• Proposition 2.5
• proof