Kitaoka's Conjecture on Ternary Quadratic Forms

• Kitaoka's Conjecture is a hypothesis stating that only finitely many totally real fields admit a universal ternary quadratic form representing every totally positive integer.
• The conjecture employs analytic and arithmetic techniques—such as indecomposable element analysis, discriminant bounds, and escalation methods—to verify universal form rarity in quadratic, biquadratic, and cyclotomic fields.
• Its verification connects sums of squares, unit group structures, and lattice theory, offering practical insights into diophantine representation challenges.

Kitaoka's Conjecture is a central hypothesis in the theory of universal quadratic forms over totally real number fields. It postulates the finiteness of such fields admitting universal ternary quadratic forms—forms of rank three that represent every totally positive element in the field’s ring of integers. The conjecture has driven the development of deep analytic and arithmetic techniques, particularly involving indecomposable elements, discriminant bounds, and the escalation method. Substantial progress in recent years has led to strong forms of verification in key families: real quadratic fields, biquadratic extensions, and real cyclotomic fields. The conjecture has also revealed intricate connections between sums of squares, the structure of totally positive units, and constraints arising from the arithmetic of the field.

1. Statement and Context of Kitaoka's Conjecture

Kitaoka's Conjecture asserts that only finitely many totally real number fields admit a universal ternary quadratic form, typically classical (with off-diagonal coefficients in $2\mathcal O_K$), representing all totally positive integers in the ring of integers $\mathcal O_K$. More generally, the conjecture anticipates that, as the degree and arithmetic complexity of a field increase, the minimal possible rank for a universal quadratic form also rises.

This principle is substantiated in quadratic, biquadratic, and cyclotomic settings, where research demonstrates that universality in rank three or four is exceptionally rare. For example, in the quadratic case, at most thirteen real quadratic fields admit a universal ternary lattice, with explicit discriminant bounds (Kala et al., 31 Jan 2025). No totally real biquadratic field supports a universal ternary classical form (Krásenský et al., 2019), and universality in real cyclotomic fields is limited strictly to low-degree examples (specifically $q=5$ or $q=8$) (Kala et al., 6 Feb 2024).

2. Universal Quadratic Forms and Indecomposable Elements

Universal quadratic forms are those that represent every totally positive element of $\mathcal O_K$. The critical obstruction is the presence of indecomposable elements—algebraic integers not expressible as the sum of two nonzero totally positive elements. These elements act as "hard cases" in universality problems.

The determination of indecomposables relies on continued fraction expansions in quadratic subfields and their associated semiconvergents. For $K=\mathbb{Q}(\sqrt{p},\sqrt{q})$, sufficient conditions are established to guarantee the persistence of indecomposability when passing from quadratic to biquadratic fields. For instance, if $a\in\mathbb{Q}(\sqrt{p})$ is a convergent of $-\omega_p$ and $V_T>M_p$ (with $M_p=\max\{u_j: j\text{ odd}\}$ in the continued fraction), then $a$ remains indecomposable in $K$ (Čech et al., 2018).

Trace and norm estimates, especially bounds on the norm of an element, are vital for the algorithmic verification of indecomposability. If $N_{K/\mathbb{Q}}(a)>A$ (where $A$ is the discriminant), then $a$ is not indecomposable (Kala et al., 2021).

3. Escalation Methodology and Lower Bounds

The escalation method is a constructive approach establishing lower bounds for the rank of universal forms. By iteratively adding variables corresponding to indecomposable elements not yet represented, one shows that the minimal rank must exceed three except in exceptional cases.

In biquadratic fields such as $\mathbb{Q}(\sqrt{2},\sqrt{3})$, algorithmic escalation demonstrates that universal forms require at least five variables, and in $\mathbb{Q}(\sqrt{6},\sqrt{19})$ at least six (Čech et al., 2018). Proposition 4.4 provides necessary conditions on non-diagonal coefficients: $a_{ij}^2\leq\mathrm{Tr}(\alpha_i\alpha_j)$. This process, exploiting finiteness from trace estimates, systematically searches all feasible cases by reducing infinite possibilities to finite computation.

In the quadratic case, explicit discriminant bounds ensure finiteness. For instance, only those $D\in\{2,3,5,6,7,10,13,17,21,33,41,65,77\}$ yield a universal ternary lattice in $\mathbb{Q}(\sqrt{D})$ (Kala et al., 31 Jan 2025).

4. Connections to Sums of Squares and Unit Structure

The existence of a ternary classical universal quadratic form is intimately related to the property that every totally positive multiple of $2$ is a sum of squares. If $K$ does not contain $\sqrt{2}$ or admits a nonsquare totally positive unit, it is shown that $2\mathcal O_K^+$ is contained in the set of sums of squares (Kala et al., 22 Oct 2025). This pattern is confirmatory: for fields of odd discriminant (where 2 is unramified), universality is achieved only in $\mathbb{Q}(\sqrt{5})$.

Moreover, the unit group structure, specifically the quotient $\mathcal U_K^+/\mathcal U_K^2$, determines the representation behavior. If this quotient has order greater than two, universality in rank three is impossible, forcing at least four variables (Kala et al., 22 Oct 2025).

5. Verification in Biquadratic and Cyclotomic Fields

Strong nonexistence results are established for biquadratic fields. No universal ternary quadratic form exists for any totally real biquadratic field, confirmed by careful analysis of indecomposables and representation conditions (Krásenský et al., 2019). The escalation strategy ensures that all critical indecomposables remain unrepresented without increasing the rank beyond three. This outcome provides strong evidence for the conjecture’s finiteness claim.

For real cyclotomic fields, a detailed enumeration using house bounds $[a]=\max_\sigma|\sigma(a)|<2+16$ allows a complete classification for degrees up to five. Computational techniques confirm that only the lowest degree cases (essentially $\mathbb{Q}$, $\mathbb{Q}(\sqrt{5})$, and $\mathbb{Q}(\sqrt{2})$) yield universality in ternary forms (Kala et al., 6 Feb 2024).

6. Finiteness Results and Discriminant Bounds

The conjecture’s essential prediction—finiteness of fields with universal ternary forms—is rigorously substantiated. For any fixed degree $d$, only finitely many totally real number fields admit such forms (Kala et al., 2021). General constructive bounds show that the rank of the universal form is less than $A(\log A)^{d-1}$, with $A$ the discriminant.

Discriminant bounds are central: if a real quadratic field has a universal ternary lattice, its discriminant must satisfy $\Delta_F<10,000$ (for $m=1$), reducing the search to finitely many explicit cases (Kala et al., 31 Jan 2025). Techniques such as block matrix decomposition, Hadamard's inequality, and Sylvester’s criterion are employed to link the Gram matrix structure to field arithmetic, ensuring that large discriminants inevitably require higher rank.

7. Implications and Broader Connections

The verification of Kitaoka's Conjecture has several far-reaching consequences. The classification of universal quadratic forms with low rank (ternary or quaternary) is exceedingly restrictive, revealing only isolated fields where such forms exist—primarily $\mathbb{Q}$, $\mathbb{Q}(\sqrt{2})$, and $\mathbb{Q}(\sqrt{5})$.

This body of work connects the arithmetic of quadratic forms, the geometry of numbers, and the structure of totally real fields. The absence of universal ternary forms in most fields highlights the delicacy of representation problems and motivates further exploration of the interplay between algebraic invariants (discriminant, unit group) and global arithmetic phenomena.

A plausible implication is that the methods—particularly trace/norm estimates, indecomposable element analysis, and escalation algorithms—serve as templates for future research in representation theory of quadratic forms in higher degrees and more exotic extensions. The synthesis of classical and computational approaches is likely to advance understanding in lattice theory, the arithmetic of sums of squares, and deep diophantine representation problems.