Lifting Problems for Universal Quadratic Forms

• The paper presents a detailed analysis of lifting universal quadratic forms across totally real field extensions using explicit local-global criteria and computational methods.
• It shows through classification and enumeration that only specific fields like Q, Q(√5), and Q(ζ7+ζ7⁻¹) admit universal Z-forms.
• The study establishes finiteness results and conjectures for universality under extensions, linking arithmetic invariants with geometric techniques.

A lifting problem for universal quadratic forms concerns the existence and transfer of positive-definite quadratic forms—especially with $\mathbb{Z}$-coefficients—capable of representing every totally positive integer in the ring of integers of a totally real number field, and explores the behavior of such forms under field extensions. The subject intricately combines techniques from arithmetic theory of quadratic forms, geometry of numbers, explicit local-global criteria, and computational enumeration. The central questions are universal existence over fixed number fields, classification for small degrees, criteria for lifting universality, and finiteness results for families of extensions (Kala et al., 2018, Kala et al., 2021, Kim et al., 2023, Kala et al., 2024, Kramer et al., 22 Jan 2026, Kala et al., 2024).

1. Preliminaries: Universal Quadratic Forms and Indecomposables

Let $K$ be a totally real number field of degree $n=[K:\mathbb{Q}]$ with ring of integers $\mathcal{O}_K$. A positive-definite quadratic form $Q$ in $m$ variables with integer coefficients ($a_{ij}\in\mathbb{Z}$) is universal over $K$ if for every totally positive $\alpha\in\mathcal{O}_K^+$, there exist $x_1,\ldots,x_m\in\mathcal{O}_K$ such that $Q(x_1,\ldots,x_m)=\alpha$. The structure and existence of universal forms are strongly influenced by arithmetic properties of $K$, notably its unit group, codifferent ideal $\mathcal{O}_K^\vee=\{\beta\in K:\Tr(\beta\,\mathcal{O}_K)\subseteq\mathbb{Z}\}$, and the set of indecomposable totally positive elements—those not expressible as the sum of two nonzero totally positive integers. Indecomposables often present the local and global obstructions to universality (Kala et al., 2018, Kala et al., 2024).

The Pythagoras number $P(R)$ of a ring $R$ (such as $\mathcal{O}_K$ or an order $\mathcal{O}$) is the minimal $m$ such that every element in $R$ that is a sum of squares is a sum of at most $m$ squares.

2. Classification of Totally Real Fields Admitting Universal $\mathbb{Z}$-Forms

For fields of degrees $d\leq 5$, complete classification results have been obtained. Only three totally real fields admit a universal quadratic form with $\mathbb{Z}$-coefficients:

• $\mathbb{Q}$: Lagrange’s four-squares theorem.
• $\mathbb{Q}(\sqrt{5})$: Universal ternary form, classification incomplete for higher ranks.
• $\mathbb{Q}(\zeta_7+\zeta_7^{-1})$: Maximal real subfield of the $7$th cyclotomic field, quaternary form $x^2+y^2+z^2+w^2+xy+xz+xw$ is universal.

No quartic, quintic, or real biquadratic fields admit universal $\mathbb{Z}$-forms. The proof utilizes a computational classification where universality for $Q$ implies that every $\alpha\in 2 \mathcal{O}_K$ must be a sum of squares. Through enumeration and “house bound” on generators, only $\mathbb{Q}$, $\mathbb{Q}(\sqrt{5})$, and $\mathbb{Q}(\zeta_7+\zeta_7^{-1})$ survive in degrees $d\leq 5$ (Kala et al., 2018, Kim et al., 2023, Kala et al., 2024).

Table: Known Fields With Universal $\mathbb{Z}$-Forms, $d\leq5$

Field	Degree	Known Universal Form (example)
$\mathbb{Q}$	1	$x_1^2+x_2^2+x_3^2+x_4^2$
$\mathbb{Q}(\sqrt{5})$	2	$x_1^2+x_2^2+x_3^2$
$\mathbb{Q}(\zeta_7+\zeta_7^{-1})$	3	$x^2+y^2+z^2+w^2+xy+xz+xw$

3. Lifting Problem: Universality Under Field Extensions

The central lifting problem asks: for a fixed totally real field $K$ and quadratic form $Q$ defined over $\mathcal{O}_K$, for which totally real extensions $L/K$ does $Q$ remain universal over $\mathcal{O}_L$? The problem generalizes to forms $Q$ defined over a subfield $F$ and their universality after scalar extension to a larger field $K$ (Kala et al., 2021, Kala et al., 2024).

A universal form $Q$ over $K$ can only be lifted universally to fields whose ring of integers admits a basis of bounded archimedean size ("short basis" property). This results in finiteness: for fixed $K$, $Q$, and extension degree $n=[L:K]$, there are only finitely many $L$ such that $Q$ is universal over $\mathcal{O}_L$ (Kala et al., 2021, Kala et al., 2024).

Indecomposable elements and local-global criteria play a decisive role in controlling universality after lifting. The presence of non-$K$-representable indecomposables in $\mathcal{O}_L$ prohibits universal lifting.

4. Universality and Sums of Squares: Local and Global Obstructions

Universal quadratic forms are intimately connected to the ability to write elements—especially multiples of 2—as sums of squares. Critical to non-existence results is the failure of every element of $2\mathcal{O}_K$ to be a sum of squares, which is checked via explicit trace, norm, and indecomposable enumeration (Kala et al., 2024).

Local criteria—especially at dyadic places—are formulated via invariants of the lattice (Jordan splitting, Beli’s BONG invariants, $R_i$, $a_i$), yielding necessary and sufficient conditions and minimal testing sets for $n$-universality over dyadic local fields (He et al., 2022, He, 2023). Arithmetic Springer’s theorem for indefinite forms ensures descent of $n$-universality under odd-degree extensions, and spinor norm principles further refine the representation theory under field extensions.

5. Conjectures, Finiteness Results, and Recent Developments

The main conjecture, confirmed in degrees $d\leq5$, posits that only $\mathbb{Q}$, $\mathbb{Q}(\sqrt{5})$, and $\mathbb{Q}(\zeta_7+\zeta_7^{-1})$ admit universal $\mathbb{Z}$-forms among totally real fields (Kala et al., 2018, Kala et al., 2024). Kitaoka’s conjecture—that only finitely many totally real fields (or extensions of fixed degree) admit universal ternary quadratic forms—has been resolved for cyclotomic and quartic families (Kala et al., 2021, Kala et al., 2024, Kramer et al., 22 Jan 2026).

The finiteness theorem for lifting (Kala–Yatsyna, Kala–Kim–Lee) establishes that for a base field $F$ and fixed extension degree, only finitely many totally real extensions $K/F$ admit universal forms defined over $F$; explicit classification is achieved for quadratic extensions over fields of class number 1 (Kala et al., 2024).

6. Essential Proof Techniques and Computational Strategies

The methodology integrates:

• Trace form and E-type lattice analysis, minimal vector bounds (Kitaoka).
• Siegel–Zagier formulas for counting minimal vectors via Dedekind zeta values.
• Computational enumeration of fields with “small house” generators and indecomposable elements.
• Explicit local criteria via BONG invariants and Jordan splitting, enabling reduction to finitely many cases.
• Geometry of numbers arguments for bounding embedding sizes and excluding candidates.

Computational approaches enumerate defining polynomials for fields of bounded degree and house bounds, check sums of squares for $2\mathcal{O}_K$, and analyze indecomposable elements via norm and trace bounds.

7. Open Problems and Future Directions

Key open questions include:

• Full classification for degrees $d>5$ and extension families not covered by current techniques.
• Classification of universal forms over $\mathbb{Q}(\sqrt{5})$ beyond ternary forms.
• Existence or non-existence of universal forms over fields with non-principal codifferent.
• Identification of infinite towers of totally real extensions with universal forms.

Advances in explicit lattice construction, refined local-global criteria, and computational techniques are expected to drive further progress in the lifting problem and universality theory for quadratic forms (Kala et al., 2018, Kala et al., 2024).