Shape of a Number Field

• Shape of a number field is the equivalence class of the trace-zero lattice under scaling, orthogonal transformations, and basis changes.
• It refines classical invariants like the discriminant, revealing nuanced arithmetic and geometric structures in different field families.
• Equidistribution results show shapes are parametrized by moduli spaces, offering a complete invariant for isomorphism classes in many settings.

The shape of a number field is a fine Euclidean invariant encoding the isomorphism class, up to scale and orthogonal transformations, of the trace-zero sublattice of the ring of integers under the Minkowski embedding. The shape serves as a refinement of classical invariants such as discriminant, elucidates the geometry of arithmetic lattices in relation to automorphic spaces, and occupies a central role in the modern arithmetic statistics of fields, particularly in equidistribution and rigidity-type questions.

1. Formal Definition and Construction

Let $K$ be a number field of degree $n$ with ring of integers $\mathcal{O}_K$. The Minkowski embedding

$$j: K \hookrightarrow \mathbb{R}^r \times \mathbb{C}^s \simeq \mathbb{R}^n$$

is defined via the $r$ real and $s$ pairs of complex embeddings ($r+2s=n$), mapping $x\in K$ to the tuple of its images under these embeddings. The full lattice $j(\mathcal{O}_K)$ in $\mathbb{R}^n$ is equipped with the positive-definite inner product

$$\langle x,y\rangle = \sum_{i=1}^r \sigma_i(x)\sigma_i(y) + 2\sum_{j=1}^s \operatorname{Re}\left(\tau_j(x)\overline{\tau_j(y)}\right)$$

where $\sigma_i$ are real, $\tau_j$ complex embeddings.

Since $j(1)$ spans a "trivial" direction present in every field of degree $n$, comparison of shapes is performed on the $(n-1)$-dimensional trace-zero subspace. Define

$$\mathcal{O}_K^0 := \{x \in \mathcal{O}_K : \operatorname{Tr}_{K/\mathbb{Q}}(x) = 0\}$$

and consider the projected lattice $j(\mathcal{O}_K^0) \subset \mathbb{R}^{n-1}$. The shape of $K$ is the equivalence class of the Gram matrix of this lattice under the group

$$\mathrm{GL}_{n-1}(\mathbb{Z}) \text{ (basis change)},\quad \mathrm{GO}_{n-1}(\mathbb{R}) \text{ (orthogonal similitude)},\quad \text{and scaling}$$

yielding a well-defined point in the locally symmetric space

$$\mathscr{S}_{n-1} = \mathrm{GL}_{n-1}(\mathbb{Z})\backslash \mathrm{GL}_{n-1}(\mathbb{R})/\mathrm{GO}_{n-1}(\mathbb{R})$$

which can be modeled as the moduli of positive-definite symmetric $(n-1)\times(n-1)$ matrices up to scaling and integral equivalence (Das et al., 30 Jun 2025, Jakhar et al., 14 Jan 2026, Hough, 2017, Harron, 2015).

2. Arithmetic and Geometric Interpretation

The shape is a geometric invariant strictly finer than the discriminant: number fields of the same degree and discriminant can have non-isomorphic shapes, particularly in families distinguished by their Galois structure or ramification type (Mantilla-Soler et al., 2019, H et al., 2019). The shape records the "angle structure" of the trace-zero sublattice and, via its parametrization, interfaces with the theory of lattices and arithmetic quotients of symmetric spaces.

For fields of small degree:

• Degree $n=3$: The shape lives in $$\mathscr{S}_2 \cong \mathrm{SL}_2(\mathbb{Z})\backslash \mathbb{H}$$, the modular surface (Harron, 2015).
• Degree $n=4$: The shape lies in $$\mathscr{S}_3 = \mathrm{GL}_3(\mathbb{Z})\backslash \mathrm{Sym}_{>0}^3/\mathbb{R}^\times$$, the moduli of oriented unimodular $3$-lattices (Das et al., 30 Jun 2025, Hough, 2017).

The analog for the group of units (log-lattice of units via Dirichlet’s theorem) yields an allied notion of shape, pertinent for Mahler measure and logarithmic regulator distribution (Corso et al., 22 Feb 2025).

3. Explicit Computation and Algebraic Invariants

The shape, concretely, is computed via the trace pairing restricted to $\mathcal{O}_K^0$: $q(x) = \operatorname{Tr}_{K/\mathbb{Q}}(x^2)$ with the Gram matrix

$G_{ij} = \operatorname{Tr}_{K/\mathbb{Q}}(v_i v_j)$

for any $\mathbb{Z}$-basis $v_1,\dots,v_{n-1}$ of $\mathcal{O}_K^0$, then passed to its similarity class (Das et al., 30 Jun 2025, Bolaños et al., 2019, Mantilla-Soler et al., 2013). In the context of Galois, cyclic, or "pure" number fields, explicit algebraic expressions for the Gram matrix and integral basis permit direct analysis of the shape locus. For tame cyclic fields, the Gram matrix is determined by the degree and discriminant (Bolaños et al., 2019). For pure prime degree fields, two families (orthorhombic and parallelogram-like) arise, depending on ramification at the prime (Holmes, 2022, Harron, 2015).

In families of pure extensions $K = \mathbb{Q}(\root{n}\of{a})$, the shape depends on congruence classes of $a$ modulo a sharp period $M(n)=n\,\mathrm{rad}(n)$ for $n$, determined entirely by local-to-global principles (Nguyen-Dang, 11 Sep 2025).

4. Classification Phenomena and the Shape as an Invariant

Recent work has established that under various natural conditions, the shape is a complete invariant for the isomorphism class of a number field within a given family:

• For pure cubic and quartic fields, and more generally, for totally real fields of fundamental discriminant and degree $n$ such that $(\mathbb{Z}/n\mathbb{Z})^*$ is cyclic, the shape distinguishes isomorphism classes (Harron, 2015, Mantilla-Soler et al., 2019, Das et al., 30 Jun 2025, Holmes, 2022).
• In Galois $V_4$-quartic and tame cyclic fields, the shape is determined by the discriminant (and the degree), and is a complete arithmetic invariant in these settings (H et al., 2019, Bolaños et al., 2019).

This rigidity fails in more general families, notably those admitting fields with coincident trace-zero forms but distinct full trace pairings, especially in non-fundamental discriminant cases (Mantilla-Soler et al., 2019).

5. Distribution of Shapes: Equidistribution and Escape of Mass

Modern arithmetic statistics investigates the distribution of shapes as families of number fields are ordered by invariants (e.g., discriminant, regulator). A paradigm emerges:

• In many families, e.g., pure cubics, pure quartics, pure sextics, shapes lie on explicit parameterized torus orbits or submanifolds within $\mathscr{S}_{n-1}$ (Das et al., 30 Jun 2025, Jakhar et al., 14 Jan 2026, Holmes, 2022, Harron, 2015).
• The asymptotic distribution of shapes (with respect to discriminant) is governed by measures of the form $\prod dx_i/x_i$ on these tori, commonly a product of a continuous density (Lebesgue over certain shape parameters) and a discrete measure reflecting local arithmetic conditions (Das et al., 30 Jun 2025, Jakhar et al., 14 Jan 2026).
• In generic ("random") families (e.g., $S_n$-fields), equidistribution in the full shape space has been established, while in Galois or abelian families, shapes may only fill lower-dimensional loci (Hough, 2017, H et al., 2019).

A representative phenomenon is the regularized equidistribution: while counts of fields with shapes in a compact set grow more slowly than the total number of fields, normalization recovers a weak convergence to a natural measure (hyperbolic in dimension 2, Haar-type on higher-dimensional tori) (Harron, 2015, Das et al., 30 Jun 2025).

Main equidistribution results include:

• Cuspidal equidistribution for quartic fields and their resolvents (Hough, 2017).
• Distribution of shapes for pure cubic and higher degree fields on parameterized curves or tori (Harron, 2015, Das et al., 30 Jun 2025, Jakhar et al., 14 Jan 2026).
• Unboundedness and (recently) density of shapes of unit lattices in totally real cubic fields (Corso et al., 22 Feb 2025).

6. Ramification, Galois Structure, and Shape Loci

Ramification data and Galois group play decisive roles in the geometry of the shape locus:

• In pure and cyclic extensions, the wild/tame ramification dichotomy produces distinct subspaces for the shape, typically orthorhombic (wild) or parallelogram-type (tame) (Mantilla-Soler et al., 2013, Holmes, 2022, H et al., 2019).
• For Galois quartic fields, $V_4$-fields exhibit shapes in 2-dimensional orthorhombic loci parameterized by quadratic subfields, while $C_4$-quartics correspond to one-dimensional tetragonal lines, often discretely distributed (H et al., 2019).
• Distribution and counting of shapes are deeply intertwined with local-to-global principles and residue class periodicities, especially in the context of integral basis structure for pure fields (Nguyen-Dang, 11 Sep 2025).

7. Open Problems and Recent Breakthroughs

Key outstanding questions include:

• The general equidistribution of shapes when ordered by discriminant in families where the ambient volume is infinite (notably beyond degree 3/4), and the associated escape of mass phenomena (Corso et al., 22 Feb 2025).
• The explicit description and classifying power of the shape for fields of composite, non-cyclic degree, particularly for non-Galois extensions (Mantilla-Soler et al., 2019).
• Structure and geometry of the closure of the set of shapes (e.g., density, topology, fractality) in the full moduli space for diverse arithmetic families (Corso et al., 22 Feb 2025, Das et al., 30 Jun 2025).

The field continues to experience rapid progress, with recent results demonstrating, for instance, the unboundedness of shapes of unit lattices in totally real cubic fields (Corso et al., 22 Feb 2025), explicit periodicity laws for integral bases in pure extensions (Nguyen-Dang, 11 Sep 2025), and the fine structure and limiting distributions for higher-degree pure fields (Jakhar et al., 14 Jan 2026, Das et al., 30 Jun 2025).

---

Table: Summary of Key Families and Shape Loci

Family	Shape Space/Locus	Invariant power
Pure cubic fields	1-parameter curves in $\mathscr{S}_2$	Complete within family
Pure quartic fields	2-parameter tori in $\mathscr{S}_3$	Complete within fixed type
Galois quartic, $V_4$	2D orthorhombic loci	Complete in tame/real
Cyclic fields, degree $n$ (tame)	Discrete by degree/discriminant	Complete (tame)
Pure prime degree, $p$	$\ell$-dim subspaces ($\ell = (p-1)/2$)	Complete within wild/tame
Unit groups in totally real cubic	Modular surface $\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}$	Not compact, unbounded

The shape of a number field thus serves as a central object at the intersection of arithmetic invariant theory, the geometry of numbers, and modern ergodic and analytic number theory, with explicit classification, equidistribution, and rigidity results now available in a range of key families. For further details and explicit computations see (Harron, 2015, Das et al., 30 Jun 2025, Hough, 2017, Bolaños et al., 2019, Holmes, 2022, Mantilla-Soler et al., 2013, H et al., 2019, Corso et al., 22 Feb 2025, Nguyen-Dang, 11 Sep 2025, Mantilla-Soler et al., 2019, Jakhar et al., 14 Jan 2026).