On the distribution of shapes of octic Kummer extensions

1. Problem Statement and Motivation

Context:
In algebraic number theory, the arithmetic properties of a number field $K$ are often studied via the geometry of its ring of integers $\mathcal{O}_K$. While the discriminant $\Delta_K$ is a standard invariant encoding ramification, it is a scalar quantity that fails to capture the finer geometric structure of the lattice $\Lambda_K$ formed by the Minkowski embedding of $\mathcal{O}_K$. To address this, the concept of the "shape" of a number field was introduced. The shape is the equivalence class of the projected lattice (onto the trace-zero hyperplane) under the action of orthogonal similitudes (rotations, reflections, and scaling).

The Specific Problem:
The authors investigate the distribution of shapes for a specific, non-generic family of number fields: Octic Kummer extensions of the form $L = \mathbb{Q}(i, \sqrt[4]{m})$, where $m \in \mathbb{Z}[i]$ is fourth-power-free.

• Non-generic nature: Unlike generic number fields (where Galois groups are symmetric groups $S_n$) which are conjectured to have equidistributed shapes in the full shape space, "thin" families with specific Galois structures often have shapes constrained to lower-dimensional submanifolds.
• Complexity: Since $L$ contains $\mathbb{Q}(i)$, the lattice $\Lambda_L$ contains the vectors $J(1)$ and $J(i)$. This imposes two linear constraints, reducing the effective rank of the shape from $n-1=7$ to $n-2=6$. The authors must define a relative shape by projecting onto the orthogonal complement of the plane spanned by $J(1)$ and $J(i)$.

2. Methodology

The paper employs a combination of algebraic number theory, lattice geometry, and analytic number theory (geometry of numbers).

A. Algebraic Setup and Integral Bases

• Field Structure: Let $K = \mathbb{Q}(i)$ and $L = K(\alpha)$ with $\alpha^4 = m$. The authors decompose $m$ into square-free, pairwise coprime Gaussian integers: $m = f g^2 h^3$.
• Integral Bases: Relying on the recent classification by Venkataraman and Kulkarni [VK24], the authors explicitly determine integral bases for $\mathcal{O}_L$ over $\mathbb{Z}[i]$. These bases depend on the congruence classes of $f, g, h$ modulo powers of $(1+i)$. There are 12 distinct cases (labeled 1–12) based on these congruences.

B. Geometric Analysis (Gram Matrices)

• Minkowski Embedding: The authors construct the Minkowski embedding $J: L \to \mathbb{C}^8 \cong \mathbb{R}^8$.
• Projection: They define the projected lattice $J(\mathcal{O}_L)'$ by orthogonally projecting onto the subspace orthogonal to $J(1)$ and $J(i)$.
• Shape Parameters: By computing the Gram matrices for the projected lattices in each of the 12 cases, they identify that the asymptotic shape is governed by two fundamental real invariants (shape parameters):
  $$\lambda_1 = \frac{|f|}{|h|}, \quad \lambda_2 = |g|$$
  where $|\cdot|$ denotes the complex norm.
• Asymptotic Stability: The authors prove that as $|m| \to \infty$ (and thus the discriminant $\Delta_L \to \infty$), the normalized Gram matrices converge to diagonal or block-diagonal forms determined solely by $\lambda_1$ and $\lambda_2$. This establishes that the shape of the field is asymptotically a function of these parameters.

C. Analytic Number Theory (Distribution)

• Counting Problem: The goal is to count the number of triples $(f, g, h)$ with bounded "height" $H(f, g, h) = |f||g|^2|h|^3 \le X$ such that the resulting shape parameters $(\lambda_1, \lambda_2)$ fall within a specific rectangle $R \subset (0, \infty)^2$.
• Strongly Carefree Condition: The authors restrict attention to "strongly carefree" triples, where $f, g, h$ are square-free and pairwise coprime.
• Geometry of Numbers: They use Davenport's lemma to estimate the number of lattice points in the region defined by the height and shape constraints.
• Sieve Methods: To handle the square-free and coprimality conditions, they employ inclusion-exclusion principles and local density calculations over the Gaussian integers $\mathbb{Z}[i]$.

3. Key Contributions

1. Definition of Relative Shape for Octic Fields: The paper successfully extends the notion of shape to octic fields containing a quadratic subfield by defining a projection that removes the constraints imposed by the subfield's units.
2. Explicit Shape Parameters: The identification of $\lambda_1 = |f|/|h|$ and $\lambda_2 = |g|$ as the governing invariants for the shape of octic Kummer extensions. This generalizes previous results for pure prime-degree fields.
3. Classification of Integral Bases: The paper provides a comprehensive computational framework linking the 12 congruence cases of $m$ to explicit Gram matrices, demonstrating that despite the complexity of the bases, the asymptotic shape behavior is uniform.
4. Limiting Distribution Measure: The derivation of an explicit limiting measure for the distribution of shapes, which factors into a continuous part (Archimedean) and a discrete part (arithmetic/congruence conditions).

4. Main Results

Theorem A (Main Distribution Result):
Let $N(R; X)$ be the number of strongly carefree triples $(f, g, h)$ with height $H(f, g, h) \le X$ such that the shape parameters $(\lambda_1, \lambda_2)$ lie in a rectangle $R$. As $X \to \infty$:
$$\lim_{X \to \infty} \frac{N(R; X)}{X} = \int_R \hat{\mu}$$
where $\hat{\mu} = \mu_\infty \times \mu_{sf}$ is a product measure on $(0, \infty)^2$.

• Continuous Measure ($\mu_\infty$): Derived from the geometry of numbers, representing the distribution of the continuous parameters.
  $$d\mu_\infty(x) = C_\infty \, dx$$
  where $C_\infty$ is a constant involving an Euler product over primes in $\mathbb{Z}[i]$.
• Discrete Measure ($\mu_{sf}$): A measure on the square-free integers $g$ (corresponding to $\lambda_2$), assigning mass:
  $$\alpha(g) = \frac{1}{|g|^2} \prod_{p|g} \left( \frac{\text{Norm}(p)}{\text{Norm}(p) + 2} \right)$$
  This reflects the local arithmetic constraints (square-freeness and coprimality).

Interpretation:
The limiting distribution is not the uniform (Haar) measure on the shape space. Instead, the shapes are equidistributed on specific loci defined by the parameters $(\lambda_1, \lambda_2)$, with a density determined by the product of a continuous geometric factor and a discrete arithmetic factor.

5. Significance

• Non-Generic Families: This work provides a rigorous example of how "thin" families of number fields (those with non-generic Galois groups) deviate from the equidistribution conjecture expected for generic families. It confirms that arithmetic constraints force shapes into lower-dimensional subsets.
• Geometric Mirror: The results illustrate the paper's thesis that the shape acts as a "precise geometric mirror" of the algebraic structure. The specific decomposition of the measure into continuous and discrete components directly reflects the interplay between the Archimedean size of the generator and the local congruence conditions of the defining polynomial.
• Extension of Literature: It extends the work of Harron (on cubic and pure fields) and Holmes (on pure fields of prime degree) to composite degrees and non-abelian Galois extensions (specifically octic Kummer extensions), filling a gap in the systematic study of shape distributions for non-abelian families.
• Methodological Framework: The explicit computation of Gram matrices for 12 distinct cases and the subsequent asymptotic analysis provide a template for studying shapes in other complex Kummer or cyclotomic extensions.