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On the distribution of shapes of sextic pure number fields

Anuj Jakhar, Ravi Kalwaniya, Anwesh Ray, Bidisha Roy

TL;DR

This work analyzes the distribution of shapes for pure sextic number fields K = \\mathbb{Q}(\\sqrt[6]{m}) ordered by discriminant. By partitioning into 20 Types via local conditions at 2 and 3 and computing explicit integral bases, the authors express each shape as a fixed Type-specific conjugation of a triple (λ1, λ2, λ3). They prove equidistribution of shapes on translated torus orbits in the shape space \\mathcal{S}_5, with limiting measures that split into continuous and discrete components, and they provide an analogous distribution result for the discrete parameters (a5/a1, a3, a2a4). The methodology blends geometry-of-numbers with analytic sieve methods to count lattice points subject to arithmetic constraints, yielding precise asymptotics and local factors. The results illuminate a new phenomenon in shape statistics and lay groundwork for extending to higher-degree pure fields or other structured families.

Abstract

The shape of a number field $K$ of degree $n$ is defined as the equivalence class of the lattice of integers with respect to linear operations that are composites of rotations, reflections, and positive scalar dilations. The shape is a point in the space of shapes $\mathscr{S}_{n-1}$, which is the double quotient $\mathrm{GL}_{n-1}(\mathbb{Z}) \backslash \mathrm{GL}_{n-1}(\mathbb{R}) / \mathrm{GO}_{n-1}(\mathbb{R})$. We investigate the distribution of shapes of pure sextic number fields $K=\mathbb{Q}(\sqrt[6]{m})$, ordered by absolute discriminant. Such fields are partitioned into $20$ distinct Types determined by local conditions at $2$ and $3$, and an explicit integral basis is given in each case. For each Type, the shape of $K$ admits an explicit description in terms of shape parameters. Fixing the sign of $m$ and a Type, we prove that the corresponding shapes are equidistributed along a translated torus orbit in the space of shapes. The limiting distribution is given by an explicit measure expressed as the product of a continuous measure and a discrete measure.

On the distribution of shapes of sextic pure number fields

TL;DR

This work analyzes the distribution of shapes for pure sextic number fields K = \\mathbb{Q}(\\sqrt[6]{m}) ordered by discriminant. By partitioning into 20 Types via local conditions at 2 and 3 and computing explicit integral bases, the authors express each shape as a fixed Type-specific conjugation of a triple (λ1, λ2, λ3). They prove equidistribution of shapes on translated torus orbits in the shape space \\mathcal{S}_5, with limiting measures that split into continuous and discrete components, and they provide an analogous distribution result for the discrete parameters (a5/a1, a3, a2a4). The methodology blends geometry-of-numbers with analytic sieve methods to count lattice points subject to arithmetic constraints, yielding precise asymptotics and local factors. The results illuminate a new phenomenon in shape statistics and lay groundwork for extending to higher-degree pure fields or other structured families.

Abstract

The shape of a number field of degree is defined as the equivalence class of the lattice of integers with respect to linear operations that are composites of rotations, reflections, and positive scalar dilations. The shape is a point in the space of shapes , which is the double quotient . We investigate the distribution of shapes of pure sextic number fields , ordered by absolute discriminant. Such fields are partitioned into distinct Types determined by local conditions at and , and an explicit integral basis is given in each case. For each Type, the shape of admits an explicit description in terms of shape parameters. Fixing the sign of and a Type, we prove that the corresponding shapes are equidistributed along a translated torus orbit in the space of shapes. The limiting distribution is given by an explicit measure expressed as the product of a continuous measure and a discrete measure.
Paper Structure (14 sections, 43 theorems, 215 equations, 1 figure)

This paper contains 14 sections, 43 theorems, 215 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation and historical context
  3. Main results
  4. Methodology
  5. Outlook
  6. Organization
  7. Preliminaries
  8. Notation
  9. Shapes of number fields
  10. Integral bases of pure number fields
  11. Integral bases of pure sextic fields
  12. Computing the shape of pure sextic fields
  13. Equidistribution results for (lambda1, lambda2, lambda3)
  14. Distribution of the parameters $(a_5/a_1,\, a_2a_4,\, a_3)$

Key Result

Theorem A

With respect to the notation above, one has where $\hat{\mu}_{i,j}$ is an explicit measure on $\mathbb{R}^3$ given by mu i,j defn which depends only on the Type $(i,j)$. Further, $\hat{\mu}_{i,j}$ is a product of continuous measures in the first and second coordinates and a discrete measure on the third coordinate.

Figures (1)

  • Figure 1: The fundamental domain $\mathcal{F}_2 \subset \mathbb{H}$ for shapes of rank-$2$ lattices.

Theorems & Definitions (66)

  • Theorem A: Theorem \ref{['main thm']}
  • Theorem B: Theorem \ref{['main thm 2']}
  • Definition 2.1
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 56 more