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On the distribution of shapes of totally real multiquadratic number fields

Anuj Jakhar, Anwesh Ray

Abstract

The shape of a number field $K$ of degree $m$ is defined as the equivalence class of the lattice of integers under linear operations generated by rotations, reflections, and positive scalar dilations. It may be viewed as a point in the space of shapes $\mathscr{S}_{m-1} = \mathrm{GL}_{m-1}(\mathbb Z)\backslash \mathrm{GL}_{m-1}(\mathbb R)/\mathrm{GO}_{m-1}(\mathbb{R})$. The double quotient space is equipped with a natural measure $μ$ which is induced from the Haar measure on $\mathrm{GL}_{m-1}(\mathbb R)$. We study the distribution of shapes of totally real multiquadratic number fields of degree $m:=2^n$ in which $2$ is unramified. We show that the distribution is governed by the restriction of $μ$ to a certain torus orbit in $\mathscr{S}_{m-1}$. Our result resolves a conjecture of Haidar.

On the distribution of shapes of totally real multiquadratic number fields

Abstract

The shape of a number field of degree is defined as the equivalence class of the lattice of integers under linear operations generated by rotations, reflections, and positive scalar dilations. It may be viewed as a point in the space of shapes . The double quotient space is equipped with a natural measure which is induced from the Haar measure on . We study the distribution of shapes of totally real multiquadratic number fields of degree in which is unramified. We show that the distribution is governed by the restriction of to a certain torus orbit in . Our result resolves a conjecture of Haidar.
Paper Structure (16 sections, 20 theorems, 211 equations)

This paper contains 16 sections, 20 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Motivation and historical context
  3. Main Results
  4. Preliminary notions
  5. Shapes of number fields
  6. Integral bases of totally real multiquadratic number fields
  7. Explicit Construction of an Integral Basis
  8. The Gram Matrices and Shapes of $K_n$
  9. Case 1: Case 1
  10. Case 2: Case 2
  11. Case 3: Case 3
  12. Density results
  13. Parameterization of totally real multiquadratic extensions
  14. Volume computations
  15. Sieving
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $n\ge1$ and $\ell=2^n-1$. Fix parameters $1\le R_2\le\dots\le R_\ell$. Then where

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5: Chatelain
  • Theorem 2.6
  • Corollary 2.7
  • ...and 30 more