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Shapes of unit lattices in $D_p$-number fields

Robert Harron, Erik Holmes, Sameera Vemulapalli

TL;DR

The paper addresses the problem of describing the unit lattice shapes of degree $p$ number fields whose Galois closure has dihedral group $D_p$ and a single real embedding. By combining Dirichlet unit theory, the logarithmic embedding, and Moser’s explicit $\mathbb{Z}[D_p]$-module structure, the authors prove that these shapes sit inside a finite union of translates of periodic torus orbits in the appropriate shape space, with the quintic case ($p=5$) yielding a single translate along a hypercycle. The results extend from $p=5$ to all primes $p$, revealing a uniform torus-orbit structure that governs unit shapes across dihedral dihedral extensions, and they connect the regulator to norm forms from the totally real subfield $\mathbf{Q}^+$. These findings illuminate constrained geometric patterns in unit lattices and have potential implications for lattice-based cryptography by clarifying the shape-geometry landscape of unit groups in higher-rank settings.

Abstract

The unit group of the ring of integers of a number field, modulo torsion, is a lattice via the logarithmic Minkowski embedding. We examine the shape of this lattice, which we call the unit shape, within the family of prime degree $p$ number fields whose Galois closure has dihedral Galois group $D_p$ and a unique real embedding. In the case $p = 5$, we prove that the unit shapes lie on a single hypercycle on the modular surface (in this case, the modular surface is the space of shapes of rank $2$ lattices). For general $p$, we show that the unit shapes are contained in a finite union of translates of periodic torus orbits in the space of shapes.

Shapes of unit lattices in $D_p$-number fields

TL;DR

The paper addresses the problem of describing the unit lattice shapes of degree number fields whose Galois closure has dihedral group and a single real embedding. By combining Dirichlet unit theory, the logarithmic embedding, and Moser’s explicit -module structure, the authors prove that these shapes sit inside a finite union of translates of periodic torus orbits in the appropriate shape space, with the quintic case () yielding a single translate along a hypercycle. The results extend from to all primes , revealing a uniform torus-orbit structure that governs unit shapes across dihedral dihedral extensions, and they connect the regulator to norm forms from the totally real subfield . These findings illuminate constrained geometric patterns in unit lattices and have potential implications for lattice-based cryptography by clarifying the shape-geometry landscape of unit groups in higher-rank settings.

Abstract

The unit group of the ring of integers of a number field, modulo torsion, is a lattice via the logarithmic Minkowski embedding. We examine the shape of this lattice, which we call the unit shape, within the family of prime degree number fields whose Galois closure has dihedral Galois group and a unique real embedding. In the case , we prove that the unit shapes lie on a single hypercycle on the modular surface (in this case, the modular surface is the space of shapes of rank lattices). For general , we show that the unit shapes are contained in a finite union of translates of periodic torus orbits in the space of shapes.
Paper Structure (24 sections, 5 theorems, 36 equations, 1 figure)

This paper contains 24 sections, 5 theorems, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Setup
  4. Statement of results
  5. The quintic case
  6. Generalizing to all primes $p$
  7. Overview of previous work
  8. Cryptographic context
  9. Acknowledgments
  10. Preliminaries
  11. Proof of \ref{['hypercycle']} and \ref{['5regulator']}
  12. The structure of $E_K$
  13. The logarithmic embedding
  14. Proof of \ref{['hypercycle']}
  15. Proof of \ref{['5regulator']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $K$ be a $D_5$-extension with a unique real embedding. Then, the unit shape of $K$ lies on the arc of the circle from $\frac{1+i\sqrt{3}}{2}$ to $-\frac{1}{2}+i\frac{5}{2\sqrt{3}}$ in $\mathrm{GL}_2(\mathbf{Z})\backslash \mathfrak{H}$.

Figures (1)

  • Figure 1: A $D_5$-extension with one real embedding has a rank $2$ unit lattice. For each of the 5422 such fields in the LMFDB (see LMFDB), we computed a certain basis for the unit lattice and plotted the corresponding point in the upper half plane above. Observe a striking pattern: the resulting points lie on an arc of the circle $\left(x+\frac{1}{2}\right)^2+\left(y-\frac{1}{2\sqrt{3}}\right)^2=\left(\frac{2}{\sqrt{3}}\right)^2$.

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Proposition 4.1
  • proof