Shapes of unit lattices in $D_p$-number fields

TL;DR

This paper studies the geometric shape of the unit lattice in certain prime degree dihedral number fields, revealing specific patterns and structures within the space of lattice shapes.

Contribution

It characterizes the shape of the unit lattice in $D_p$-number fields, showing they lie on a hypercycle for p=5 and in finite unions of torus orbit translates for general p.

Findings

Unit shapes for p=5 lie on a hypercycle on the modular surface.

For general p, unit shapes are contained in finite unions of torus orbit translates.

The work connects algebraic number theory with geometric structures in the shape space.

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.