Shapes of unit lattices and escape of mass

TL;DR

This paper investigates the distribution of units in totally real cubic fields on the modular surface, conjecturing density, identifying explicit curves in the closure, and exploring escape of mass phenomena.

Contribution

It introduces explicit families of orders generalizing simplest cubic fields and analyzes their unit groups to study distribution and escape of mass on the modular surface.

Findings

Closure of the set contains countably many explicit curves.

Conjecture that the set of points is dense.

Provides a strategy to prove the set has non-empty interior.

Abstract

We study the collection of points on the modular surface obtained from the logarithm embeddings of the groups of units in totally real cubic number fields. We conjecture that this set is dense and show that its closure contains countably many explicit curves and give a strategy to show that it has non-empty interior. The results are obtained by constructing explicit families of orders (generalizing the so called simplest cubic fields) and calculating their groups of units. We also address the question of escape of mass for the compact orbits of the diagonal group associated to these orders.