Unboundedness of shapes of unit lattices in totally real cubic fields

TL;DR

This paper proves that the set of shapes of orders of totally real cubic fields is unbounded in the modular surface, addressing a longstanding question in the distribution of unit lattice shapes in number fields.

Contribution

It establishes the unboundedness of shapes of unit lattices in totally real cubic fields, providing new insights into their distribution in the modular surface.

Findings

Shapes of orders in totally real cubic fields are unbounded in the modular surface.

Addresses a longstanding open problem in number theory.

Advances understanding of the distribution of unit lattice shapes.

Abstract

The question of the distribution of shapes of unit lattices in number fields, pioneered by Margulis and Gromov, has lately attracted considerable interest, not least because of the lack of available results. Here we prove that the set of shapes of orders of totally real cubic fields is unbounded in the modular surface.