Unboundedness of shapes of unit lattices in totally real cubic fields
Emilio Corso, Federico Rodriguez Hertz
TL;DR
The paper proves that the set of shapes of unit lattices for orders in totally real cubic fields is unbounded in the modular surface. It builds a concrete family of cubic orders with linear fundamental units by examining polynomials $f_{a,b,t}(X)=f_0(X)+tX(aX-b)$ and tracking two key roots that approximate $0$ and $a^{-1}$ as $t\to\infty$. By selecting $a_t$ on the scale $t^{\alpha}$ with $0<\alpha<1/4$ and establishing sharp root asymptotics, the authors construct a curve of shapes escaping to the cusp, using a Cusick-type discriminant/regulator analysis to certify fundamental-unit structure. This yields an explicit topological witness to the unboundedness of shapes in the modular surface, contributing to the broader understanding of unit lattices in number fields. The approach complements related density results and highlights a smooth-curve accumulation mechanism toward the cusp in the shape space.
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.