Unit lattices of $D_4$-quartic number fields with signature $(2,1)$

TL;DR

This paper studies the geometric distribution of unit lattices in certain quartic number fields, revealing they correspond to transcendental boundary points and identifying explicit limit points for these lattices.

Contribution

It characterizes the boundary points of unit lattice distributions in $D_4$-quartic fields with signature $(2,1)$ and provides explicit algebraic limit points.

Findings

Unit lattices correspond to transcendental boundary points.

Explicit algebraic limit points of the lattice set are identified.

The distribution of these lattices is linked to boundary behavior in a fundamental domain.

Abstract

There has been a recent surge of interest on distributions of shapes of unit lattices in number fields, due to both their applications to number theory and the lack of known results. In this work we focus on $D_4$-quartic fields with signature $(2,1)$; such fields have a rank $2$ unit group. Viewing the unit lattice as a point of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$, we prove that every lattice which arises this way must correspond to a transcendental point on the boundary of a certain fundamental domain of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$. Moreover, we produce three explicit (algebraic) points of $GL_2(\mathbb{Z})\backslash \mathfrak{h}$ which are limit points of the set of (points associated to) unit lattices of $D_4$-quartic fields with signature $(2,1)$.