Geometry of Biquadratic and Cyclic Cubic Log-Unit Lattices

TL;DR

This paper explores the geometric structure of log-unit lattices in biquadratic and cyclic cubic number fields, revealing conditions for orthogonality and demonstrating that these lattices are always equilateral triangles.

Contribution

It characterizes the geometry of log-unit lattices in specific number fields, identifying orthogonality conditions and proving the universal equilateral triangular structure in cyclic cubic cases.

Findings

Log-unit lattice is orthogonal in certain biquadratic fields.

Log-unit lattice is always an equilateral triangle in cyclic cubic fields.

Provides geometric criteria for lattice properties in these fields.

Abstract

By Dirichlet's Unit Theorem, under the log embedding the units in the ring of integers of a number field form a lattice, called the log-unit lattice. We investigate the geometry of these lattices when the number field is a biquadratic or cyclic cubic extension of $\mathbb{Q}$. In the biquadratic case, we determine when the log-unit lattice is orthogonal. In the cyclic cubic case, we show that the log-unit lattice is always equilateral triangular.