# Twisters and signed fundamental domains for number fields

## TL;DR

This paper constructs an explicit signed fundamental domain for the action of totally positive units on a specific real and complex space of a number field, using twisters to control cone geometry and intersections.

## Contribution

It introduces a novel method using twisters to explicitly construct signed fundamental domains for unit actions in number fields.

## Key findings

- Constructed explicit signed fundamental domains for number field units.
- Used twisters to control cone geometry and intersections.
- Provided a method applicable to non-totally complex fields.

## Abstract

We give a signed fundamental domain for the action on $\mathbb{R}^{r_1}_+\times{\mathbb{C}^*}^{r_2}$ of the totally positive units $E_+$ of a number field $k$ of degree $n=r_1+2r_2$ which we assume is not totally complex. Here $r_1$ and $r_2$ denote the number of real and complex places of $k$ and $\mathbb{R}_+$ denotes the positive real numbers. The signed fundamental domain consists of $n$-dimensional $k$-rational cones $C_\alpha$, each equipped with a sign $\mu_\alpha=\pm1$, with the property that the net number of intersections of the cones with any $E_+$-orbit is 1. The cones $C_\alpha$ and the signs $\mu_\alpha$ are explicitly constructed from any set of fundamental totally positive units and a set of $3^{r_2}$ "twisters", i.e. elements of $k$ whose arguments at the $r_2$ complex places of $k$ are sufficiently varied. Introducing twisters gives us the right number of generators for the cones $C_\alpha $ and allows us to make the $C_\alpha$ turn in a controlled way around the origin at each complex embedding.