# Collusions in Teichm\"uller expansions

## TL;DR

This paper investigates recurring patterns in Teichmüller expansions within cyclotomic extensions, developing a unified theory based on affine group dynamics to explain these phenomena.

## Contribution

It identifies three patterns in Teichmüller expansions and introduces a new group action framework to unify and explain these observations.

## Key findings

- Identified three recurring patterns in Teichmüller expansions.
- Developed a unifying theory using affine group actions.
- Explained the patterns through the dynamics of group actions.

## Abstract

If $\mathfrak{p} \subseteq \mathbb{Z}[\zeta]$ is a prime ideal over $p$ in the $(p^d - 1)$th cyclotomic extension of $\mathbb{Z}$, then every element $\alpha$ of the completion $\mathbb{Z}[\zeta]_\mathfrak{p}$ has a unique expansion as a power series in $p$ with coefficients in $\mu_{p^d -1} \cup \{0\}$ called the Teichm\"uller expansion of $\alpha$ at $\mathfrak{p}$. We observe three peculiar and seemingly unrelated patterns that frequently appear in the computation of Teichm\"uller expansions, then develop a unifying theory to explain these patterns in terms of the dynamics of an affine group action on $\mathbb{Z}[\zeta]$.