Collusions in Teichmüller expansions

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 Teichmüller expansion of $\alpha$ at $\mathfrak{p}$. We observe three peculiar and seemingly unrelated patterns that frequently appear in the computation of Teichmüller expansions, then develop a unifying theory to explain these patterns in terms of the dynamics of an affine group action on $\mathbb{Z}[\zeta ]$.

arXiv comment12 pages, comments welcome