Prime-to-$p$ \'etale fundamental groups of punctured projective lines over strictly Henselian fields

Abstract

Let $K$ be the fraction field of a strictly Henselian DVR of characteristic $p \geq 0$ with algebraic closure $\bar{K}$, and let $\alpha_{1}, ..., \alpha_{d} \in \mathbb{P}_{K}^{1}(K)$. In this paper, we give explicit generators and relations for the prime-to-$p$ \'etale fundamental group of $\mathbb{P}_K^1\smallsetminus\{\alpha_1,...,\alpha_d\}$ that depend (solely) on their intersection behavior. This is done by a comparison theorem that relates this situation to a topological one. Namely, let $a_{1}, ..., a_{d}$ be distinct power series in $\mathbb{C}[[x]]$ with the same intersection behavior as the $\alpha_i$'s, converging on an open disk centered at $0$, and choose a point $z_{0} \neq 0$ lying in this open disk. We compare the natural action of $\mathrm{Gal}(K)$ on the prime-to-$p$ \'etale fundamental group of $\mathbb{P}_{\bar{K}} \smallsetminus \{\alpha_{1}, ..., \alpha_{d}\}$ to the topological action of looping $z_0$ around the origin on the fundamental group of $\mathbb{P}_{\mathbb{C}}^1 \smallsetminus \{a_1(z_0),...,a_d(z_0)\}$. This latter action is, in turn, interpreted in terms of Dehn twists. A corollary of this result is that every prime-to-$p$ $G$-Galois cover of $\mathbb{P}_{\bar K}^1 \smallsetminus \{\alpha_1,...,\alpha_d\}$ satisfies that its field of moduli (as a $G$-Galois cover) has degree over $K$ dividing the exponent of $G / Z(G)$.