Prime-to-$p$ étale fundamental groups of punctured projective lines over strictly Henselian fields

Abstract:

Let $K$ be the fraction field of a strictly Henselian DVR of residue characteristic $p \geq 0$ with algebraic closure $\bar {K}$, and let $\alpha _{1}, \ldots , \alpha _{d} \in \mathbb {P}_{K}^{1}(K)$. In this paper, we give explicit generators and relations for the prime-to-$p$ étale fundamental group of $\mathbb {P}_K^1\smallsetminus \{\alpha _1,\ldots ,\alpha _d\}$ that depend (solely) on their intersection behavior. This is done by a comparison theorem that relates this situation to a topological one. Namely, let $a_{1}, \ldots , a_{d}$ be a 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 $\operatorname {Gal}(\bar {K} / K)$ on the prime-to-$p$ étale fundamental group of $\mathbb {P}_{\bar {K}} \smallsetminus \{\alpha _{1}, \ldots , \alpha _{d}\}$ to the topological action of looping $z_0$ around the origin on the fundamental group of $\mathbb {P}_{\mathbb {C}}^1\smallsetminus \{a_1(z_0),\ldots ,a_d(z_0)\}$. 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 $\mathbb {P}_{\bar K}^1\smallsetminus \{\alpha _1,\ldots ,\alpha _d\}$ satisfies that its field of moduli (as a $G$-Galois cover) has degree over $K$ dividing the exponent of $G / Z(G)$.