Fields of moduli of three-point $G$-covers with cyclic $p$-Sylow, I

Abstract

We examine in detail the stable reduction of $G$-Galois covers of the projective line over a complete discrete valuation field of mixed characteristic $\left(0,p\right)$, where $G$ has a cyclic $p$-Sylow subgroup of order $p^n$. If $G$ is further assumed to be $p$-solvable (that is, $G$ has no nonabelian simple composition factors with order divisible by $p$), we obtain the following consequence: Suppose $f:Y\to {\mathbb{P}}^1$ is a three-point $G$-Galois cover defined over $ℂ$. Then the $n$-th higher ramification groups above $p$ for the upper numbering for the extension $K∕\mathbb{Q}$ vanish, where $K$ is the field of moduli of $f$. This extends work of Beckmann and Wewers. Additionally, we completely describe the stable model of a general three-point $\mathbb{Z}∕p^n$-cover, where $p>2$.