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

TL;DR

This paper studies the stable reduction of three-point Galois covers with cyclic p-Sylow subgroups over fields of mixed characteristic, extending previous results and describing stable models for certain covers.

Contribution

It extends the understanding of the fields of moduli for three-point G-Galois covers with cyclic p-Sylow groups, especially for p-solvable groups, and describes stable models for Z/p^n-covers.

Findings

Higher ramification groups above p vanish for the field of moduli.

Complete description of stable models for Z/p^n-covers.

Extension of Beckmann and Wewers' work on ramification.

Abstract

We examine in detail the stable reduction of Galois covers of the projective line over a complete discrete valuation field of mixed characteristic (0, p), where G has a cyclic p-Sylow subgroup of order p^n. If G is further assumed to be p-solvable (i.e., G has no nonabelian simple composition factors with order divisible by p), we obtain the following consequence: Suppose f: Y --> P^1 is a three-point G-Galois cover defined over the complex numbers. Then the nth higher ramification groups above p for the upper numbering of the (Galois closure of the) extension K/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 Z/p^n-cover, where p > 2.