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

TL;DR

This paper investigates the fields of moduli for three-point G-Galois covers with cyclic p-Sylow subgroups, showing that certain higher ramification groups vanish under specific conditions, advancing understanding of their arithmetic properties.

Contribution

It extends previous work by analyzing the stable reduction and ramification behavior of Galois covers with cyclic p-Sylow subgroups, revealing new vanishing results for ramification groups.

Findings

Higher ramification groups above p vanish for the Galois closure extension

Results apply to covers with normalizer acting via an involution

Advances understanding of fields of moduli in mixed characteristic

Abstract

We continue the examination of the stable reduction and fields of moduli of G-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 P of order p^n. Suppose further that the normalizer of P acts on P via an involution. Under mild assumptions, if f: Y --> P^1 is a three-point G-Galois cover defined over C, 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.