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

Obus, Andrew

doi:10.2140/ant.2012.6.833

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

Andrew Obus

Vol. 6 (2012), No. 5, 833–883

DOI: 10.2140/ant.2012.6.833

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 $(0, p)$ , 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 $f : Y \to ℙ^{1}$ 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 $K ∕ ℚ$ 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 $p > 2$ .

Keywords

Field of moduli, stable reduction, Galois cover

Mathematical Subject Classification 2000

Primary: 14H30

Secondary: 14G20, 14G25, 14H25, 11G20, 11S20

Milestones

Received: 9 December 2009

Revised: 22 September 2011

Accepted: 4 November 2011

Published: 31 July 2012

Authors

Andrew Obus
Columbia University Department of Mathematics MC4403 2990 Broadway New York, NY 10027 United States

Vol. 6, No. 5, 2012