Vol. 6, No. 5, 2012

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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
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 $f:Y\to {ℙ}^{1}$ 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