Conductors of wildly ramified covers. III.
type: Item
instance of: scholarly article
title: Conductors of wildly ramified covers. III.
zbMATH Open document ID: 1058.14051
author: Rachel Pries
published in: Pacific Journal of Mathematics
publication date: 2004-09-16T00:00:00Z
review text: Let \(X\) be a smooth connected proper curve with marked points \(\{x_i\}\) over an algebraically closed field \(k\) of characteristic \(p\). Consider a Galois cover \(\phi : Y \to X\) of smooth connected curves branched only at \(\{x_i\}\). Abhyankar's conjecture (proved by \textit{M. Raynaud}, [Invent. Math. 116, 425--462 (1994; Zbl 0798.14013)] and \textit{D. Harbater} [Invent. Math. 117, 1--25 (1994; Zbl 0805.14014)]) determines exactly which groups \(G\) can be the Galois group of \(\phi\). An open problem is to determine which inertia groups and filtrations of higher ramification groups can be realized for such a cover \(\phi\). In this article, the author deals with the case \(\phi : Y \to \mathbb{P}^1\) is branched at only one point. Such covers exist if and only if \(G\) is a quasi-\(p\) group, which means that \(G\) is generated by \(p\)-groups. \textit{D. Harbater} [Am. J. Math. 115, 487--508 (1993; Zbl 0790.14027)] proved that the Sylow \(p\)-subgroups of \(G\) can be realized as the inertia groups of such a cover \(\phi\). Under the assumption that the Sylow \(p\)-subgroups of \(G\) have order \(p\), the filtration of higher ramification groups is determined by one integer \(j\) for which \(p \nmid j\) namely by the lower jump or conductor. The author shows that all sufficiently large conductors occur and for \(p \neq 2\) gives an explicit bound for which conductors are sufficiently large. Theorem. Let \(G\) be a finite quasi-\(p\) group whose Sylow \(p\)-subgroups have order \(p \neq 2\). There exists an integer \(J\) depending explicitly on \(p\), the \(p\)-weight of \(G\), and the exponent of the normalizer of a Sylow \(p\)-subgroup of \(G\) with the following property: if \(j \geq J\) and \(p \nmid j\) then there exists a \(G\)-Galois cover \(\phi : Y \to \mathbb{P}^1\) branched at only one point over which it has inertia group \(\mathbb{Z}/p\) and conductor \(j\). The \(p\)-weight of \(G\) is defined in the following way. Let \(S\) be a chosen Sylo…
reviewed by: Christophe Ritzenthaler
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 14E22
zbMATH DE Number: 2101105
zbMATH Keywords: Galois cover
zbMATH Keywords: \(p\)-groups
zbMATH Keywords: sylow \(p\)-subgroups
MaRDI profile type: Publication
full work available at URL: pjm.2003.211.163
OpenAlex ID: W2074126087
DOI: 10.2140/PJM.2003.211.163
description: scientific article; zbMATH DE number 2101105