Rigidity, reduction, and ramification
type: Item
instance of: scholarly article
title: Rigidity, reduction, and ramification
zbMATH Open document ID: 1029.14010
author: Irene I. Bouw
author: Rachel Pries
published in: Mathematische Annalen
publication date: 2003-09-16T00:00:00Z
review text: Let \(k\) be an algebraically closed field of characteristic \(p>2\). Let \(G\) be a finite quasi-\(p\) group. The ``inertia conjecture'' asserts that if \(I\) is a subgroup of \(G\) which is an extension of a cyclic group of order prime-to-\(p\) by a \(p\)-group \(P\), whose conjugates generate \(G\), then \(I\) occurs as inertia group for some \(G\)-Galois cover \(f : Y \rightarrow \mathbb{P}^1_k\) branched only at \(\infty\). The authors prove that the inertia conjecture is true for the groups \(A_p\) and \(\text{PSL}_2(p)\), giving the first serious evidence for this conjecture. In addition, they prove that the set of conductors that can be realized depends on the group. The main technique is to study covers \(f\) in characteristic zero such that \(p\) divides the ramification indices. Such a cover has bad reduction to characteristic \(p\). Under some conditions on the Galois group of \(f\), one component of the stable reduction of \(f\) yields a \(G\)-Galois cover of \(P_k^1\) which is wildly ramified above only one point. For \(G=\text{PSL}_2(p)\), the authors use the reduction of the modular curve \(X(2p)\), and for \(G=A_p\), they use the existence of subgroups with index \(p\).
reviewed by: Christophe Ritzenthaler
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14G32
zbMATH DE Number: 1982592
zbMATH Keywords: inertia conjecture
zbMATH Keywords: Galois cover
zbMATH Keywords: conductor
zbMATH Keywords: modular curve
zbMATH Keywords: bad reduction
zbMATH Keywords: characteristic \(p\)
zbMATH Keywords: Galois group
MaRDI profile type: Publication
full work available at URL: s00208-003-0441-x
OpenAlex ID: W1987956467
DOI: 10.1007/S00208-003-0441-X
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description: scientific article; zbMATH DE number 1982592