Conjugacy classes of series in positive characteristic and Witt vectors
type: Item
instance of: scholarly article
title: Conjugacy classes of series in positive characteristic and Witt vectors
zbMATH Open document ID: 1233.12011
author: Sandrine Jean
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2009-10-26T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10880
review text: Let \(k\) be an algebraic closure of the finite field \(\mathbb{F}_p\), and \(K=k((t))\) the field of formal Laurent series over \(k\). In this context, Oort's conjecture states that every cyclic field extension \(L|K\) of finite degree should lift smoothly to mixed characteristic. Note that a choice of uniformizer \(s\in L\) yields the isomorphism \(L\cong k((s))\), and then a finite-order continuous \(k\)-automorphism of \(L\) corresponds to a power series \(\sigma(s)\in ks\) having finite order for the composition law. Hence, Oort's conjecture can be reformulated as the statement that every invertible power series of order \(n\) for the composition law in \(ks\) can be lifted to a series of order \(n\) for the composition law in \(RS\), where \(R\) is a mixed characteristic discrete valuation ring with residue field \(k\). Oort's conjecture is known to hold for all tamely-ramified extensions, so the essential case is that of totally ramified cyclic extensions of degree \(p^m\). The case \(m=1\) was established using global methods by \textit{T. Sekiguchi, F. Oort}, and \textit{N. Suwa} in [``On the deformation of Artin-Schreier to Kummer'', Ann. Sci. Éc. Norm. Supér. (4) 22, No. 3, 345--375 (1989; Zbl 0714.14024)], and the \(m=1,2\) cases were obtained via local methods in [\textit{B. Green} and \textit{M. Matignon}, ``Lifting Galois covers of smooth curves'', Compos. Math. 113, 237--272 (1998; Zbl 0923.14006)]. Recently, A. Obus and S. Wewers have announced a proof of the \(m=3\) case: [{A. Obus}, ``The (local) lifting problem for curves'', \texttt{arXiv:1105.1530}]. In order to establish the conjecture in general, it suffices to lift a single representative from each conjugacy class of series of order \(p^m\) in \(ks\). The present paper contributes to this area by providing (Theorem 5.6) an explicit bijection between the set \(\mathcal{Y}_m\) of conjugacy classes of series of order \(p^m\) and the orbits of a certain group action on a particula…
reviewed by: Scott Corry
Mathematics Subject Classification ID: 12J99
Mathematics Subject Classification ID: 13F35
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 11S20
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20F28
zbMATH DE Number: 5620650
zbMATH Keywords: lifting problem
zbMATH Keywords: Oort conjecture
zbMATH Keywords: power series
zbMATH Keywords: ramification
MaRDI profile type: Publication
OpenAlex ID: W1997325154
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DOI: 10.5802/JTNB.670
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description: scientific article; zbMATH DE number 5620650