Conjugacy classes of series in positive characteristic and Witt vectors.

Sandrine Jean[1]

  • [1] XLIM UMR 6172 Département de Mathématiques et Informatique Université de Limoges 123 avenue Albert Thomas 87 060 Limoges Cedex, France

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 2, page 263-284
  • ISSN: 1246-7405

Abstract

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Let k be the algebraic closure of 𝔽 p and K be the local field of formal power series with coefficients in k . The aim of this paper is the description of the set 𝒴 n of conjugacy classes of series of order p n for the composition law. This work is concerned with the formal power series with coefficients in a field of characteristic p which are invertible and of finite order p n for the composition law. In order to investigate Oort’s conjecture, I give a description of conjugacy classes of series by means of Witt vectors of finite length. We develop some tools which permit us to construct a bijection between a set 𝒜 n of Witt vectors and a set 𝒳 n of pairs constituted by a cyclic totally ramified extension L / K of degree p n and a generator of its Galois group. We are able to define for any element of 𝒜 n a sequence of ramification breaks. We also describe another bijection between 𝒴 n and the orbits of 𝒜 n under a certain group action. Ramification breaks of a series belonging to 𝒴 n can be recovered from the components of a corresponding vector in 𝒜 n .

How to cite

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Jean, Sandrine. "Conjugacy classes of series in positive characteristic and Witt vectors.." Journal de Théorie des Nombres de Bordeaux 21.2 (2009): 263-284. <http://eudml.org/doc/10880>.

@article{Jean2009,
abstract = {Let $k$ be the algebraic closure of $\mathbb\{F\}_p$ and $K$ be the local field of formal power series with coefficients in $k$. The aim of this paper is the description of the set $\mathcal\{Y\}_n$ of conjugacy classes of series of order $p^n$ for the composition law. This work is concerned with the formal power series with coefficients in a field of characteristic $p$ which are invertible and of finite order $p^n$ for the composition law. In order to investigate Oort’s conjecture, I give a description of conjugacy classes of series by means of Witt vectors of finite length. We develop some tools which permit us to construct a bijection between a set $\mathcal\{A\}_n$ of Witt vectors and a set $\mathcal\{X\}_n$ of pairs constituted by a cyclic totally ramified extension $L/K$ of degree $p^n$ and a generator of its Galois group. We are able to define for any element of $\mathcal\{A\}_n$ a sequence of ramification breaks. We also describe another bijection between $\mathcal\{Y\}_n$ and the orbits of $\mathcal\{A\}_n$ under a certain group action. Ramification breaks of a series belonging to $\mathcal\{Y\}_n$ can be recovered from the components of a corresponding vector in $\mathcal\{A\}_n$.},
affiliation = {XLIM UMR 6172 Département de Mathématiques et Informatique Université de Limoges 123 avenue Albert Thomas 87 060 Limoges Cedex, France},
author = {Jean, Sandrine},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {lifting problem; Oort conjecture; power series; ramification},
language = {eng},
number = {2},
pages = {263-284},
publisher = {Université Bordeaux 1},
title = {Conjugacy classes of series in positive characteristic and Witt vectors.},
url = {http://eudml.org/doc/10880},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Jean, Sandrine
TI - Conjugacy classes of series in positive characteristic and Witt vectors.
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 2
SP - 263
EP - 284
AB - Let $k$ be the algebraic closure of $\mathbb{F}_p$ and $K$ be the local field of formal power series with coefficients in $k$. The aim of this paper is the description of the set $\mathcal{Y}_n$ of conjugacy classes of series of order $p^n$ for the composition law. This work is concerned with the formal power series with coefficients in a field of characteristic $p$ which are invertible and of finite order $p^n$ for the composition law. In order to investigate Oort’s conjecture, I give a description of conjugacy classes of series by means of Witt vectors of finite length. We develop some tools which permit us to construct a bijection between a set $\mathcal{A}_n$ of Witt vectors and a set $\mathcal{X}_n$ of pairs constituted by a cyclic totally ramified extension $L/K$ of degree $p^n$ and a generator of its Galois group. We are able to define for any element of $\mathcal{A}_n$ a sequence of ramification breaks. We also describe another bijection between $\mathcal{Y}_n$ and the orbits of $\mathcal{A}_n$ under a certain group action. Ramification breaks of a series belonging to $\mathcal{Y}_n$ can be recovered from the components of a corresponding vector in $\mathcal{A}_n$.
LA - eng
KW - lifting problem; Oort conjecture; power series; ramification
UR - http://eudml.org/doc/10880
ER -

References

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  10. P. Samuel, Groupes finis d’automorphismes des anneaux de séries formelles. Bull. Sc. math. 90 (1966), 97–101. Zbl0142.01002MR209278
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  13. L. Thomas, Arithmétique des extensions d’Artin-Schreier-Witt. Thèse de doctorat, Toulouse, 2005. 
  14. L. Thomas, Ramification groups in Artin-Schreier-Witt extensions. Journal de théorie des Nombres de Bordeaux 17 (2005), 689–720. Zbl1207.11109MR2211314

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