Ramification groups in Artin-Schreier-Witt extensions

Lara Thomas[1]

  • [1] Equipe GRIMM Université Toulouse II 5, allées A. Machado 31058 Toulouse, France Chaire de Structures Algébriques et Géométriques Ecole Polytechnique Fédérale de Lausanne SB - IMB (Bâtiment MA) Station 8 CH-1015 Lausanne

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 2, page 689-720
  • ISSN: 1246-7405

Abstract

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Let K be a local field of characteristic p > 0 . The aim of this paper is to describe the ramification groups for the pro- p abelian extensions over K with regards to the Artin-Schreier-Witt theory. We shall carry out this investigation entirely in the usual framework of local class field theory. This leads to a certain non-degenerate pairing that we shall define in detail, generalizing in this way the Schmid formula to Witt vectors of length n . Along the way, we recover a result of Brylinski but with a different proof which is more explicit and requires less technical machinery. A first attempt is finally made to extend these computations to the case where the perfect field of K is merely perfect.

How to cite

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Thomas, Lara. "Ramification groups in Artin-Schreier-Witt extensions." Journal de Théorie des Nombres de Bordeaux 17.2 (2005): 689-720. <http://eudml.org/doc/249428>.

@article{Thomas2005,
abstract = {Let $K$ be a local field of characteristic $p&gt;0$. The aim of this paper is to describe the ramification groups for the pro-$p$ abelian extensions over $K$ with regards to the Artin-Schreier-Witt theory. We shall carry out this investigation entirely in the usual framework of local class field theory. This leads to a certain non-degenerate pairing that we shall define in detail, generalizing in this way the Schmid formula to Witt vectors of length $n$. Along the way, we recover a result of Brylinski but with a different proof which is more explicit and requires less technical machinery. A first attempt is finally made to extend these computations to the case where the perfect field of $K$ is merely perfect.},
affiliation = {Equipe GRIMM Université Toulouse II 5, allées A. Machado 31058 Toulouse, France Chaire de Structures Algébriques et Géométriques Ecole Polytechnique Fédérale de Lausanne SB - IMB (Bâtiment MA) Station 8 CH-1015 Lausanne},
author = {Thomas, Lara},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {ramification groups for pro- abelian extensions},
language = {eng},
number = {2},
pages = {689-720},
publisher = {Université Bordeaux 1},
title = {Ramification groups in Artin-Schreier-Witt extensions},
url = {http://eudml.org/doc/249428},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Thomas, Lara
TI - Ramification groups in Artin-Schreier-Witt extensions
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 2
SP - 689
EP - 720
AB - Let $K$ be a local field of characteristic $p&gt;0$. The aim of this paper is to describe the ramification groups for the pro-$p$ abelian extensions over $K$ with regards to the Artin-Schreier-Witt theory. We shall carry out this investigation entirely in the usual framework of local class field theory. This leads to a certain non-degenerate pairing that we shall define in detail, generalizing in this way the Schmid formula to Witt vectors of length $n$. Along the way, we recover a result of Brylinski but with a different proof which is more explicit and requires less technical machinery. A first attempt is finally made to extend these computations to the case where the perfect field of $K$ is merely perfect.
LA - eng
KW - ramification groups for pro- abelian extensions
UR - http://eudml.org/doc/249428
ER -

References

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  2. J.-L. Brylinski, Théorie du corps de classes de Kato et revêtements abéliens de surfaces. Ann. Inst. Fourier, 33.3, Grenoble (1983), 23–38. Zbl0524.12008MR723946
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  5. H. Hasse, Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichem Konstantenkörper. J. Reine Angew. Math. 172 (1934), 37–54. Zbl0010.00501
  6. M. Hazewinkel, Abelian extensions of local fields, Ph.D. thesis. Universiteit Nijmegen, Holland (1969). Zbl0193.35001MR248110
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  8. L. Ribes, P. Zalesskii, Profinite groups. A series of Modern Surveys in Mathematics, Volume 40, Springer (2000). Zbl0949.20017MR1775104
  9. P. Roquette, Class Field Theory in characteristic p . Its origin and development. K. Miyake (ed.), Class Field Theory- Its Centenary and Prospect, Advanced Studies In Pure Mathematics, vol. 30, Tokyo (2000), 549–631. Zbl1068.11073MR1846477
  10. H.L. Schmid, Über das Reziprozitätsgezsetz in relativ-zyklischen algebraischen Funktionkörpern mit endlichem Konstantenkörper. Math. Z. 40 (1936), no. 1, 94–109. Zbl0011.14604MR1545545
  11. H.L. Schmid, Zyklischen algebraische Funktionkörper vom Grade p n über endlichem Konstantenkörper der Charakteristik p . J. Reine Angew. Math. 175, (1936), 108–123. Zbl0014.00402
  12. H.L. Schmid, Zur Arithmetik der zyklischen p -Körper. J. Reine Angew. Math. 176 (1937), 161–167 Zbl0016.05205
  13. J.-P. Serre, Corps Locaux. Hermann, Paris (1962). [English translation : Local Fields. Graduate Texts in Math. 67, Springer, New York (1979)]. Zbl0137.02601MR150130
  14. S.S. Shatz, Profinite groups, arithmetic and geometry. Princeton university press and university of Tokyo press (1972). Zbl0236.12002MR347778
  15. O. Teichmüller, Zerfallende zyklische p -Algebren. J. Reine Angew. Math., 174 (1936), 157–160. Zbl0016.05201
  16. L. Thomas, Arithmétique des extensions d’Artin-Schreier-Witt, Ph.D. (2005). Université Toulouse II Le Mirail. 
  17. E. Witt, Zyklische Körper und Algebren der Charakteristik vom Grad p n . J. Reine Angew. Math., 174 (1936), 126–140. Zbl0016.05101

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