Wintenberger’s functor for abelian extensions

Kevin Keating[1]

  • [1] Department of Mathematics University of Florida Gainesville, FL 32611 USA

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 3, page 665-678
  • ISSN: 1246-7405

Abstract

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Let k be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian p -adic Lie extensions E / F , where F is a local field with residue field k , and a category whose objects are pairs ( K , A ) , where K k ( ( T ) ) and A is an abelian p -adic Lie subgroup of Aut k ( K ) . In this paper we extend this equivalence to allow Gal ( E / F ) and A to be arbitrary abelian pro- p groups.

How to cite

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Keating, Kevin. "Wintenberger’s functor for abelian extensions." Journal de Théorie des Nombres de Bordeaux 21.3 (2009): 665-678. <http://eudml.org/doc/10903>.

@article{Keating2009,
abstract = {Let $k$ be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian $p$-adic Lie extensions $E/F$, where $F$ is a local field with residue field $k$, and a category whose objects are pairs $(K,A)$, where $K\cong k((T))$ and $A$ is an abelian $p$-adic Lie subgroup of $\mathrm\{Aut\}_k(K)$. In this paper we extend this equivalence to allow $\mathrm\{Gal\}(E/F)$ and $A$ to be arbitrary abelian pro-$p$ groups.},
affiliation = {Department of Mathematics University of Florida Gainesville, FL 32611 USA},
author = {Keating, Kevin},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {ramification; field of norms; extensions of local fields; automorphisms of local fields.},
language = {eng},
number = {3},
pages = {665-678},
publisher = {Université Bordeaux 1},
title = {Wintenberger’s functor for abelian extensions},
url = {http://eudml.org/doc/10903},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Keating, Kevin
TI - Wintenberger’s functor for abelian extensions
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 3
SP - 665
EP - 678
AB - Let $k$ be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian $p$-adic Lie extensions $E/F$, where $F$ is a local field with residue field $k$, and a category whose objects are pairs $(K,A)$, where $K\cong k((T))$ and $A$ is an abelian $p$-adic Lie subgroup of $\mathrm{Aut}_k(K)$. In this paper we extend this equivalence to allow $\mathrm{Gal}(E/F)$ and $A$ to be arbitrary abelian pro-$p$ groups.
LA - eng
KW - ramification; field of norms; extensions of local fields; automorphisms of local fields.
UR - http://eudml.org/doc/10903
ER -

References

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  1. K. Keating, Extensions of local fields and truncated power series. J. Number Theory 116 (2006), 69–101. Zbl1161.11407MR2197861
  2. F. Laubie, Extensions de Lie et groupes d’automorphismes de corps locaux. Compositio Math. 67 (1988), 165–189. Zbl0649.12012MR951749
  3. F. Laubie, Ramification des groupes abéliens d’automorphismes de 𝔽 q ( ( X ) ) . Canad. Math. Bull. 50 (2007), 594–597. Zbl1204.11178MR2364208
  4. J.-P. Serre, Corps Locaux. Hermann, 1962. MR354618
  5. J.-P. Wintenberger, Automorphismes des corps locaux de charactéristique p . J. Théor. Nombres Bordeaux 16 (2004), 429–456. Zbl1194.11103MR2143563
  6. J.-P. Wintenberger, Extensions abéliennes et groupes d’automorphismes de corps locaux. C. R. Acad. Sci. Paris Sér. A-B 290 (1980), A201–A203. Zbl0428.12012MR564309
  7. J.-P. Wintenberger, Le corps des normes de certaines extensions infinies de corps locaux; applications. Ann. Sci. École Norm. Sup. (4) 16 (1983), 59–89. Zbl0516.12015MR719763

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