Lifting the field of norms

Laurent Berger[1]

  • [1] UMPA de l’ENS de Lyon, UMR 5669 du CNRS, IUF 46 allée d’Italie, 69007 Lyon, France

Journal de l’École polytechnique — Mathématiques (2014)

  • Volume: 1, page 29-38
  • ISSN: 2270-518X

Abstract

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Let K be a finite extension of Q p . The field of norms of a p -adic Lie extension K ∞ / K is a local field of characteristic p which comes equipped with an action of Gal ( K ∞ / K ) . When can we lift this action to characteristic 0 , along with a compatible Frobenius map? In this note, we formulate precisely this question, explain its relevance to the theory of ( ϕ , Γ ) -modules, and give a condition for the existence of certain types of lifts.

How to cite

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Berger, Laurent. "Lifting the field of norms." Journal de l’École polytechnique — Mathématiques 1 (2014): 29-38. <http://eudml.org/doc/275440>.

@article{Berger2014,
abstract = {Let $K$ be a finite extension of $\mathbf\{Q\}_p$. The field of norms of a $p$-adic Lie extension $K_\infty /K$ is a local field of characteristic $p$ which comes equipped with an action of $\mathrm\{Gal\}(K_\infty /K)$. When can we lift this action to characteristic $0$, along with a compatible Frobenius map? In this note, we formulate precisely this question, explain its relevance to the theory of $(\varphi ,\Gamma )$-modules, and give a condition for the existence of certain types of lifts.},
affiliation = {UMPA de l’ENS de Lyon, UMR 5669 du CNRS, IUF 46 allée d’Italie, 69007 Lyon, France},
author = {Berger, Laurent},
journal = {Journal de l’École polytechnique — Mathématiques},
keywords = {Field of norms; $(\phi ,\Gamma )$-module; $p$-adic representation; anticyclotomic extension; Cohen ring; non-Archimedean dynamical system; field of norms; -module; -adic representation},
language = {eng},
pages = {29-38},
publisher = {École polytechnique},
title = {Lifting the field of norms},
url = {http://eudml.org/doc/275440},
volume = {1},
year = {2014},
}

TY - JOUR
AU - Berger, Laurent
TI - Lifting the field of norms
JO - Journal de l’École polytechnique — Mathématiques
PY - 2014
PB - École polytechnique
VL - 1
SP - 29
EP - 38
AB - Let $K$ be a finite extension of $\mathbf{Q}_p$. The field of norms of a $p$-adic Lie extension $K_\infty /K$ is a local field of characteristic $p$ which comes equipped with an action of $\mathrm{Gal}(K_\infty /K)$. When can we lift this action to characteristic $0$, along with a compatible Frobenius map? In this note, we formulate precisely this question, explain its relevance to the theory of $(\varphi ,\Gamma )$-modules, and give a condition for the existence of certain types of lifts.
LA - eng
KW - Field of norms; $(\phi ,\Gamma )$-module; $p$-adic representation; anticyclotomic extension; Cohen ring; non-Archimedean dynamical system; field of norms; -module; -adic representation
UR - http://eudml.org/doc/275440
ER -

References

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  5. J.-M. Fontaine, J.-P. Wintenberger, Le “corps des normes” de certaines extensions algébriques de corps locaux, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), A367-A370 Zbl0475.12020MR526137
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  7. L. Fourquaux, B. Xie, Triangulable 𝒪 F -analytic ( ϕ q , Γ ) -modules of rank 2 , Algebra Number Theory 7 (2013), 2545-2592 Zbl1297.11145MR3194651
  8. M. Kisin, W. Ren, Galois representations and Lubin-Tate groups, Doc. Math. 14 (2009), 441-461 Zbl1246.11112MR2565906
  9. J. Lubin, Non-Archimedean dynamical systems, Compositio Math. 94 (1994), 321-346 Zbl0843.58111MR1310863
  10. A. J. Scholl, Higher fields of norms and ( φ , Γ ) -modules, Doc. Math. (2006), 685-709 Zbl1186.11070MR2290602
  11. S. Sen, Ramification in p -adic Lie extensions, Invent. Math. 17 (1972), 44-50 Zbl0242.12012MR319949
  12. 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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