Prime to p fundamental groups and tame Galois actions

Mark Kisin

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 4, page 1099-1126
  • ISSN: 0373-0956

Abstract

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We show that for a local, discretely valued field F , with residue characteristic p , and a variety 𝒰 over F , the map ρ : Gal ( F sep / F ) → Out ( π 1 , geom ( p ' ) ( 𝒰 ) ) to the outer automorphisms of the prime to p geometric étale fundamental group of 𝒰 maps the wild inertia onto a finite image. We show that under favourable conditions ρ depends only on the reduction of 𝒰 modulo a power of the maximal ideal of F . The proofs make use of the theory of logarithmic schemes.

How to cite

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Kisin, Mark. "Prime to $p$ fundamental groups and tame Galois actions." Annales de l'institut Fourier 50.4 (2000): 1099-1126. <http://eudml.org/doc/75450>.

@article{Kisin2000,
abstract = {We show that for a local, discretely valued field $F$, with residue characteristic $p$, and a variety $\{\cal U\}$ over $F$, the map $\rho : \{\rm Gal\}(F^\{\{\rm sep\}\}/F)\rightarrow \{\rm Out\}(\pi _\{1,\{\rm geom\}\}^\{(p^\{\prime \})\}(\{\cal U\}))$ to the outer automorphisms of the prime to $p$ geometric étale fundamental group of $\{\cal U\}$ maps the wild inertia onto a finite image. We show that under favourable conditions $\rho $ depends only on the reduction of $\{\cal U\}$ modulo a power of the maximal ideal of $F$. The proofs make use of the theory of logarithmic schemes.},
author = {Kisin, Mark},
journal = {Annales de l'institut Fourier},
keywords = {local field; absolute Galois group; rigid analysis},
language = {eng},
number = {4},
pages = {1099-1126},
publisher = {Association des Annales de l'Institut Fourier},
title = {Prime to $p$ fundamental groups and tame Galois actions},
url = {http://eudml.org/doc/75450},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Kisin, Mark
TI - Prime to $p$ fundamental groups and tame Galois actions
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 4
SP - 1099
EP - 1126
AB - We show that for a local, discretely valued field $F$, with residue characteristic $p$, and a variety ${\cal U}$ over $F$, the map $\rho : {\rm Gal}(F^{{\rm sep}}/F)\rightarrow {\rm Out}(\pi _{1,{\rm geom}}^{(p^{\prime })}({\cal U}))$ to the outer automorphisms of the prime to $p$ geometric étale fundamental group of ${\cal U}$ maps the wild inertia onto a finite image. We show that under favourable conditions $\rho $ depends only on the reduction of ${\cal U}$ modulo a power of the maximal ideal of $F$. The proofs make use of the theory of logarithmic schemes.
LA - eng
KW - local field; absolute Galois group; rigid analysis
UR - http://eudml.org/doc/75450
ER -

References

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