Prime to p fundamental groups and tame Galois actions
Annales de l'Institut Fourier, Tome 50 (2000) no. 4, pp. 1099-1126.

Soit F un corps ayant une valuation complète et discrète, et de caractéristique résiduelle p. Si 𝒰 est une variété sur F, notons π1,geom(p′)(𝒰) le quotient maximal du groupe étale fondamental de 𝒰 qui est premier à p. Nous considérons l’application ρ:Gal(Fsep/F)→Out(π1,geom(p′)(𝒰)) au groupe des automorphismes extérieurs, et nous montrons qu’elle applique le groupe de ramification sauvage sur un groupe fini. Nous montrons que sous certaines conditions ρ dépend seulement de la réduction de 𝒰 modulo une puissance de l’idéal maximal de F. Les preuves utilisent la théorie des schémas logarithmiques.

We show that for a local, discretely valued field F, with residue characteristic p, and a variety 𝒰 over F, the map ρ:Gal(Fsep/F)→Out(π1,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.

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     author = {Kisin, Mark},
     title = {Prime to $p$ fundamental groups and tame {Galois} actions},
     journal = {Annales de l'Institut Fourier},
     pages = {1099--1126},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {50},
     number = {4},
     year = {2000},
     doi = {10.5802/aif.1786},
     mrnumber = {2001j:14035},
     zbl = {0961.14014},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1786/}
}
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Kisin, Mark. Prime to $p$ fundamental groups and tame Galois actions. Annales de l'Institut Fourier, Tome 50 (2000) no. 4, pp. 1099-1126. doi : 10.5802/aif.1786. https://www.numdam.org/articles/10.5802/aif.1786/

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