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

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.

Soit F un corps ayant une valuation complète et discrète, et de caractéristique résiduelle p. Si 𝒰 est une variété sur F, notons π 1, geom (p ) (𝒰) le quotient maximal du groupe étale fondamental de 𝒰 qui est premier à p. Nous considérons l’application ρ: Gal (F sep /F) Out (π 1, 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.

@article{AIF_2000__50_4_1099_0,
     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},
     zbl = {0961.14014},
     mrnumber = {2001j:14035},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1786/}
}
TY  - JOUR
AU  - Kisin, Mark
TI  - Prime to $p$ fundamental groups and tame Galois actions
JO  - Annales de l'Institut Fourier
PY  - 2000
SP  - 1099
EP  - 1126
VL  - 50
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1786/
DO  - 10.5802/aif.1786
LA  - en
ID  - AIF_2000__50_4_1099_0
ER  - 
%0 Journal Article
%A Kisin, Mark
%T Prime to $p$ fundamental groups and tame Galois actions
%J Annales de l'Institut Fourier
%D 2000
%P 1099-1126
%V 50
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1786/
%R 10.5802/aif.1786
%G en
%F AIF_2000__50_4_1099_0
Kisin, Mark. Prime to $p$ fundamental groups and tame Galois actions. Annales de l'Institut Fourier, Volume 50 (2000) no. 4, pp. 1099-1126. doi : 10.5802/aif.1786. https://aif.centre-mersenne.org/articles/10.5802/aif.1786/

[deJ1] A.J. De Jong, Smoothness, semi-stability and alterations, Inst. des Hautes Etudes Sci. Publ. Math., 83 (1996), 51-93. | EuDML | Numdam | MR | Zbl

[deJ2] A.J. De Jong, Crystalline Dieudonné module theory via formal and rigid geometry, Inst. des Hautes Etudes Sci. Publ. Math., 82 (1995), 5-96. | EuDML | Numdam | MR | Zbl

[EGA] A. Grothendieck, J. Dieudonné, Eléments de géométrie algébrique I, II, III, IV, Inst. des Hautes Etudes Sci. Publ. Math., 4, 8, 11, 17, 20, 24, 28, 32 (1961-1967). | Numdam | MR

[FK] K. Fujiwara, K. Kato, Logarithmic Étale Topology Theory, preprint.

[Ful] W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies 131, Princeton University Press, Princeton, 1993. | MR | Zbl

[Ka1] K. Kato, Logarithmic Structures of Fontaine-Illusie, Algebraic Analysis, Geometry, and Number Theory, Proceedings of the JAMI Inaugural Conference, The John Hopkins University Press, Baltimore and London, 191-224, 1989. | MR | Zbl

[Ka2] K. Kato, Toric Singularities, Amer. J. Math., 116 (1994), 1073-1099. | MR | Zbl

[Ki1] M. Kisin, Local Constancy in Families of Non-Abelian Galois Representations, To appear in Math. Z., 16 pages. | Zbl

[Ki2] M. Kisin, Local Constancy in p-adic Familes of Galois Representations, Math. Z., 230(3) (1999), 569-593. | MR | Zbl

[Ki3] M. Kisin, Endomorphisms of Logarithimic Schemes, SFB 478, Münster (1999).

[Lut] W. Lütkebohmert, Riemann's existence problem for a p-adic field, Invent. Math., 111 (1993), 309-330. | MR | Zbl

[Mat] H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, 1986. | MR | Zbl

[Na] C. Nakayama, Nearby Cycles for log smooth families, Compositio Math., 112 (1998), 45-75. | MR | Zbl

[Na2] C. Nakayama, Logarithmic étale cohomology, Math. Ann., 308 (1997), 365-404. | MR | Zbl

[RZ] M. Rapoport, Th. Zink, Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik., Invent. Math., 68 (1982), 21-101. | Zbl

[SGA1] A. Grothendieck, Revetment Étale et Groupe Fundamental, Lect. Notes in Math. 224, Springer, Heidelberg, 1970.

[SGA2] A. Grothendieck, Cohomologie Locale des Faisceaux Cohérents et Théorèmes des Lefschetz Locaux et Globaux, North-Holland, Amsterdam, 1962. | Zbl

Cited by Sources: