Geodesic
The $K(\pi,1)$-property for smooth marked curves over finite fields
Izvestiya. Mathematics , Tome 79 (2015) no. 5, pp. 1043-1050.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the case of smooth marked curves $(X,T)$ over finite fields of characteristic $p$, we study the $K(\pi,1)$-property for $p$. We prove that $(X,T)$ has the $K(\pi,1)$-property if $X$ is affine, and give positive and negative examples in the case when $X$ is proper. We also consider the case of unmarked proper curves over a finite field of characteristic different from $p$.
Keywords: Galois cohomology, étale cohomology, restricted ramification. 
@article{IM2_2015_79_5_a6,
     author = {Ph. Lebacque and A. Schmidt},
     title = {The $K(\pi,1)$-property for smooth marked curves over finite fields},
     journal = {Izvestiya. Mathematics },
     pages = {1043--1050},
     publisher = {mathdoc},
     volume = {79},
     number = {5},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2015_79_5_a6/}
}
TY  - JOUR
AU  - Ph. Lebacque
AU  - A. Schmidt
TI  - The $K(\pi,1)$-property for smooth marked curves over finite fields
JO  - Izvestiya. Mathematics 
PY  - 2015
SP  - 1043
EP  - 1050
VL  - 79
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2015_79_5_a6/
LA  - en
ID  - IM2_2015_79_5_a6
ER  - 
%0 Journal Article
%A Ph. Lebacque
%A A. Schmidt
%T The $K(\pi,1)$-property for smooth marked curves over finite fields
%J Izvestiya. Mathematics 
%D 2015
%P 1043-1050
%V 79
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2015_79_5_a6/
%G en
%F IM2_2015_79_5_a6
Ph. Lebacque; A. Schmidt. The $K(\pi,1)$-property for smooth marked curves over finite fields. Izvestiya. Mathematics , Tome 79 (2015) no. 5, pp. 1043-1050. http://geodesic.mathdoc.fr/item/IM2_2015_79_5_a6/

[1] A. Schmidt, “Rings of integers of type $K(\pi,1)$”, Doc. Math., 12 (2007), 441–471 (electronic) | MR | Zbl

[2] A. Schmidt, “On the $K(\pi,1)$-property for rings of integers in the mixed case”, Algebraic number theory and related topics 2007, RIMS Kôkyûroku Bessatsu, B12, Res. Inst. Math. Sci. (RIMS), Kyoto, 2009, 91–100 | MR | Zbl

[3] A. Schmidt, “Über Pro-$p$-Fundamentalgruppen markierter arithmetischer Kurven”, J. Reine Angew. Math., 2010:640 (2010), 203–235 | DOI | MR | Zbl

[4] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number fields, Grundlehren Math. Wiss., 323, 2nd ed., 2nd corr. print., Springer-Verlag, Berlin, 2013, 826 pp. | DOI | MR | Zbl

[5] M. Artin, P. Deligne, {T}héorie des topos et cohomologie étale des schémas. Tome 3, Séminaire de géométrie algébrique du Bois-Marie 1963–1964 (SGA 4). Dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de P. Deligne, B. Saint-Donat, Lecture Notes in Math., 305, Springer-Verlag, Berlin–New York, 1973, vi+640 pp. | DOI | MR | Zbl

[6] Y. Ihara, “How many primes decompose completely in an infinite unramified Galois extension of a global field?”, J. Math. Soc. Japan, 35:4 (1983), 693–709 | DOI | MR | Zbl