Geodesic
On abelian birational sections
Esnault, Hélène 1   ; Wittenberg, Olivier 2
1 Universität Duisburg–Essen, Mathematik, 45117 Essen, Germany
2 Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, 75320 Paris Cedex 05, France
Journal of the American Mathematical Society, Tome 23 (2010) no. 3, pp. 713-724 Cet article a éte moissonné depuis la source American Mathematical Society

Voir la notice de l'article

For a smooth and geometrically irreducible variety $X$ over a field $k$, the quotient of the absolute Galois group $G_{k(X)}$ by the commutator subgroup of $G_{\bar k(X)}$ projects onto $G_k$. We investigate the sections of this projection. We show that such sections correspond to “infinite divisions” of the elementary obstruction of Colliot-Thélène and Sansuc. If $k$ is a number field and the Tate–Shafarevich group of the Picard variety of $X$ is finite, then such sections exist if and only if the elementary obstruction vanishes. For curves this condition also amounts to the existence of divisors of degree $1$. Finally we show that the vanishing of the elementary obstruction is not preserved by extensions of scalars.
DOI : 10.1090/S0894-0347-10-00660-0

Esnault, Hélène  1   ; Wittenberg, Olivier  2

1 Universität Duisburg–Essen, Mathematik, 45117 Essen, Germany
2 Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, 75320 Paris Cedex 05, France 
@article{10_1090_S0894_0347_10_00660_0,
     author = {Esnault, H\'el\`ene and Wittenberg, Olivier},
     title = {On abelian birational sections},
     journal = {Journal of the American Mathematical Society},
     pages = {713--724},
     year = {2010},
     volume = {23},
     number = {3},
     doi = {10.1090/S0894-0347-10-00660-0},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-10-00660-0/}
}
TY  - JOUR
AU  - Esnault, Hélène
AU  - Wittenberg, Olivier
TI  - On abelian birational sections
JO  - Journal of the American Mathematical Society
PY  - 2010
SP  - 713
EP  - 724
VL  - 23
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-10-00660-0/
DO  - 10.1090/S0894-0347-10-00660-0
ID  - 10_1090_S0894_0347_10_00660_0
ER  - 
%0 Journal Article
%A Esnault, Hélène
%A Wittenberg, Olivier
%T On abelian birational sections
%J Journal of the American Mathematical Society
%D 2010
%P 713-724
%V 23
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-10-00660-0/
%R 10.1090/S0894-0347-10-00660-0
%F 10_1090_S0894_0347_10_00660_0
Esnault, Hélène; Wittenberg, Olivier. On abelian birational sections. Journal of the American Mathematical Society, Tome 23 (2010) no. 3, pp. 713-724. doi: 10.1090/S0894-0347-10-00660-0

[1] Borovoi, M., Colliot-Thélène, J.-L., Skorobogatov, A. N. The elementary obstruction and homogeneous spaces Duke Math. J. 2008 321 364

[2] Cartan, Henri, Eilenberg, Samuel Homological algebra 1956

[3] Colliot-Thélène, Jean-Louis Conjectures de type local-global sur l’image des groupes de Chow dans la cohomologie étale 1999 1 12

[4] Colliot-Thélène, Jean-Louis, Sansuc, Jean-Jacques La descente sur les variétés rationnelles. II Duke Math. J. 1987 375 492

[5] Deligne, P. Le groupe fondamental de la droite projective moins trois points 1989 79 297

[6] Grothendieck, Alexander Brief an G. Faltings 1997 49 58

[7] Grothendieck, Alexander Le groupe de Brauer. II. Théorie cohomologique 1968 67 87

[8] Harari, David, Szamuely, Tamás Local-global principles for 1-motives Duke Math. J. 2008 531 557

[9] Harari, David, Szamuely, Tamás Galois sections for abelianized fundamental groups Math. Ann. 2009 779 800

[10] Koenigsmann, Jochen On the ‘section conjecture’ in anabelian geometry J. Reine Angew. Math. 2005 221 235

[11] Lichtenbaum, Stephen Duality theorems for curves over 𝑝-adic fields Invent. Math. 1969 120 136

[12] Milne, J. S. Arithmetic duality theorems 1986

[13] Saito, S. Some observations on motivic cohomology of arithmetic schemes Invent. Math. 1989 371 404

[14] Scheiderer, Claus Real and étale cohomology 1994

[15] Serre, Jean-Pierre Cohomologie galoisienne 1994

[16] Skorobogatov, Alexei Torsors and rational points 2001

[17] Wittenberg, Olivier On Albanese torsors and the elementary obstruction Math. Ann. 2008 805 838

Cité par Sources :