. Ch, 3. -Variétés abéliennes sur les corps de fonctions de courbes sur des corps locaux supérieurs
URL : https://hal.archives-ouvertes.fr/hal-01344034

C. Soient and C. Une-courbe-projective-lisse-sur-k-telle-que, X) sur k. D'après le théorème 7.1, le groupe X 1 (K, J C × k K) est égal à Br Y /Br X. Par ailleurs, nous savons que Br 1 Y = H 1 (k, Pic Y k ) car Br k = 0. D'après la proposition 1.7 de [SZ14], le morphisme naturel H 1 (k, Pic C k ) × H 1 (k, Pic X k ) ? H 1 (k, Pic Y k ) a un noyau et un conoyau nis

C. Berge, Théorie des graphes et ses applications, deuxième édition, 1963.

S. Bloch, Algebraic cycles and higher K-theory, Advances in Mathematics, vol.61, issue.3, pp.267-304, 1986.
DOI : 10.1016/0001-8708(86)90081-2

URL : http://doi.org/10.1016/0001-8708(86)90081-2

M. Borovoi, Abelian Galois cohomology of reductive groups, Memoirs of the American Mathematical Society, vol.132, issue.626, p.50, 1998.
DOI : 10.1090/memo/0626

M. Borovoi, J. Van, and H. , Extended Picard complexes and linear algebraic groups, Journal f??r die reine und angewandte Mathematik (Crelles Journal), vol.2009, issue.627, p.5382, 2009.
DOI : 10.1515/CRELLE.2009.011

URL : http://arxiv.org/abs/math/0612156

S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Mathematics and Related Areas, 1990.
DOI : 10.1007/978-3-642-51438-8

L. Breen, Extensions of abelian sheaves and Eilenberg-Maclane algebras, Inventiones Mathematicae, vol.31, issue.1, p.1544, 1969.
DOI : 10.1007/BF01389887

[. Bruhat and J. Tits, Groupes algébriques sur un corps local. Chapitre III. Compléments et applications à la cohomologie galoisienne, J. Fac. Sci. Univ. Tokyo Sect. IA Math, vol.34, issue.3, p.671698, 1987.
DOI : 10.1007/978-3-642-87942-5_3

[. Chernousov, A remark on the (mod 5)-invariant of Serre for groups of type E 8, Mat. Zametki, vol.56, issue.157, p.116121, 1994.

[. Chernousov, On the Kernel of the Rost Invariant for E 8 Modulo 3, Quadratic forms, linear algebraic groups, and cohomology, 2010.
DOI : 10.1007/978-1-4419-6211-9_11

J. Colliot-thélène, Birational invariants, purity and the Gersten conjecture in K-theory and algebraic geometry : connections with quadratic forms and division algebras, Proc. Sympos. Pure Math.. Amer. Math. Soc, vol.58, 1992.

J. Colliot, Thélène Points rationnels sur les brations, Higher dimensional varieties and rational points, p.171221, 2001.

J. Colliot-thélène, Résolutions asques des groupes linéaires connexes, J. reine angew. Math, vol.618, pp.77133-221, 2008.

J. Colliot-thélène, P. Gille, and R. Parimala, Arithmetic of linear algebraic groups over 2-dimensional geometric elds, Duke Math. J, vol.121, issue.2, p.285341, 2004.

J. Colliot-thélène and D. Harari, Dualité et principe local-global pour les tores sur une courbe au-dessus de C, Proc. Lond, p.11014751516, 2015.

J. Colliot-thélène, M. Ojanguren, and R. Parimala, Quadratic forms over fraction elds of two-dimensional Henselian rings and Brauer groups of related schemes, Algebra, arithmetic and geometry, p.185217, 2000.

J. Colliot-thélène, R. Parimala, and V. Suresh, Patching and local-global principles for homogeneous spaces over function elds of p-adic curves, Comment. Math. Helv, vol.87, issue.4, p.10111033, 2012.

J. Colliot-thélène, R. Parimala, and V. Suresh, Lois de r??ciprocit?? sup??rieures et points rationnels, Transactions of the American Mathematical Society, vol.368, issue.6, p.42194255, 2016.
DOI : 10.1090/tran/6519

J. Colliot-thélène and J. Sansuc, La $R$-??quivalence sur les tores, Annales scientifiques de l'??cole normale sup??rieure, vol.10, issue.2, p.175229, 1977.
DOI : 10.24033/asens.1325

J. Sansuc, La descente sur les variétés rationnelles II, Duke Mathematical Journal, vol.54, p.375492, 1987.

R. François, I. V. Cossec, and . Dolgachev, Enriques surfaces I, Progress in Mathematics, vol.76, 1989.

. Springer-verlag, Séminaire de Géométrie Algébrique du Bois-Marie SGA 4 1, 1977.

J. Douai and C. Touibi, Courbes dénies sur les corps de séries formelles et loi de réciprocité, Acta Arith, vol.42, issue.1, p.10110683, 1982.

J. Douai and C. Touibi, Courbes dénies sur les corps de séries formelles et loi de réciprocité, Errata. Acta Arith, vol.46, p.197, 1985.

E. Freitag and R. Kiehl, Étale cohomology and the Weil conjecture
DOI : 10.1007/978-3-662-02541-3

L. Fu, Etale cohomology theory, Nankai Tracts in Mathematics, vol.13
DOI : 10.1142/7773

L. Fuchs, Innite abelian groups, Vol. I. Pure and Applied Mathematics, vol.36, 1970.

R. S. Garibaldi, The Rost invariant has trivial kernel for quasi-split groups of low rank, Commentarii Mathematici Helvetici, vol.76, issue.4, p.684711, 2001.
DOI : 10.1007/s00014-001-8325-8

T. Geisser, Motivic cohomology over Dedekind rings, Mathematische Zeitschrift, vol.7, issue.4, pp.773-794, 2004.
DOI : 10.1007/s00209-004-0680-x

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.170.4191

T. Geisser, Motivic Cohomology, K-Theory and Topological Cyclic Homology
DOI : 10.1007/978-3-540-27855-9_6

T. Geisser and M. Levine, The K-theory of elds in characteristic p, Invent. Math, vol.139, issue.3, p.459493, 2000.

T. Geisser and M. Levine, The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky, Journal f??r die reine und angewandte Mathematik (Crelles Journal), vol.2001, issue.530, p.55103, 2001.
DOI : 10.1515/crll.2001.006

P. Gille and A. Pianzola, Isotriviality and ??tale cohomology of Laurent polynomial rings, Journal of Pure and Applied Algebra, vol.212, issue.4, p.780800, 2008.
DOI : 10.1016/j.jpaa.2007.07.005

D. Cristian and . González-avilés, Arithmetic duality theorems for 1-motives over function elds, J. reine angew. Math, vol.632, p.203231, 2009.

D. Cristian and . González-avilés, Quasi-abelian crossed modules and nonabelian cohomology, J. Algebra, vol.369, p.235255, 2012.

A. Grothendieck, Schémas en groupes (SGA 3), Lecture Notes in Mathematics, vol.151, issue.153, 1970.

A. Grothendieck, Groupes de monodromie en géométrie algébrique. I (SGA 7 I), Dirigé par A. Grothendieck, 1972.

D. Harari, Méthode des brations et obstruction de Manin, Duke Math. J, vol.75, p.221260, 1994.
DOI : 10.1215/s0012-7094-94-07507-8

D. Harari, C. Scheiderer, and T. Szamuely, Weak approximation for tori over p-adic function elds, Int. Math. Res. Not, vol.10, p.27512783, 2015.

D. Harari and T. Szamuely, Arithmetic duality theorems for 1-motives, J. reine angew. Math, vol.578, p.93128, 2005.

D. Harari and T. Szamuely, Local-global principles for 1-motives. Duke Math, J, vol.143, issue.3, p.531557, 2008.

D. Harari and T. Szamuely, Local-global questions for tori over $p$-adic function fields, Journal of Algebraic Geometry, vol.25, issue.3, pp.571-605, 2016.
DOI : 10.1090/jag/661

[. Harbater, J. Hartmann, and D. Krashen, Applications of patching to quadratic forms and central simple algebras Local-global principles for Galois cohomology, Invent. Math. Comment. Math. Helv, vol.178, issue.891, p.231263215253, 2009.

[. Harbater, J. Hartmann, and D. Krashen, Local-global principles for torsors over arithmetic curves, American Journal of Mathematics, vol.137, issue.6, pp.15591612-223, 2015.
DOI : 10.1353/ajm.2015.0039

E. Hewitt and K. A. Ross, Abstract harmonic analysis, of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences, 1979.

Y. Hu, A cohomological Hasse principle over two-dimensional local rings Prépublication sur http, 2014.

K. Hulek and M. Schütt, Enriques surfaces and Jacobian elliptic K3 surfaces, Mathematische Zeitschrift, vol.476, issue.1, p.10251056, 2011.
DOI : 10.1007/s00209-010-0708-3

URL : http://arxiv.org/abs/0912.0608

[. Illusie, Y. Laszlo, and F. Orgogozo, Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasi-excellents. Astérisque, No. 363-364 Séminaire à l'École Polytechnique, Avec la collaboration de Frédéric Déglise, 20062008.
DOI : 10.1007/bfb0069278

D. Izquierdo, Th??or??mes de dualit?? pour les corps de fonctions sur des corps locaux sup??rieurs, Mathematische Zeitschrift, vol.174, issue.1, 2014.
DOI : 10.1007/s00209-016-1672-3

D. Izquierdo, Principe local???global pour les corps de fonctions sur des corps locaux sup??rieurs I, Journal of Number Theory, vol.157, p.250270, 2015.
DOI : 10.1016/j.jnt.2015.05.005

D. Izquierdo, Dualité et principe local-global sur des corps locaux de dimension 2. 2016. Prépublication sur http
DOI : 10.1016/j.jnt.2015.05.005

P. Jaworski, On the strong Hasse principle for elds of quotients of power series rings in two variables, Math. Z, vol.236, issue.3, p.531566, 2001.

B. Kahn, The decomposable part of motivic cohomology and bijectivity of the norm residue homomorphism, Contemp. Math, vol.126, pp.79-87, 1989.
DOI : 10.1090/conm/126/00502

B. Kahn, Classes de cycles motiviques ??tales, Algebra & Number Theory, vol.6, issue.7, p.13691407, 2012.
DOI : 10.2140/ant.2012.6.1369

URL : http://arxiv.org/abs/1102.0375

K. Kato, A generalization of local class eld theory by using K-groups I, J. Fac. Sci. Univ. Tokyo Sect. IA Math, vol.26, issue.2, p.303376, 1979.

K. Kato, A Hasse principle for two-dimensional global elds, J. reine angew. Math, vol.366, p.142183, 1986.

K. Kato, Existence theorem for higher local elds, Invitation to higher local elds, p.165195, 1999.

A. Victor and . Kolyvagin, Finiteness of E(Q) and SH(E, Q) for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat, vol.52670671, issue.3, p.522540, 1988.

[. Koya, On a duality theorem of abelian varieties over higher dimensional local elds, Kodai Math. J, vol.2, p.297308, 2000.

S. Lang, On quasi algebraic closure, Ann. of Math, vol.55, issue.2, p.373390, 1952.

S. Lang and J. Tate, Principal homogeneous spaces over abelian varieties
DOI : 10.2307/2372778

J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Publications math??matiques de l'IH??S, vol.62, issue.1, 1969.
DOI : 10.1007/BF02684604

J. Lipman, Desingularization of Two-Dimensional Schemes, The Annals of Mathematics, vol.107, issue.2, p.151207, 1978.
DOI : 10.2307/1971141

Q. Liu, Algebraic geometry and arithmetic curves, volume 6 of Oxford Graduate Texts in Mathematics, 2002.

Y. I. Manin, LE GROUPE DE BRAUER-GROTHENDIECK EN G??OM??TRIE DIOPHANTIENNE
DOI : 10.1142/9789812830517_0009

Y. I. Manin, Cubic forms Algebra, geometry, arithmetic, 1986.

J. Marot, Limite inductive plate de P-anneaux, Journal of Algebra, vol.57, issue.2, pp.484-496, 1979.
DOI : 10.1016/0021-8693(79)90234-5

URL : http://doi.org/10.1016/0021-8693(79)90234-5

A. Mattuck, Abelian varieties over p-adic ground elds, Ann. of Math, vol.62, issue.2, p.92119, 1955.
DOI : 10.2307/2007101

J. S. Milne, Étale cohomology, volume 33 of Princeton Mathematical Series, 1980.

J. S. Milne, Abelian Varieties, Arithmetic geometry, p.103150, 1984.
DOI : 10.1007/978-1-4613-8655-1_5

J. S. Milne, Arithmetic duality theorems, 2006.

S. James, K. Milne, and . Shih, Conjugates of Shimura varieties, Hodge Cycles, Motives and Shimura Varieties, p.280356, 1982.

M. Nagata, Note on a paper of Lang concerning quasi algebraic closure, Memoirs of the College of Science, University of Kyoto. Series A: Mathematics, vol.30, issue.3
DOI : 10.1215/kjm/1250777008

P. Yu, A. A. Nesterenko, and . Suslin, Homology of the general linear group over a local ring, and Milnor's K-theory, Izv. Akad. Nauk SSSR Ser. Mat, vol.53, issue.1, p.121146, 1989.

J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number elds, of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences, 2008.

P. Andrew and . Ogg, Cohomology of abelian varieties over function elds, Ann. of Math, vol.76, issue.2, p.185212, 1962.

T. Ono, Arithmetic of Algebraic Tori, The Annals of Mathematics, vol.74, issue.1, p.101139, 1961.
DOI : 10.2307/1970307

F. Oort, Commutative group schemes, Lecture Notes in Mathematics, vol.15, 1966.
DOI : 10.1007/BFb0097479

R. Parimala, Arithmetic of linear algebraic groups over two-dimensional elds

J. Riou, La conjecture de Bloch-Kato (d'après M. Rost & V. Voevodsky) Séminaire Bourbaki, exposé n o 1073, 2013.

K. Rubin, Tate-Shafarevich groups andL-functions of elliptic curves with complex multiplication, Inventiones Mathematicae, vol.115, issue.3, pp.527559-225, 1987.
DOI : 10.1007/BF01388984

J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. reine angew. Math, vol.327, p.1280, 1981.

S. Saito, Arithmetic on two dimensional local rings, Inventiones Mathematicae, vol.30, issue.2, pp.379-414, 1986.
DOI : 10.1007/BF01389096

C. Scheiderer, J. Van, and H. , Cohomology of tori over p -adic curves, Mathematische Annalen, vol.326, issue.1, p.155183, 2003.
DOI : 10.1007/s00208-003-0416-y

M. Schütt and T. Shioda, Elliptic surfaces, Algebraic geometry in East AsiaSeoul, p.51160, 2008.

N. Semenov, Motivic construction of cohomological invariants, Commentarii Mathematici Helvetici, vol.91, issue.1, p.163202, 2016.
DOI : 10.4171/CMH/382

J. Serre, Lie algebras and Lie groups, Second edition, 1964 lectures given at Harvard University, Lecture Notes in Mathematics, vol.1500, 1992.

J. Serre, Galois cohomology

S. Stephen and N. J. Shatz, Pronite groups, arithmetic, and geometry, Annals of Mathematics Studies, issue.67, 1972.

A. Skorobogatov, Beyond the Manin obstruction, Inventiones Mathematicae, vol.135, issue.2, pp.399-424, 1999.
DOI : 10.1007/s002220050291

A. N. Skorobogatov and Y. G. Zarhin, The Brauer group and the BrauerManin set of products of varieties, J. Eur. Math. Soc, vol.16, p.749768, 2014.

A. Andrei, S. Suslin, and . Joukhovitski, Norm varieties, J. Pure Appl. Algebra, vol.206, issue.12, p.245276, 2006.

A. Andrei, V. Suslin, and . Voevodsky, Bloch-Kato conjecture and motivic cohomology with nite coecients. In The arithmetic and geometry of algebraic cycles, NATO Sci. Ser. C Math. Phys. Sci, vol.548, p.117189, 1998.

G. Tamme, Introduction to étale cohomology. Universitext, 1994.

J. Tate, WC-groups over p-adic elds, Séminaire Bourbaki, vol.156, 1957.

J. Tate, Duality theorems in Galois cohomology over number elds, Proc. Internat. Congr. Mathematicians (Stockholm, p.288295, 1962.

J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, 1964.

B. Totaro, Milnor K-theory is the simplest part of algebraic K-theory. K-Theory, p.177189, 1992.

V. Voevodsky, On motivic cohomology with Z/l-coecients, Ann. of Math, vol.174, issue.21, p.401438, 2011.

C. A. Weibel, An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics, 1994.