Corps de classes des schémas arithmétiques

Szamuely, Tamás

Séminaire Bourbaki : volume 2008/2009 exposés 997-1011 - Avec table par noms d'auteurs de 1848/49 à 2008/09, Astérisque, no. 332 (2010), Exposé no. 1006, 30 p.

[1] E. Artin & J. Tate - Class field theory, W. A. Benjamin, Inc., New York- Amsterdam, 1968. | MR | Zbl

[2] M. Artin, A. Grothendieck & J.-L. Verdier (éds.) - Théorie des topos et cohomologie étale des schémas. Tome 3 (SGA 4), Lecture Notes in Math., vol. 305, Springer, 1973.

[3] S. Bloch - Algebraic $K$-theory and classfield theory for arithmetic surfaces, Ann. of Math. 114 (1981), p. 229-265. | MR | Zbl | DOI

[4] S. Bloch, Algebraic cycles and higher $K$-theory, Adv. in Math. 61 (1986), p. 267- 304. | MR | Zbl | DOI

[5] J.-L. Colliot-Thélène, J.-J. Sansuc & C. Soulé - Torsion dans le groupe de Chow de codimension deux, Duke Math. J. 50 (1983), p. 763-801. | MR | Zbl | DOI

[6] I. Fesenko & M. Kurihara (éds.) - Invitation to higher local fields, Geometry & Topology Monographs, vol. 3, Geometry & Topology Publications, Coventry, 2000. | MR | Zbl | DOI

[7] O. Gabber - On space filling curves and Albanese varieties, Geom. Fund. Anal. 11 (2001), p. 1192-1200. | MR | Zbl | DOI

[8] T. Geisser & M. Levine - The $K$-theory of fields in characteristic $p$, Invent. Math. 139 (2000), p. 459-493. | MR | Zbl | DOI

[9] T. Geisser & M. Levine, The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky, J. reine angew. Math. 530 (2001), p. 55-103. | MR | Zbl

[10] M. Gros - Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique, Mém. Soc. Math. France 21 (1985). | MR | Zbl | EuDML | Numdam

[11] M. Gros, Sur la partie $p$-primaire du groupe de Chow de codimension deux, Comm. Algebra 13 (1985), p. 2407-2420. | MR | Zbl | DOI

[12] A. Grothendieck (éd.) - Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Math., vol. 224, Springer, 1971.

A. Grothendieck (éd.) - Revêtements étales et groupe fondamental (SGA 1). Nouvelle édition annotée : Documents Mathématiques, vol. 3, Soc. Math. France, 2003. | Zbl

[13] L. Illusie - Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. 12 (1979), p. 501-661. | MR | Zbl | EuDML | Numdam | DOI

[14] U. Jannsen - Hasse principles for higher-dimensional fields, prépublication, 2005.

[15] U. Jannsen & S. Saito - Kato homology of arithmetic schemes and higher class field theory over local fields, Doc. Math. (2003), p. 479-538. | MR | Zbl | EuDML

[16] A. J. De Jong - Smoothness, semi-stability and alterations, Publ. Math. I.H.É.S. 83 (1996), p. 51-93. | MR | Zbl | EuDML | Numdam | DOI

[17] K. Kato - A generalization of local class field theory by using $K$-groups. I, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26 (1979), p. 303-376. | MR | Zbl

K. Kato - A generalization of local class field theory by using $K$-groups. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), p. 603-683. | MR | Zbl

K. Kato - A generalization of local class field theory by using $K$-groups. III, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 29 (1982), p. 31-43. | MR | Zbl

[18] K. Kato, A Hasse principle for two-dimensional global fields, J. reine angew. Math. 366 (1986), p. 142-183. | MR | Zbl | EuDML

[19] K. Kato, Class field theory, $𝒟$-modules, and ramification on higher-dimensional schemes. I, Amer. J. Math. 116 (1994), p. 757-784. | MR | Zbl | DOI

[20] K. Kato & S. Saito - Two-dimensional class field theory, in Galois groups and their representations (Nagoya, 1981), Adv. Stud. Pure Math., vol. 2, North- Holland, 1983, p. 103-152. | MR | Zbl | DOI

[21] K. Kato & S. Saito, Unramified class field theory of arithmetical surfaces, Ann. of Math. 118 (1983), p. 241-275. | MR | Zbl | DOI

[22] K. Kato & S. Saito, Global class field theory of arithmetic schemes, in Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), Contemp. Math., vol. 55, Amer. Math. Soc., 1986, p. 255-331. | MR | Zbl | DOI

[23] N. M. Katz & S. Lang - Finiteness theorems in geometric classfield theory, Enseign. Math. 27 (1981), p. 285-319. | MR | Zbl

[24] M. Kerz - Higher class field theory and the connected component, prépublication arXiv:0711.4485. | MR | Zbl | DOI

[25] M. Kerz & A. Schmidt - Covering data and higher dimensional global class field theory, J. Number Theory 129 (2009), p. 2569-2599. | MR | Zbl | DOI

[26] M. Kertz & A. Schmidt, On different notions of tameness in arithmetic geometry, Math. Ann. 346 (2010), p. 641-668. | MR | Zbl | DOI

[27] S. Lang - Sur les séries $L$ d'une variété algébrique, Bull. Soc. Math. France 84 (1956), p. 385-407. | MR | Zbl | EuDML | Numdam | DOI

[28] S. Lang, Unramified class field theory over function fields in several variables, Ann. of Math. 64 (1956), p. 285-325. | MR | Zbl | DOI

[29] A. Merkurjev & A. Suslin - $K$-cohomologie des variétés de Severi-Brauer et l'homomorphisme de résidu normique (en russe), Izv. Akad. Nauk SSSR 46 (1982) ; traduction anglaise : Math. USSR Izvestiya 21 (1983), p. 307-340. | MR | Zbl

[30] J. S. Milne - Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton Univ. Press, 1980. | MR | Zbl

[31] J. Neukirch - Algebraic number theory, Grund. Math. Wiss., vol. 322, Springer, 1999. | MR | Zbl

[32] F. Orgogozo & I. Vidal - Le théorème de spécialisation du groupe fondamental, in Courbes semi-stables et groupe fondamental en géométrie algébrique (Luminy, 1998), Progr. Math., vol. 187, Birkhäuser, 2000, p. 169-184. | MR | Zbl

[33] A. N. Parshin - Revêtements abéliens de schémas arithmétiques (en russe), Dokl. Akad. Nauk SSSR 243 (1978), p. 855-858 ; traduction anglaise : Math. Dokl 19 (1978), p. 1438-1442. | Zbl

[34] B. Poonen - Bertini theorems over finite fields, Ann. of Math. 160 (2004), p. 1099-1127. | MR | Zbl | DOI

[35] W. Raskind - Abelian class field theory of arithmetic schemes, in K-theory and algebraic geometry : connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., 1995, p. 85-187. | MR | Zbl

[36] S. Saito - Unramified class field theory of arithmetical schemes, Ann. of Math. 121 (1985), p. 251-281. | MR | Zbl | DOI

[37] A. Schmidt - Tame coverings of arithmetic schemes, Math. Ann. 322 (2002), p. 1-18. | MR | Zbl | DOI

[38] A. Schmidt, Tame class field theory for arithmetic schemes, Invent. Math. 160 (2005), p. 527-565. | MR | Zbl | DOI

[39] A. Schmidt, Singular homology of arithmetic schemes, Algebra Number Theory 1 (2007), p. 183-222. | MR | Zbl | DOI

[40] A. Schmidt, Some consequences of Wiesend's higher dimensional class field theory, Math. Z. 256 (2007), p. 731-736. | MR | Zbl | DOI

[41] A. Schmidt & M. Spiess - Singular homology and class field theory of varieties over finite fields, J. reine angew. Math. 527 (2000), p. 13-36. | MR | Zbl

[42] J-P. Serre - Groupes algébriques et corps de classes, Publications de l'institut de mathématique de l'université de Nancago, VII Hermann, Paris, 1959. | MR | Zbl

[43] J-P. Serre, Zeta and $L$ functions, in Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, 1965, p. 82-92. | MR | Zbl

[44] J-P. Serre, Corps locaux, 2e éd., Hermann, 1968. | MR

[45] J-P. Serre, Théorie du corps de classes pour les revêtements non ramifiés de variétés algébriques (d'après S. Lang), Séminaire Bourbaki, vol. 1955/56, exposé n° 133, Collection hors série, vol. 3, Soc. Math. France, 1995, p. 347-355. | MR | Zbl | EuDML | Numdam

[46] A. Suslin & V. Voevodsky - Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996), p. 61-94. | MR | Zbl | EuDML | DOI

[47] A. Suslin & V. Voevodsky, Bloch-Kato conjecture and motivic cohomology with finite coefficients, in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., 2000, p. 117-189. | MR | Zbl | DOI

[48] T. Szamuely - Sur la théorie des corps de classes pour les variétés sur les corps $p$-adiques, J. reine angew. Math. 525 (2000), p. 183-212. | MR | Zbl | DOI

[49] T. Szamuely, Galois groups and fundamental groups, Cambridge Studies in Advanced Math., vol. 117, Cambridge Univ. Press, 2009. | MR | Zbl

[50] B. L. Van Der Waerden - Algebra II, 5e éd., Springer, 1967. | MR | Zbl

[51] G. Wiesend - A construction of covers of arithmetic schemes, J. Number Theory 121 (2006), p. 118-131. | MR | Zbl | DOI

[52] G. Wiesend, Class field theory for arithmetic schemes, Math. Z. 256 (2007), p. 717- 729. | MR | Zbl | DOI

[53] G. Wiesend, Tamely ramified covers of varieties and arithmetic schemes, Forum Math. 20 (2008), p. 515-522. | MR | Zbl | DOI