Équivalences rationnelle et numérique sur certaines variétés de type abélien sur un corps fini
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 36 (2003) no. 6, pp. 977-1002.
@article{ASENS_2003_4_36_6_977_0,
     author = {Kahn, Bruno},
     title = {\'Equivalences rationnelle et num\'erique sur certaines vari\'et\'es de type ab\'elien sur un corps fini},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {977--1002},
     publisher = {Elsevier},
     volume = {4e s{\'e}rie, 36},
     number = {6},
     year = {2003},
     doi = {10.1016/j.ansens.2003.02.002},
     zbl = {1073.14034},
     mrnumber = {2032532},
     language = {fr},
     url = {http://archive.numdam.org/articles/10.1016/j.ansens.2003.02.002/}
}
TY  - JOUR
AU  - Kahn, Bruno
TI  - Équivalences rationnelle et numérique sur certaines variétés de type abélien sur un corps fini
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2003
DA  - 2003///
SP  - 977
EP  - 1002
VL  - 4e s{\'e}rie, 36
IS  - 6
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.ansens.2003.02.002/
UR  - https://zbmath.org/?q=an%3A1073.14034
UR  - https://www.ams.org/mathscinet-getitem?mr=2032532
UR  - https://doi.org/10.1016/j.ansens.2003.02.002
DO  - 10.1016/j.ansens.2003.02.002
LA  - fr
ID  - ASENS_2003_4_36_6_977_0
ER  - 
%0 Journal Article
%A Kahn, Bruno
%T Équivalences rationnelle et numérique sur certaines variétés de type abélien sur un corps fini
%J Annales scientifiques de l'École Normale Supérieure
%D 2003
%P 977-1002
%V 4e s{\'e}rie, 36
%N 6
%I Elsevier
%U https://doi.org/10.1016/j.ansens.2003.02.002
%R 10.1016/j.ansens.2003.02.002
%G fr
%F ASENS_2003_4_36_6_977_0
Kahn, Bruno. Équivalences rationnelle et numérique sur certaines variétés de type abélien sur un corps fini. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 36 (2003) no. 6, pp. 977-1002. doi : 10.1016/j.ansens.2003.02.002. http://archive.numdam.org/articles/10.1016/j.ansens.2003.02.002/

[1] André Y., Cycles de Tate et cycles motivés sur les variétés abéliennes en caractéristique p>0, prépublication, 2003.

[2] André Y., Kahn B., Nilpotence, radicaux et structures monoïdales (avec un appendice de Peter O'Sullivan), Rend. Sem. Math. Univ. Padova 108 (2002) 107-291. | Numdam | MR | Zbl

[3] Beilinson A.A., Height pairings between algebraic cycles, in: Lect. Notes in Math., vol. 1289, Springer, 1987, pp. 1-26. | MR | Zbl

[4] Bloch S., Torsion algebraic cycles and a theorem of Roǐtman, Compositio Math. 39 (1979) 107-127. | Numdam | MR | Zbl

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

[6] Bloch S., Algebraic cycles and the Beĭlinson conjectures, in: The Lefschetz Centennial Conference (Mexico City, 1984), Contemp. Math., vol. 58(I), Amer. Math. Society, Providence, RI, 1986, pp. 65-79. | MR | Zbl

[7] Bloch S., The moving lemma for higher Chow groups, J. Alg. Geom. 3 (1994) 537-568. | MR | Zbl

[8] Bloch S., Lichtenbaum S., A spectral sequence for motivic cohomology, prépublication, 1996.

[9] Colliot-Thélène J.-L., Sansuc J.-J., Soulé C., Torsion dans le groupe de Chow de codimension 2, Duke Math. J. 50 (1983) 763-801. | MR | Zbl

[10] Colliot-Thélène J.-L., Hoobler R.T., Kahn B., The Bloch-Ogus-Gabber theorem, in: Fields Institute for Research in Mathematical Sciences Communications Series, vol. 16, Amer. Math. Society, Providence, RI, 1997, pp. 31-94. | MR | Zbl

[11] Deligne P., La conjecture de Weil, I, Publ. Math. IHÉS 43 (1974) 5-77. | MR

[12] Deligne P., Katz N., Groupes de Monodromie en géométrie algébrique (SGA 7) II, Lect. Notes in Math., vol. 340, Springer, 1973. | MR | Zbl

[13] Deligne P. et al. , Cohomologie étale (SGA 4 1/2), in: Lect. Notes in Math., vol. 569, Springer, 1977. | MR | Zbl

[14] Friedlander E., Suslin A., The spectral sequence relating algebraic K-theory and motivic cohomology, Ann. Sci. Éc. Norm. Sup. 35 (2002) 773-875. | Numdam | MR | Zbl

[15] Gabber O., Sur la torsion dans la cohomologie l-adique d'une variété, C. R. Acad. Sci. Paris 297 (1983) 179-182. | MR | Zbl

[16] Geisser T., Tate's conjecture, algebraic cycles and rational K-theory in characteristic p, 13 (1998) 109-122. | MR | Zbl

[17] Geisser T., Weil-étale motivic cohomology, prépublication, 2002 , http://www.math.uiuc.edu/K-theory#565.

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

[19] Geisser T., Levine M., The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky, J. Reine Angew. Math. 530 (2001) 55-103. | MR | Zbl

[20] Gillet H., Riemann-Roch theorems for higher algebraic K-theory, Adv. Math. 40 (1981) 203-289. | MR | Zbl

[21] Gillet H., Gersten's conjecture for the K-theory with torsion coefficients of a discrete valuation ring, J. Algebra 103 (1986) 377-380. | MR | Zbl

[22] Grayson D., Finite generation of the K-groups of a curve over a finite field (after D. Quillen), in: Lect. Notes in Math., vol. 966, Springer, 1982, pp. 69-90. | MR | Zbl

[23] Harder G., Die Kohomologie S-arithmetischer Gruppen über Funktionenkörpern, Invent. Math. 42 (1977) 135-175. | MR | Zbl

[24] Izhboldin O., On p-torsion in KM for fields of characteristic p, in: Algebraic K-Theory, Soviet Math., vol. 4, Amer. Math. Society, Providence, RI, 1991, pp. 129-144. | Zbl

[25] Jannsen U., Continuous étale cohomology, Math. Ann. 280 (1988) 207-245. | MR | Zbl

[26] Jannsen U., Motives, numerical equivalence and semi-simplicity, Invent. Math. 107 (1992) 447-452. | MR | Zbl

[27] De Jeu R., On K4(3) of curves over number fields, Invent. Math. 125 (1996) 523-556. | MR | Zbl

[28] De Jong A.J., Smoothness, semi-stability and alterations, Publ. Math. IHÉS 83 (1996) 51-93. | Numdam | MR | Zbl

[29] Kahn B., K3 d'un schéma régulier, C. R. Acad. Sci. Paris 315 (1992) 433-436. | MR | Zbl

[30] Kahn B., Deux théorèmes de comparaison en cohomologie étale ; applications, Duke Math. J. 69 (1993) 137-165. | MR | Zbl

[31] Kahn B., Résultats de “pureté” pour les variétés lisses sur un corps fini, in: Algebraic K-Theory and Algebraic Topology, NATO ASI Series, Ser. C, vol. 407, Kluwer, 1993, pp. 57-62. | Zbl

[32] Kahn B., Applications of weight-two motivic cohomology, Doc. Math. 1 (1996) 395-416. | MR | Zbl

[33] Kahn B., A sheaf-theoretic reformulation of the Tate conjecture, prépublication de l'Institut de Mathématiques de Jussieu no 150, 1998 , math.AG/9801017.

[34] Kahn B., K-theory of semi-local rings with finite coefficients and étale cohomology, 25 (2002) 99-139. | MR | Zbl

[35] Kahn B., The Geisser-Levine method revisited and algebraic cycles over a finite field, Math. Ann. 324 (2002) 581-617. | MR | Zbl

[36] Kahn B., Some finiteness results for étale cohomology, J. Number Theory 99 (2002) 57-73. | MR | Zbl

[37] Katsura T., Shioda T., On Fermat varieties, Tôhoku Math. J. 31 (1979) 97-115. | MR | Zbl

[38] Katz N., Messing W., Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974) 73-77. | MR | Zbl

[39] Kimura S.I., Chow motives can be finite-dimensional, in some sense, J. Alg. Geom., à paraître.

[40] Kratzer C., λ-structure en K-théorie algébrique, Comment. Math. Helv. 55 (1970) 233-254. | Zbl

[41] Lenstra H.W., Zarhin Y.G., The Tate conjecture for almost ordinary abelian varieties over finite fields, in: Advances in Number Theory (Kingston, ON, 1991), Oxford Univ. Press, 1993, pp. 179-194. | MR | Zbl

[42] Levine M., K-theory and motivic cohomology of schemes, I, prépublication, 2001.

[43] Lichtenbaum S., Values of zeta functions at non-negative integers, in: Lect. Notes in Math., vol. 1068, Springer, 1984, pp. 127-138. | MR | Zbl

[44] Lichtenbaum S., The Weil-étale topology, prépublication, 2001.

[45] Merkurjev A., Suslin A., K-cohomologie des variétés de Severi-Brauer et homomorphisme de reste normique, Izv. Akad. Nauk SSSR 46 (1982) 1011-1046, (en russe), Trad. anglaise , Math. USSR Izvestiya 21 (1983) 307-340. | Zbl

[46] Merkurjev A.S., Suslin A.A., Le groupe K3 pour un corps, Izv. Akad. Nauk SSSR 54 (1990) 339-356, (en russe), Trad. anglaise , Math. USSR Izv. 36 (1990) 541-565. | Zbl

[47] Milne J.S., Etale Cohomology, Princeton Univ. Press, Princeton, 1980. | MR | Zbl

[48] Milne J.S., Values of zeta functions of varieties over finite fields, Amer. J. Math. 108 (1986) 297-360. | MR | Zbl

[49] Milne J.S., Motivic cohomology and values of the zeta function, Compositio Math. 68 (1988) 59-102. | Numdam | MR | Zbl

[50] Milne J.S., Motives over finite fields, in: Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55(1), Amer. Math. Society, Providence, RI, 1994, pp. 401-459. | MR | Zbl

[51] Milne J.S., Lefschetz motives and the Tate conjecture, Compositio Math. 117 (1999) 47-81. | MR | Zbl

[52] Milne J.S., The Tate conjecture for certain abelian varieties over finite fields, Acta Arith. 100 (2001) 135-166. | MR | Zbl

[53] Milne J.S., Ramachandran N., Integral motives and special values of zeta functions, prépublication, 2002 (version préliminaire) , math.NT/0204065.

[54] Quillen D., On the cohomology and the K-theory of the general linear group over a finite field, Ann. of Math. 96 (1972) 179-198. | MR | Zbl

[55] Raskind W., A finiteness theorem in the Galois cohomology of algebraic number fields, Trans. Amer. Math. Soc. 303 (1987) 743-749. | MR | Zbl

[56] Rost M., Chow groups with coefficients, Doc. Math. 1 (1996) 319-393. | MR | Zbl

[57] Soulé C., Groupes de Chow et K-théorie de variétés sur un corps fini, Math. Ann. 268 (1984) 317-345. | MR | Zbl

[58] Soulé C., Opérations en K-théorie algébrique, Can. Math. J. 37 (1985) 488-550. | MR | Zbl

[59] Spiess M., Proof of the Tate conjecture for products of elliptic curves over finite fields, Math. Ann. 314 (1999) 285-290. | MR | Zbl

[60] Suslin A., Algebraic K-theory of fields, in: Proceedings of the International Congress of Mathematicians, Berkeley, 1986, pp. 222-244. | MR | Zbl

[61] Tate J.T., Algebraic cycles and poles of zeta functions, in: Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), 1965, pp. 93-110. | MR | Zbl

[62] Tate J.T., Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966) 134-144. | MR | Zbl

[63] Tate J.T., Conjectures on algebraic cycles in l-adic cohomology, in: Motives, Proc. Symposia Pure Math., vol. 55(1), Amer. Math. Society, Providence, RI, 1994, pp. 71-83. | MR | Zbl

[64] Voevodsky V., Motivic cohomology with Z/2 coefficients, Publ. Math. IHÉS, à paraître. | Numdam | Zbl

[65] Weil A., Courbes algébriques et variétés abéliennes, Hermann, 1954, rééd. 1971. | Zbl

[66] Zarhin Y.G., Abelian varieties of K3 type, in: Séminaire de théorie des nombres, Paris, 1990-1991, Progr. Math., vol. 108, Birkhäuser, 1992, pp. 263-279. | MR | Zbl

Cited by Sources: