Équivalences rationnelle et numérique sur certaines variétés de type abélien sur un corps fini

Kahn, Bruno

doi:10.1016/j.ansens.2003.02.002

Kahn, Bruno

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 36 (2003) no. 6, pp. 977-1002.

MR Zbl | 2 citations dans Numdam

DOI : 10.1016/j.ansens.2003.02.002

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     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},
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Kahn, Bruno. Équivalences rationnelle et numérique sur certaines variétés de type abélien sur un corps fini. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 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/

Bibliographie
Cité par

[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 K^M_∗ 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

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