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ISSN 1088-6834(e) ISSN 0894-0347(p)

Integral motives and special values of zeta functions

Author(s): James S. Milne; Niranjan Ramachandran
Journal: J. Amer. Math. Soc. 17 (2004), 499-555.
MSC (2000): Primary 11G09; Secondary 14F42, 14G10
Posted: April 26, 2004
MathSciNet review: 2053950
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Abstract: For each field , we define a category of rationally decomposed mixed motives with $\mathbb{Z}$ -coefficients. When is finite, we show that the category is Tannakian, and we prove formulas relating the behaviour of zeta functions near integers to certain $\operatorname{Ext}$ groups.

References:

1.: André, Yves, Pour une théorie inconditionnelle des motifs. Inst. Hautes Études Sci. Publ. Math. No. 83, (1996), 5-49. MR 98m:14022
2.: André, Yves; Kahn, Bruno, Construction inconditionnelle de groupes de Galois motiviques. C. R. Math. Acad. Sci. Paris 334 (2002), no. 11, 989-994. MR 2003k:14020
3.: Atiyah, M. F., and Hirzebruch, F., Analytic cycles on complex manifolds. Topology 1 (1962), 25-45. MR 26:3091
4.: Ballico, E., Ciliberto, C., and Catanese F., Trento examples. In: Classification of irregular varieties. Minimal models and abelian varieties. Proceedings of the conference held in Trento, December 17-21, 1990. Edited by E. Ballico, F. Catanese and C. Ciliberto. Lecture Notes in Mathematics, 1515. Springer-Verlag, Berlin, 1992, 134-139. MR 93d:14005
5.: Bloch, Spencer, and Esnault, Hélène, The coniveau filtration and non-divisibility for algebraic cycles. Math. Ann. 304 (1996), no. 2, 303-314. MR 97c:14003
6.: Brosnan, Patrick, Steenrod operations in Chow theory. Trans. Amer. Math. Soc. 355 (2003), no. 5, 1869-1903.
7.: Deligne, P., Le groupe fondamental de la droite projective moins trois points. Galois groups over $\mathbb{Q} {}$ (Berkeley, CA, 1987), 79-297, Math. Sci. Res. Inst. Publ., 16, Springer, New York-Berlin, 1989. MR 90m:14016
8.: Deligne, P., Catégories tannakiennes. The Grothendieck Festschrift, Vol. II, 111-195, Progr. Math., 87, Birkhäuser Boston, Boston, MA, 1990. MR 92d:14002
9.: Deligne, Pierre, À quoi servent les motifs? Motives (Seattle, WA, 1991), 143-161, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.MR 95c:14013
10.: Deligne, Pierre, and Milne, James, Tannakian Categories. In: Hodge cycles, motives, and Shimura varieties. Lecture Notes in Mathematics, 900. Springer-Verlag, Berlin-New York, 1982, 101-228. MR 84m:14046
11.: Demazure, Michel, Lectures on -divisible groups. Lecture Notes in Mathematics, 302. Springer-Verlag, Berlin-New York, 1972. MR 49:9000
12.: Dwyer, William G., and Friedlander, Eric M., Algebraic and étale -theory. Trans. Amer. Math. Soc. 292 (1985), no. 1, 247-280. MR 87h:18013
13.: Ekedahl, Torsten, Diagonal complexes and -gauge structures. Travaux en Cours. Hermann, Paris, 1986. MR 88g:14015
14.: Faltings, Gerd, and Wüstholz, Gisbert (Eds). Rational points. Papers from the seminar held at the Max-Planck-Institut für Mathematik, Bonn, 1983/1984. Aspects of Mathematics, E6. Friedr. Vieweg & Sohn, Braunschweig; distributed by Heyden & Son, Inc., Philadelphia, Pa., 1984. MR 87h:14016
15.: Fontaine, Jean-Marc, and Mazur, Barry, Geometric Galois representations. Elliptic curves, modular forms, and Fermat's last theorem (Hong Kong, 1993), 41-78, Ser. Number Theory, I, Internat. Press, Cambridge, MA, 1995. MR 96h:11049
16.: Fontaine, J.-M., and Illusie, L., -adic periods: a survey. Proceedings of the Indo-French Conference on Geometry (Bombay, 1989), 57-93, Hindustan Book Agency, Delhi, 1993. MR 95e:14013
17.: Gabber, Ofer, Sur la torsion dans la cohomologie -adique d'une variété. C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 3, 179-182. MR 85f:14018
18.: Giraud, Jean. Méthode de la descente. Bull. Soc. Math. France Mém. 2 (1964), viii+150 pp. MR 32:7556
19.: Giraud, Jean, Cohomologie non abélienne, Springer-Verlag, Berlin, 1971. MR 49:8992
20.: Griffiths, Phillip, and Harris, Joe, On the Noether-Lefschetz theorem and some remarks on codimension-two cycles. Math. Ann. 271 (1985), no. 1, 31-51. MR 87a:14030
21.: Hanamura, Masaki, Mixed motives and algebraic cycles. I. Math. Res. Lett. 2 (1995), no. 6, 811-821. MR 97d:14014
22.: Hanamura, Masaki, Mixed motives and algebraic cycles. III. Math. Res. Lett. 6 (1999), no. 1, 61-82. MR 2000d:14011
23.: Hiller, Howard L., $\lambda$ -rings and algebraic -theory. J. Pure Appl. Algebra 20 (1981), no. 3, 241-266. MR 82e:18016
24.: Huber, Annette, Calculation of derived functors via Ind-categories. J. Pure Appl. Algebra 90 (1993), no. 1, 39-48. MR 94i:18001
25.: Jacobson, Nathan, The Theory of Rings. American Mathematical Society Mathematical Surveys, vol. I. American Mathematical Society, New York, 1943. MR 5:31f
26.: Jannsen, Uwe, Motives, numerical equivalence, and semi-simplicity. Invent. Math. 107 (1992), 447-452. MR 93g:14009
27.: Jensen, C. U. Les foncteurs dérivés de $\varprojlim$ et leurs applications en théorie des modules. Lecture Notes in Mathematics, Vol. 254. Springer-Verlag, Berlin-New York, 1972. MR 53:10874
28.: de Jong, A. J., Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. No. 83, (1996), 51-93. MR 98e:14011
29.: de Jong, A. J., Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic. Invent. Math. 134 (1998), no. 2, 301-333, Erratum ibid. 138 (1999), no. 1, 225. MR 2000f:14070a; MR 2000f:14070b
30.: Katz, Nicholas M. and Messing, William, Some consequences of the Riemann hypothesis for varieties over finite fields. Invent. Math. 23 (1974), 73-77. MR 48:11117
31.: Kleiman, Steven L., The standard conjectures. Motives (Seattle, WA, 1991), 3-20, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994. MR 95k:14010
32.: Kratzer, Ch., $\lambda$ -structure en -théorie algébrique. Comment. Math. Helv. 55 (1980), no. 2, 233-254. MR 81m:18011
33.: Lang, Serge, and Tate, John, Principal homogeneous spaces over abelian varieties. Amer. J. Math. 80 (1958), 659-684. MR 21:4960
34.: Levine, Marc, Mixed motives. Mathematical Surveys and Monographs, 57. American Mathematical Society, Providence, RI, 1998. MR 99i:14025
35.: Lichtenbaum, Stephen, The Weil-étale topology. Preprint (2002) (available at www.math. brown.edu/ $\sim$ slicht/) Compositio Math. (to appear).
36.: Milne, J. S., Extensions of abelian varieties defined over a finite field. Invent. Math. 5 (1968), 63-84. MR 37:5226
37.: Milne, J. S., Étale cohomology. Princeton Mathematical Series, 33. Princeton University Press, Princeton, NJ, 1980. MR 81j:14002
38.: Milne, J. S., Values of zeta functions of varieties over finite fields. Amer. J. Math. 108 (1986), no. 2, 297-360. MR 87g:14019
39.: Milne, J. S., Arithmetic duality theorems. Perspectives in Mathematics, 1. Academic Press, Inc., Boston, Mass., 1986. MR 88e:14028
40.: Milne, J. S., Motives over finite fields. Motives (Seattle, WA, 1991), 401-459, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.MR 95g:11053
41.: Milne, J. S., Lefschetz motives and the Tate conjecture, Compositio Math. 117 (1999), 47-81.MR 2000i:11090
42.: Milne, J. S., Gerbes and abelian motives, preprint 2003, arXiv:math.AG/0301304.
43.: Mitchell, Barry, Theory of categories. Pure and Applied Mathematics, Vol. XVII Academic Press, New York-London, 1965. MR 34:2647
44.: Oort, F., Yoneda extensions in abelian categories. Math. Ann. 153 (1964), 227-235. MR 29:140
45.: Quillen, Daniel, On the cohomology and -theory of the general linear groups over a finite field. Ann. of Math. (2) 96 (1972), 552-586. MR 47:3565
46.: Roos, Jan-Erik. Bidualité et structure des foncteurs dérivés de $\varprojlim$ dans la catégorie des modules sur un anneau régulier. C. R. Acad. Sci. Paris 254 (1962), 1720-1722.MR 25:106b
47.: Saavedra Rivano, Neantro, Catégories Tannakiennes. Lecture Notes in Mathematics, Vol. 265. Springer-Verlag, Berlin-New York, 1972. MR 49:2769
48.: Schoen, Chad, An integral analog of the Tate conjecture for one-dimensional cycles on varieties over finite fields. Math. Ann. 311 (1998), no. 3, 493-500. MR 99e:14005
49.: Serre, Jean-Pierre, Groupes de Grothendieck des schémas en groupes réductifs déployés. Inst. Hautes Études Sci. Publ. Math. No. 34 (1968), 37-52. MR 38:159
50.: Soulé, Christophe, -théorie des anneaux d'entiers de corps de nombres et cohomologie étale. Invent. Math. 55 (1979), no. 3, 251-295. MR 81i:12016
51.: Soulé, Christophe, -theory and values of zeta functions. Algebraic -theory and its applications (Trieste, 1997), 255-283, World Sci. Publishing, River Edge, NJ, 1999. MR 2000g:11063
52.: Tate, John, Endomorphisms of abelian varieties over finite fields. Invent. Math. 2 (1966), 134-144. MR 34:5829
53.: Tate, John, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki: Vol. 1965/66, Exposé 306, 1966. MR 99f:00041
54.: Tate, John, Relations between $K_{2}$ and Galois cohomology. Invent. Math. 36 (1976), 257-274.MR 55:2847
55.: Tate, John, Conjectures on algebraic cycles in $\ell$ -adic cohomology. Motives (Seattle, WA, 1991), 71-83, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.MR 95a:14010
56.: Verdier, Jean-Louis, Des catégories dérivées des catégories abéliennes. Astérisque No. 239 , 1996.MR 98c:18007
57.: Voevodsky, V., Triangulated categories and motives over a field, in Cycles, transfers, and motivic homology theories. Annals of Mathematics Studies, 143. Princeton University Press, Princeton, NJ, 2000, 188-238. MR 2001d:14026
58.: Zarhin, Ju. G., Endomorphisms of Abelian varieties over fields of finite characteristic. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 2, 272-277, 471. MR 51:8114

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