zbMATH — the first resource for mathematics

Rationalité et valeurs de fonctions L en cohomologie cristalline. (Rationality and values of L-functions in crystalline cohomology). (English) Zbl 0624.14016
In Sémin. Bourbaki 1971-72, Exposé 409, Lect. Notes Math. 317, 167- 200 (1973; Zbl 0259.14007), N. Katz conjectured the p-adic meromorphy of the function L(X,E,t) attached to a smooth variety X over a finite field \({\mathbb{F}}_ q (q=p^ a)\) and to an F-crystal E on X. If X is proper and smooth over \({\mathbb{F}}_ q\) we prove that L is rational and given by the usual formula using the action of Frobenius on crystalline cohomology with coefficients in E; this result was only known, via “Weil conjectures”, for particular unit-root F-crystals: those issued of a representation of \(\pi _ 1(X)\) through a finite quotient. The proof of the theorem involves the formalism of a cohomology class associated to a morphism of crystals, extending the fundamental class of an algebraic cycle, and leading to a Lefschetz trace formula. When E is a unit-root F- crystal the link between crystalline cohomology and De Rham-Witt complex with coefficients in E enables us to interpret zeroes and poles of L of the form \(t=q^{-r}\), r an integer. Under certain hypotheses this complex yields also equivalents of the L function in the neighbourhood of the preceding poles: these results extend those of Milne for zeta functions.

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14F30 \(p\)-adic cohomology, crystalline cohomology
14F40 de Rham cohomology and algebraic geometry
14G20 Local ground fields in algebraic geometry
Full Text: DOI Numdam EuDML
[1] P. BAYER, J. NEUKIRCH, On values of zeta functions and l-adic Euler characteristics, Invent. Math., 50 (1978), 35-64. · Zbl 0409.12018
[2] P. BERTHELOT, Cohomologie cristalline des schémas de caractéristique p > 0, Lecture Notes in Math., n° 407, Springer Verlag, 1974. · Zbl 0298.14012
[3] P. BERTHELOT, Sur le “Théorème de Lefschetz faible” en cohomologie cristalline, C.R.A.S. Paris, t. 277, 12 nov. 1973, série A, p. 955-958. · Zbl 0268.14007
[4] P. BERTHELOT, Le théorème de dualité plate pour LES surfaces, d’après J. S. Milne, dans “Surfaces algébriques”, Lecture Notes in Math., n° 868, Springer Verlag, 1981. · Zbl 0469.14010
[5] P. BERTHELOT, W. MESSING, Théorie de Dieudonné cristalline I, Astérisque, 63 (1979), 17-38. · Zbl 0414.14014
[6] P. BERTHELOT, A. OGUS, Notes on crystalline cohomology, Mathematical Notes, n° 21, Princeton University Press, (1978). · Zbl 0383.14010
[7] P. BERTHELOT, A. OGUS, F-isocrystals and de Rham cohomology I, Invent. Math,. 72 (1983), 159-199. · Zbl 0516.14017
[8] N. BOURBAKI, Algèbre et algèbre commutative, Hermann.
[9] H. CARTAN, S. EILENBERG, Homological algebra, Princeton University Press, 1956. · Zbl 0075.24305
[10] R. CREW: L-functions of p-adic characters and geometric Iwasawa theory, Invent. Math., 88 (1987), 395-403. · Zbl 0615.14013
[11] R. CREW, F-isocrystals and p-adic representations. Algebraic geometry-Bowdoin 1985, PSPM, 46, Part 2 (1987), 111-138. · Zbl 0639.14011
[12] J.-L. COLLIOT-THELENE, J.-J. SANSUC, C. SOULÉ, Torsion dans le groupe de Chow de codimension deux, Duke Math. Journal, vol. 50, n° 3 (sept. 1983). · Zbl 0574.14004
[13] P. DELIGNE, La conjecture de Weil I, Publ. Math. I.H.E.S., n° 43.
[14] J.-Y. ETESSE, Complexe de de Rham-Witt à coefficients dans un F-cristal unité et dualité plate pour LES surfaces. Thèse 3e cycle, Rennes 1981.
[15] J.-Y. ETESSE, Complexe de de Rham-Witt à coefficients dans un cristal Compositio Math., 66 (1988), 57-120. · Zbl 0708.14013
[16] J.-Y. ETESSE, Dualité plate pour LES surfaces, à coeffients dans un groupe de type multiplicatif. Preprint Rennes. (A paraître au Bulletin de la SMF.).
[17] A. GROTHENDIECK, J. DIEUDONNÉ, Éléments de géométrie algébrique, Publ. Math. I.H.E.S. 8, 11, 17, 20, 24, 28, 32. EGA I, Grundlehren n° 166, Springer-Verlag, 1971. · Zbl 0203.23301
[18] M. GROS, Classes de Chern et classes de cycles en cohomologie logarithmique, Bull. Soc. Math. Fr., 113, Fasc. 4 (1985). · Zbl 0615.14011
[19] O. GABBER, Sur la torsion dans la cohomologie l-adique d’une variété, C.R. Acad. Sc. Paris, t. 297, Série I (1983), 179-182. · Zbl 0574.14019
[20] A. GROTHENDIECK, Formule de Lefschetz et rationalité des fonctions L, Séminaire Bourbaki n° 279, décembre 1964. · Zbl 0199.24802
[21] L. ILLUSIE, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Scient. Ec. Norm. Sup., 4e série, t. 12 (1979), 501-661. · Zbl 0436.14007
[22] L. ILLUSIE, M. RAYNAUD, LES suites spectrales associées au complexe de de Rham-Witt, Publ. Math. I.H.E.S., n° 57. · Zbl 0538.14012
[23] N. KATZ, Travaux de dwork, Séminaire Bourbaki n° 409, février 1972, Lecture Notes in Math., n° 383, Springer Verlag.
[24] N. KATZ, Crystalline cohomology, Dieudonné modules and Jacobi sums, in Automorphic forms, Bombay 1979, Springer Verlag, 1981. · Zbl 0502.14007
[25] N. KATZ, P-adic properties of modular schemes and modular forms, dans “Modular Functions of one Variable III”, Lecture Notes in Math., n° 350, Springer Verlag (1973). · Zbl 0271.10033
[26] N. KATZ, W. MESSING, Some consequences of the Riemann hypothesis for varieties over finite fields, Inventiones Math., 23 (1974), 73-77. · Zbl 0275.14011
[27] S. LICHTENBAUM, Values of zeta and L-functions at zero, Astérisque, (1975), 24-25. · Zbl 0312.12016
[28] S. LICHTENBAUM, Zeta functions of varieties over finite fields at s = 1, Arithmetic and Geometry, Progress in Math. 35, Birkhäuser (1983), 173-194. · Zbl 0567.14015
[29] S. LICHTENBAUM, Values of zeta-functions at non-negative integers, Journées arithmétiques, Noordwijkerhout, July 1983. · Zbl 0591.14014
[30] J.-S. MILNE, On a conjecture of Artin and Tate, Annals of Math., 102 (1975), 517-533. · Zbl 0343.14005
[31] J.-S. MILNE, Duality in the flat cohomology of a surface, Ann. Scient. Ec. Norm., Sup. 4e série, t. 9 (1976), 171-202. · Zbl 0334.14010
[32] J.-S. MILNE, Etale cohomology, Princeton University Press, 1980. · Zbl 0433.14012
[33] J.-S. MILNE, Values of zeta functions of varieties over finite fields, Am. Jour. of Math., 108 (1986), 297-360. · Zbl 0611.14020
[34] J.-S. MILNE, Values of zeta functions over finite fields : complements and corrections to, Amer. J. Math., 108 (1986), 297-360 5th May, 1986. · Zbl 0611.14020
[35] A. OGUS, F-crystals and Griffiths transversality, Int. Symposium on Algebraic Geometry, Kyoto (1977), 15-44. · Zbl 0427.14007
[36] N. SAAVEDRA, Catégories tanakiennes, Lecture Notes in Math., n° 265, Springer Verlag (1972). · Zbl 0241.14008
[37] P. SCHNEIDER, On the values of the zeta function of a variety over a finite field, Compositio Math., 46 (1982), 133-143. · Zbl 0505.14020
[38] J.-P. SERRE, Corps locaux, Hermann, 1968.
[39] J.-P. SERRE, Groupes pro-algébriques, Publ. Math. I.H.E.S. n° 7 (1960).
[40] J.-P. SERRE, Représentations linéaires des groupes finis, Hermann, 1967. · Zbl 0189.02603
[41] J.-P. SERRE, Sur la topologie des variétés algébriques en caractéristique p, Symposium internacional de topologia algebraica, Mexico (1958), 24-53. · Zbl 0098.13103
[42] Cohomologie étale, Lecture Notes in Math., n° 569, Springer Verlag (1977). · Zbl 0345.00010
[43] Cohomologie l-adique et fonctions L, Lecture Notes in Math. n° 589, Springer Verlag (1977). · Zbl 0345.00011
[44] Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Math., n° 225, Springer Verlag (1971).
[45] J. TATE, On a conjecture of Birch and Swinnerton-Dyer and a geometric analogue, Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam (1968), 189-214. · Zbl 0199.55604
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.