p-adic variation of the zeta-function over families of varieties defined over finite fields
Koblitz, Neal
Compositio Mathematica, Tome 31 (1975), p. 119-218 / Harvested from Numdam
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@article{CM_1975__31_2_119_0,
     author = {Koblitz, Neal},
     title = {$p$-adic variation of the zeta-function over families of varieties defined over finite fields},
     journal = {Compositio Mathematica},
     volume = {31},
     year = {1975},
     pages = {119-218},
     mrnumber = {414557},
     zbl = {0332.14008},
     language = {en},
     url = {http://dml.mathdoc.fr/item/CM_1975__31_2_119_0}
}

Koblitz, Neal. $p$-adic variation of the zeta-function over families of varieties defined over finite fields. Compositio Mathematica, Tome 31 (1975) pp. 119-218. http://gdmltest.u-ga.fr/item/CM_1975__31_2_119_0/

[1] M. Artin: Supersingular K3 Surfaces (to appear). | Numdam | MR 371899 | Zbl 0322.14014

[2] J. Ax: Zeroes of Polynomials over Finite Fields. Amer. J. Math., 86, 1964. | MR 160775 | Zbl 0121.02003

[3] N. Bourbaki: Algèbre commutative. Paris: Hermann, 1961.

[4] P. Cartier: Une nouvelle opération sur les formes différentielles. C. R. Acad. Sci. Paris 244, 1957. | MR 84497 | Zbl 0077.04502

[5] P. Deligne: Théorème de Lefschetz et critères de dégénérescence de suites spectrales. Publ. Math. I.H.E.S. 35, 1969. | Numdam | MR 244265 | Zbl 0159.22501

[6] P. Deligne and D. Mumford: The Irreducibility of the Space of Curves of Given Genus. Publ. Math. I.H.E.S. 36, 1969. | Numdam | MR 262240 | Zbl 0181.48803

[7] M. Demazure: Lectures on p-Divisible Groups, Lecture Notes in Mathematics 302, Berlin- Heidelberg-New York: Springer, 1972. | MR 883960 | Zbl 0247.14010

[8] B. Dwork: On the Congruence Properties of the Zeta Function of Algebraic Varieties. J. für die Reine und Angewandte Math. 203, 1960. | MR 140493 | Zbl 0119.36804

[9] B. Dwork: Normalized Period Matrices I and II. Annals of Math. 94, 2nd series, 1971, and Annals of Math. 98, 2nd series, 1973. | Zbl 0241.14011 | Zbl 0265.14008

[10] B. Dwork: On the Rationality of the Zeta Function of an Algebraic Variety. Amer. J. Math. 82, 1960. | MR 140494 | Zbl 0173.48501

[11] B. Dwork: On the Zeta Function of a Hypersurface II. Annals of Math. 80, 2nd series, 1964. | MR 188215 | Zbl 0173.48601

[12] A. Grothendieck: Fondements de la géométrie algébrique, Secrétariat Mathématique, 11 rue Pierre Curie, Paris 5e, France, 1962. | MR 146040 | Zbl 0239.14002

[13] A. Grothendieck: Séminaire de géométrie algébrique, 1, Lecture Notes in Mathematics 224, Berlin-Heidelberg-New York: Springer, 1971. | MR 354651 | Zbl 0234.14002

[14] A. Grothendieck: Séminaire de géométrie algébrique, 2, I.H.E.S., 1962. | Zbl 0197.47202

[15] H. Hasse: Existenz Separabler Zyklischer Unverzweigter Erweiterungskörper vom Primzahlgrade p über Elliptischen Funktionenkörpern der Charakteristik p. J. Reine Angew. Math. 172, 1934. | JFM 60.0910.02 | Zbl 0010.14803

[16] H. Hasse and E. Witt: Zyklische Unverzweigte Erweiterungskörper vom Primzahlgrade p über einem Algebraischen Funktionenkörper der Charakteristik p. Monatsh. für Math. u. Phys. 43, 1936. | JFM 62.0112.01 | Zbl 0013.34102

[17] T. Honda: Isogeny Classes of Abelian Varieties over Finite Fields. J. Math. Soc. Japan 20, 1968. | MR 229642 | Zbl 0203.53302

[18] J. Igusa: Arithmetic Variety of Moduli for Genus Two. Annals of Math. 72, 2nd series, 1960. | MR 114819 | Zbl 0122.39002

[19] J. Igusa: Class Number of a Definite Quaternion with Prime Discriminant. Proc. Natl. Acad. Sci. 44, 1958. | MR 98728 | Zbl 0081.03601

[20] N. Katz: Algebraic Solutions of Differential Equations (p-Curvature and the Hodge Filtration), Inventiones Math. 18, 1972. | MR 337959 | Zbl 0278.14004

[21] N. Katz: Une formule de congruence pour la fonction ζ. S.G.A. 7 II, Lecture Notes in Mathematics 340, Berlin-Heidelberg-New York: Springer, 1973. | Zbl 0275.14015

[22] N. Katz: Nilpotent Connections and the Monodromy Theorem: Applications of a Result of Turrittin. Publ. Math. I.H.E.S. 39, 1971. | Numdam | MR 291177 | Zbl 0221.14007

[23] N. Katz: Le niveau de la cohomologie des intersections complètes, S.G.A. 7 II, Lecture Notes in Mathematics 340, Berlin-Heidelberg-New York: Springer, 1973. | Zbl 0288.14001

[24] N. Katz: A Note on a Theorem of Ax. Amer. J. Math., 93, 1971. | MR 288099 | Zbl 0237.12012

[25] N. Katz: Pinceaux de Lefschetz: théorème d'existence. S.G.A. 7 II, Lecture Notes in Mathematics 340, Berlin-Heidelberg-New York: Springer, 1973. | Zbl 0284.14006

[26] N. Katz: Review of Gauss Sums à la Stickelberger (unpublished notes).

[27] N. Katz: Travaux de Dwork. Exp. 409. Séminaire Bourbaki 1971/72, Lecture Notes in Mathematics 317, Berlin-Heidelberg-New York: Springer, 1973. | Numdam | MR 498577 | Zbl 0259.14007

[28] N. Katz and T. Oda: On the Differentiation of De Rham Cohomology Classes with Respect to Parameters. J. Math. Kyoto Univ. 8, 1968. | MR 237510 | Zbl 0165.54802

[29] S. Kleiman: Weil Cohomologies in Dix exposés sur la théorie des schémas. Amsterdam: North-Holland, 1969.

[30] S. Lang: Algebra. Reading, Mass.:Addison-Wesley, 1965. | MR 197234 | Zbl 0193.34701

[31] Ju. Manin: Algebraic Curves over Fields with Differentiation. A.M.S. Translations (2) 37, 1964. | Zbl 0151.27601

[32] Ju. Manin: The Hasse-Witt Matrix of an Algebraic Curve. A.M.S. Translations (2) 45, 1965. | Zbl 0148.28002

[33] Ju. Manin: Lekcii po Algebraičeskoĭ Geometrii. Moscow: Izd. Mosk. Univ., 1970.

[34] Ju. Manin: The Theory of Commutative Formal Groups over Fields of Finite Characteristic. Translated in Russian Math. Surv. 18, 1963. | MR 157972 | Zbl 0128.15603

[35] B. Mazur: Frobenius and the Hodge Filtration. Bull. A.M.S. 78, 1972. | MR 330169 | Zbl 0258.14006

[36] B. Mazur: Frobenius and the Hodge Filtration (Estimates), Annals of Math. 98, 2nd series, 1973. | MR 321932 | Zbl 0261.14005

[37] B. Mazur and W. Messing: Universal Extensions and One Dimensional Crystalline Cohomology. Lecture Notes in Mathematics 370, Berlin-Heidelberg-New York: Springer, 1974. | MR 374150 | Zbl 0301.14016

[38] W. Messing: The Crystals Associated to Barsotti-Tate Groups : With Applications to Abelian Schemes. Lecture Notes in Mathematics 264, Berlin-Heidelberg -New York: Springer, 1972. | MR 347836 | Zbl 0243.14013

[39] L. Miller: Curves over a Finite Field with Invertible Hasse-Witt Matrix. Math. Annalen 197, 1972. | Zbl 0235.14009

[40] L. Miller: Über Gewöhnliche Projektive Hyperflächen (preprint).

[41] P. Monsky: P-adic Analysis and Zeta Functions. Lectures in Mathematics No. 4, Tokyo: Kinokuniya Book-Store Ltd., 1970. | MR 282981 | Zbl 0256.14009

[42] D. Mumford: Abelian Varieties. Bombay: Oxford University Press, 1970. | MR 282985 | Zbl 0223.14022

[43] D. Mumford: Geometric Invariant Theory. New York: Academic Press, 1965. | MR 214602

[44] D. Mumford: Introduction to Algebraic Geometry (preliminary version of first three chapters). Harvard University.

[45] D. Mumford: Prym Varieties (to appear).

[46] D. Mumford and K. Suominen: Introduction to the Theory of Moduli, in Proc. 5th Nordic Summer Sch. The Netherlands: Wolters-Noordhoff Pub., 1970. | MR 437531 | Zbl 0242.14004

[47] T. Oda: The First De Rham Cohomology Group and Dieudonné Modules. Annales sci. de l'École Norm. Sup., 4e serie 2, 1969. | Numdam | MR 241435 | Zbl 0175.47901

[48] A. Ogus and G. Bergman: Nakayama's Lemma for Half-Exact Functors. Proc. of the A.M.S. 31, 1972. | MR 302633 | Zbl 0206.33201

[49] F. Oort: Finite Group Schemes, Local Moduli for Abelian Varieties, and Lifting Problems, in Proc. 5th Nordic Summer Sch. The Netherlands: Wolters-Noordhoff Pub., 1970. | Zbl 0239.14018

[50] F. Oort: Subvarieties of Moduli Spaces. Inventiones math. 24, 1974. | MR 424813 | Zbl 0259.14011

[51] I. Šafarevič: Foundations of Algebraic Geometry. translated in Russian Math. Surv. 24, 1969. | MR 265356 | Zbl 0204.21301

[52] M. Schlessinger: Functors of Artin Rings. Trans. A.M.S. 130, 1968. | MR 217093 | Zbl 0167.49503

[53] J.-P. Serre: Algèbre Locale. Multiplicités. Lecture Notes in Mathematics 11, Berlin- Heidelberg-New York: Springer, 1965. | MR 201468 | Zbl 0142.28603

[54] J.-P. Serre: Faisceaux algébriques cohérents. Annals of Math. 61, 2nd series, 1955. | MR 68874 | Zbl 0067.16201

[55] J.-P. Serre: Sur la topologie des variétés algébriques en caractéristique p. Symposio Intern. de Top. Alg., Mexico, 1958. | MR 98097 | Zbl 0098.13103

[56] L. Stickelberger: Über eine Verallgemeinerung der Kreistheilung. Math. Ann. 37, 1890. | JFM 22.0100.01 | MR 1510649

[57] J. Tate: Algebraic Cycles and Poles of Zeta Functions, in Arithmetical Algebraic Geometry (Proc. of Purdue Conf.), New York: Harper and Row, 1965. | MR 225778 | Zbl 0213.22804

[58] J. Tate: Classes d'isogénie des variétés abéliennes sur un corpsfini (d'après T. Honda). Exp. 352, Séminaire Bourbaki 1968/69, Lecture Notes in Mathematics 179, Berlin- Heidelberg-New York: Springer, 1971. | Numdam | Zbl 0212.25702

[59] J. Tate: Endomorphisms of Abelian Varieties over Finite Fields. Inventiones Math. 2, 1966. | MR 206004 | Zbl 0147.20303

[60] E. Warning: Bemerkung zur Verstehenden Arbeit von Herr Chevalley, Abh. Math. Sem. Univ. Hamburg, 11, 1936. | JFM 61.1043.02 | Zbl 0011.14601

[61] A. Weil: Jacobi Sums as 'Grössencharaktere', Trans. A.M.S. 73, 1952. | MR 51263 | Zbl 0048.27001

[62] A. Weil: Numbers of Solutions of Equations in Finite Fields. Bull. A.M.S. 55, 1949. | MR 29393 | Zbl 0032.39402

[63] P. Deligne and M. Rapoport: Les schémas de modules de courbes elliptiques, in Lecture Notes in Mathematics 349 (Proc. Intern. Summer Sch. on Modular Functions), Berlin -Heidelberg-New York: Springer, 1973. | MR 337993 | Zbl 0281.14010

[64] H.W. Lenstra Jr. and F. Oort: Simple Abelian Varieties Having a Prescribed Formal Isogeny Type. Univ. of Amsterdam Math. Inst. Report 73-02, 1973.