Algebraic solutions of differential equations (p-curvature and the Hodge filtration). (English) Zbl 0278.14004


14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
34G99 Differential equations in abstract spaces
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
Full Text: DOI EuDML


[1] Atiyah, M., Hodge, W.: Integrals of the second kind on an algebraic variety. Annals of Math.62, 56-91 (1955). · Zbl 0068.34401 · doi:10.2307/2007100
[2] Bateman Manuscript Project (Erdelyi Ed.): Higher transcendental functions, vol. 1. New-York: McGraw-Hill 1953. · Zbl 0052.29502
[3] Cartier, P.: Une nouvelle opération sur les formes différentielles. C. R. Acad. Sci. Paris244, 426-428 (1957). · Zbl 0077.04502
[4] Cartier, P.: Questions de rationalité des diviseurs en géométrie algébrique. Bull. Soc. Math. France86, 177-251 (1958). · Zbl 0091.33501
[5] Clemens, H.: Picard-Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities. Trans. Amer. Math. Soc.136, 93-108 (1969). · Zbl 0185.51302 · doi:10.1090/S0002-9947-1969-0233814-9
[6] Deligne, P.: Théorème de Lefschetz et critères de dégénérescence de suites spectrales. Publ. Math. I. H. E. S.35, 107-126 (1969). · Zbl 0159.22501
[7] Deligne, P.: Cohomologie des intersections complètes. Exposé XI, SGA 7, 1969. Multigraph available from I. H. E. S, 91-Bures-sur-Y vette, France.
[8] Deligne, P.: Travaux de Griffiths. Exposé 376, Séminaire N. Bourbaki, 1969/1970. Lectures Notes in Mathematics180. Berlin-Heidelberg-New York: Springer 1971.
[9] Deligne, P.: Equations différentielles à points singuliers réguliers. Lecture Notes in Mathematics163. Berlin-Heidelberg-New York: Springer 1970.
[10] Deligne, P.: Théorie de Hodge, Publ. Math. I. H. E. S.40 (1971). · Zbl 0219.14006
[11] Dwork, B.: P-adic cycles. Publ. Math. I. H. E. S.37, 27-115 (1970). · Zbl 0284.14008
[12] Goursat, E.: L’Equation d’Euler et de Gauss. Paris: Hermann 1936. · Zbl 0014.06202
[13] Goursat, E.: Intégrales Algébriques. Paris: Hermann 1938.
[14] Griffiths, P.: Periods of integrals on algebraic manifolds: summary of main results and discussion of open problems. Bull. Amer. Math. Soc.75, 2, 228-296 (1970). · Zbl 0214.19802 · doi:10.1090/S0002-9904-1970-12444-2
[15] Grothendieck, A., Dieudonne, J.: Eléments de géométrie algébrique, Publ. Math. I. H. E. S.4, 8, 11, 17, 20, 24, 28, 32.
[16] Grothendieck, A.: Fondements de la géométrie algébrique. Secrétariat mathématique, 11, rue Pierre Curie, Paris 5o, 1962.
[17] Grothendieck, A.: On the de Rham cohomology of algebraic varieties. Publ. Math. I. H. E. S.29 (1966). · Zbl 0145.17602
[18] Hartshorne, R.: Residues and duality. Lecture Notes in Mathematics20. Berlin-Heidelberg-New York: Springer 1966. · Zbl 0212.26101
[19] Hasse, H.: Existenz separabler zyklischer unverzweigter Erweiterungskörper vom Primzahlgradep über elliptischen Funktionenkörpern der Charakteristikp. J. Reine angew. Math.172, 77-85 (1934).
[20] Hasse, H., Witt, E.: Zyklische unverzweigte Erweiterungskörper vom Primzahlgradep über einem algebraischen Funktionenkörper der Charakteristikp. Monatsh. für Math. u. Phys.43, 477-492 (1936). · Zbl 0013.34102 · doi:10.1007/BF01707628
[21] Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero I, II. Annals of Math.79, 109-326 (1964). · Zbl 0122.38603 · doi:10.2307/1970486
[22] Hironaka, H.: Bimeromorphic smoothing of a complex-analytic space. Preprint available from Mathematics Institute, University of Warwick (1971). · Zbl 0407.32006
[23] Hochschild, G.: Simple algebras with purely inseparable splitting fields of exponent one. Trans. Amer. Math. Soc.79, 477-489 (1955). · Zbl 0065.01902
[24] Honda, T.: Differential equations and formal groups. (Preprint). Presented at the 1971 U.S.-Japan Number Theory conference. · Zbl 0337.14031
[25] Igusa, J.: Class number of a definite quaternion with prime discriminant. Proc. Nat’l. Acad. Sci.44, 312-314 (1958). · Zbl 0081.03601 · doi:10.1073/pnas.44.4.312
[26] Ihara, Y.: Schwarzian equations 1. Preprint, 1971. · Zbl 0353.14010
[27] Katz, N.: On the differential equations satisfied by period matrices. Publ. Math. I. H. E. S.35 (1968). · Zbl 0159.22502
[28] Katz, N.: Nilpotent connections and the monodromy theorem: application of a result of Turrittin. Publ. Math. I. H. E. S.39, 355-232 (1970). · Zbl 0221.14007
[29] Katz, N.: Une formule de congruence pour la fonction ?. Exposé XXII, SGA 7, 1969. multigraph available from I. H. E. S. 91-Bures-sur-Y vette, France.
[30] Kodaira, K., Spencer, D. C.: On deformations of complex structures, I, II, Annals of Math.67, 328-466 (1958). · Zbl 0128.16901 · doi:10.2307/1970009
[31] Manin, Ju.: Algebraic curves over fields with differentiation. AMS Translations (2),37, 59-78 (1964). · Zbl 0151.27601
[32] Manin, Ju.: The Hasse-Witt matrix of an algebraic curve. AMS Translations (2)45, 245-264 (1965). · Zbl 0148.28002
[33] Manin, Ju.: Rational points of algebraic curves over function fields. AMS Translations (2),50, 189-234 (1966). · Zbl 0178.55102
[34] Mazur, B.: Frobenius and the Hodge filtration, Preprint available form Mathematics Institute, University of Warwick (1971). · Zbl 0258.14006
[35] Mazur, B.: Frobenius and the Hodge filtration. (To appear.) · Zbl 0258.14006
[36] Messing, W.: On the nilpotence of the hypergeometric equation. (To appear.) · Zbl 0275.14002
[37] Messing, W.: The crystals associated to Barsoth-Tate groups; with applications to abelian schemes. Thesis, Princeton, 1971. (To appear.)
[38] Messing, W., Mazur, B.: Cristalline cohomology and the universal extension of an abelian scheme. (To appear.) · Zbl 0301.14016
[39] Mumford, D.: Abelian varieties. Bombay: Oxford University Press 1971. · Zbl 0222.14023
[40] Oda, T., Katz, N.: On the differentiation of de Rham cohomology classes with respect to parameters. J. Math. Kyoto Univer.8, 199-213 (1968). · Zbl 0165.54802
[41] Raynaud, M.: Geométrie algébrique et géométrie analytique. Exposé XII, SGA 1. Lecture Notes in Mathematics224. Berlin-Heidelberg-New York: Springer 1971.
[42] Serre, J.P.: Représentations linéaires des groupes finis. Paris: Hermann 1967.
[43] Serre, J.P.: Sur la topologie des variétés algébriques en caractéristiquep. Symposio International de Topologia Algebraica. Mexico, 1958.
[44] Serre, J.P.: Geométrie algébrique et géométrie analytique. Ann Inst. Fourier. Grenoble6, 1-42 (1956).
[45] Tate, J.:W-C groups overp-adic fields. Exposé 156. Séminaire BOurbaki 1957/1958. New-York: W. A. Benjamin 1966.
[46] Tessier, B., Lejeune, M.: Quelques calculs utiles pour la résolution des singularités. Multigraph available from Centre de Mathématiques, Ecole Polytechnique, 17, rue Descartes, Paris 5o, (1971).
[47] Verdier, J. L.: Base change for twisted inverse image of coherent sheaves. Algebraic geometry, Bombay: Oxford University Press 1968.
[48] Weil, A.: Variétés Kählériennes. Act. Sci. et Ind. 1267, Paris: Hermann 1968.
[49] Weil, A.: Jacobi sums as ?Größencharaktere?. Trans. Amer. Math. Soc.73, 487-495 (1952). · Zbl 0048.27001
[50] Whittaker, E. T., Watson, G.N.: A course of modern analysis. Cambridge: Cambridge University Press 1962. · Zbl 0105.26901
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