Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 

 

Exponential sums over finite fields and differential equations over the complex numbers: Some interactions


Author: Nicholas M. Katz
Journal: Bull. Amer. Math. Soc. 23 (1990), 269-309
MSC (1985): Primary 10G05, 11L40, 14F99, 32C38, 22E47
MathSciNet review: 1032857
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • Alan Adolphson and Steven Sperber, Exponential sums and Newton polyhedra: cohomology and estimates, Ann. of Math. (2) 130 (1989), no. 2, 367–406. MR 1014928, 10.2307/1971424
  • E. Artin, Quadratische Körper im Gebiete der höheren Kongruenzen. I, Math. Z. 19 (1924), no. 1, 153–206 (German). MR 1544651, 10.1007/BF01181074
  • James Ax, Zeroes of polynomials over finite fields, Amer. J. Math. 86 (1964), 255–261. MR 0160775
  • A. A. Beĭlinson, On the derived category of perverse sheaves, 𝐾-theory, arithmetic and geometry (Moscow, 1984–1986) Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 27–41. MR 923133, 10.1007/BFb0078365
  • A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
  • G. V. Belyĭ, Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 2, 267–276, 479 (Russian). MR 534593
  • [Ber] J. Bernstein, Six lectures on the algebraic theory of D-modules, Xeroxed notes, E.T.H., Zurich, 1983.
  • Pierre Berthelot, Cohomologie cristalline des schémas de caractéristique 𝑝>0, Lecture Notes in Mathematics, Vol. 407, Springer-Verlag, Berlin-New York, 1974 (French). MR 0384804
  • Alan Adolphson and Steven Sperber, Exponential sums and Newton polyhedra: cohomology and estimates, Ann. of Math. (2) 130 (1989), no. 2, 367–406. MR 1014928, 10.2307/1971424
  • E. Artin, Quadratische Körper im Gebiete der höheren Kongruenzen. I, Math. Z. 19 (1924), no. 1, 153–206 (German). MR 1544651, 10.1007/BF01181074
  • James Ax, Zeroes of polynomials over finite fields, Amer. J. Math. 86 (1964), 255–261. MR 0160775
  • A. A. Beĭlinson, On the derived category of perverse sheaves, 𝐾-theory, arithmetic and geometry (Moscow, 1984–1986) Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 27–41. MR 923133, 10.1007/BFb0078365
  • A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
  • G. V. Belyĭ, Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 2, 267–276, 479 (Russian). MR 534593
  • [Ber] J. Bernstein, Six lectures on the algebraic theory of D-modules, Xeroxed notes, E.T.H., Zurich, 1983.
  • Pierre Berthelot, Cohomologie cristalline des schémas de caractéristique 𝑝>0, Lecture Notes in Mathematics, Vol. 407, Springer-Verlag, Berlin-New York, 1974 (French). MR 0384804
  • F. Beukers and G. Heckman, Monodromy for the hypergeometric function _{𝑛}𝐹_{𝑛-1}, Invent. Math. 95 (1989), no. 2, 325–354. MR 974906, 10.1007/BF01393900
  • Frits Beukers, W. Dale Brownawell, and Gert Heckman, Siegel normality, Ann. of Math. (2) 127 (1988), no. 2, 279–308. MR 932298, 10.2307/2007054
  • B. J. Birch, How the number of points of an elliptic curve over a fixed prime field varies, J. London Math. Soc. 43 (1968), 57–60. MR 0230682
  • A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers, Algebraic 𝐷-modules, Perspectives in Mathematics, vol. 2, Academic Press, Inc., Boston, MA, 1987. MR 882000
  • [Bour1 ] N. Bourbaki, Groupes et algebres de Lie, Chapter 1, Diffusion CCLS, Paris, 1971.
  • Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1981 (French). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]. MR 647314
  • [Bour3] N. Bourbaki, Groupes et algebres de Lie, Chapters 7 and 8, Diffusion CCLS, Paris, 1975.
  • Nicolas Bourbaki, Éléments de mathématique: groupes et algèbres de Lie, Masson, Paris, 1982 (French). Chapitre 9. Groupes de Lie réels compacts. [Chapter 9. Compact real Lie groups]. MR 682756
  • Jean-Luc Brylinski, Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Astérisque 140-141 (1986), 3–134, 251 (French, with English summary). Géométrie et analyse microlocales. MR 864073
  • Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. MR 0144979
  • Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders; Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 632548
  • [De1] P. Deligne, Rapport sur la formule des traces, in Cohomologie Etale (SGA4½), Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin, 1977, pp. 76-109.
  • [De2] P. Deligne, Application de la formule des traces aux sommes trigonométriques, in Cohomologie Etale (SGA4½), Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin, 1977, pp. 168-232.
  • P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 111–195 (French). MR 1106898
  • Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
  • P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, Vol. 569, Springer-Verlag, Berlin-New York, 1977. Séminaire de Géométrie Algébrique du Bois-Marie SGA 41\over2; Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier. MR 0463174
  • Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 0340258
  • [De7] P. Deligne, La conjecture de WeilII, Publ. Math. I.H.E.S. 52 (1981), 313-428.
  • Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982. MR 654325
  • Max Deuring, Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. III, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa. 1956 (1956), 37–76 (German). MR 0079611
  • B. Dwork, Bessel functions as 𝑝-adic functions of the argument, Duke Math. J. 41 (1974), 711–738. MR 0387281
  • [EGA] A. Grothendieck, Rédigé avec la collaboration de Dieudonné, J., Elements de géométrie algébrique, Publ. Math. I.H.E.S. 4, 8, 11, 17, 20, 24, 28, 32, (1960-1967).
  • Torsten Ekedahl, On the adic formalism, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 197–218. MR 1106899
  • [Er] A. Erdelyi, Higher transcendental functions, vol. I (Bateman Manuscript Project), McGraw-Hill, New York, 1953.
  • Leonard Evens, A generalization of the transfer map in the cohomology of groups, Trans. Amer. Math. Soc. 108 (1963), 54–65. MR 0153725, 10.1090/S0002-9947-1963-0153725-1
  • I. M. Gel′fand, General theory of hypergeometric functions, Dokl. Akad. Nauk SSSR 288 (1986), no. 1, 14–18 (Russian). MR 841131
  • John Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc. 301 (1987), no. 1, 77–101. MR 879564, 10.1090/S0002-9947-1987-0879564-8
  • A. Grothendieck, Crystals and the de Rham cohomology of schemes, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 306–358. Notes by I. Coates and O. Jussila. MR 269663
  • [Gro2] A. Grothendieck, Formule de Lefschetz et rationalité des fonctions L, Séminaire Bourbaki 1964-65, Exposé 279, reprinted in Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968.
  • [Gr3] A. Grothendieck, et al. Séminaire de géomètre algébrique du Bois-Marie, SGA 1; SGA 4 Parts I, II, and III; SGA 41/2; SGA 5; SGA 7 Parts I and II, Lecture Notes in Math., Vols. 224, 269-270, 305, 569, 589, 288-340, Springer-Verlag, Berlin, 1971-1977.
  • [Ha] H. Hasse, Beweis des Analogons der Riemannschen Vermutung fur die Artinsche und F. K. Schmidschen Kongruenz-zetafunktionen in gewissen elliptischen Fallen, Ges. d. Wiss. Nach. Math-Phys. Klasse 3 (1933), 253-262.
  • D. R. Heath-Brown and S. J. Patterson, The distribution of Kummer sums at prime arguments, J. Reine Angew. Math. 310 (1979), 111–130. MR 546667
  • P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, Vol. 569, Springer-Verlag, Berlin-New York, 1977. Séminaire de Géométrie Algébrique du Bois-Marie SGA 41\over2; Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier. MR 0463174
  • Luc Illusie, Deligne’s 𝑙-adic Fourier transform, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 151–163. MR 927979
  • Masaki Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci. 20 (1984), no. 2, 319–365. MR 743382, 10.2977/prims/1195181610
  • Nicholas M. Katz, Algebraic solutions of differential equations (𝑝-curvature and the Hodge filtration), Invent. Math. 18 (1972), 1–118. MR 0337959
  • Nicholas M. Katz, On the calculation of some differential Galois groups, Invent. Math. 87 (1987), no. 1, 13–61. MR 862711, 10.1007/BF01389152
  • Nicholas M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, vol. 116, Princeton University Press, Princeton, NJ, 1988. MR 955052
  • Nicholas M. Katz, Local-to-global extensions of representations of fundamental groups, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 4, 69–106 (English, with French summary). MR 867916
  • Nicholas M. Katz, On the monodromy groups attached to certain families of exponential sums, Duke Math. J. 54 (1987), no. 1, 41–56. MR 885774, 10.1215/S0012-7094-87-05404-4
  • Nicholas M. Katz, Perversity and exponential sums, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 209–259. MR 1097616
  • Nicholas M. Katz, Exponential sums and differential equations, Annals of Mathematics Studies, vol. 124, Princeton University Press, Princeton, NJ, 1990. MR 1081536
  • Nicholas M. Katz, A conjecture in the arithmetic theory of differential equations, Bull. Soc. Math. France 110 (1982), no. 2, 203–239 (English, with French summary). MR 667751
  • Nicholas M. Katz, Sommes exponentielles, Astérisque, vol. 79, Société Mathématique de France, Paris, 1980 (French). Course taught at the University of Paris, Orsay, Fall 1979; With a preface by Luc Illusie; Notes written by Gérard Laumon; With an English summary. MR 617009
  • Nicholas M. Katz, Travaux de Laumon, Astérisque 161-162 (1988), Exp. No. 691, 4, 105–132 (1989). Séminaire Bourbaki, Vol. 1987/88. MR 992205
  • Nicholas M. Katz and Gérard Laumon, Transformation de Fourier et majoration de sommes exponentielles, Inst. Hautes Études Sci. Publ. Math. 62 (1985), 361–418 (French). MR 823177
  • Nicholas M. Katz and Richard Pink, A note on pseudo-CM representations and differential Galois groups, Duke Math. J. 54 (1987), no. 1, 57–65. MR 885775, 10.1215/S0012-7094-87-05405-6
  • Neal Koblitz, The number of points on certain families of hypersurfaces over finite fields, Compositio Math. 48 (1983), no. 1, 3–23. MR 700577
  • E. R. Kolchin, Algebraic groups and algebraic dependence, Amer. J. Math. 90 (1968), 1151–1164. MR 0240106
  • Bertram Kostant, A characterization of the classical groups, Duke Math. J. 25 (1958), 107–123. MR 0095883
  • Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947
  • G. Laumon, Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 131–210 (French). MR 908218
  • G. Laumon, Semi-continuité du conducteur de Swan (d’après P. Deligne), The Euler-Poincaré characteristic (French), Astérisque, vol. 83, Soc. Math. France, Paris, 1981, pp. 173–219 (French). MR 629128
  • A. H. M. Levelt, Jordan decomposition for a class of singular differential operators, Ark. Mat. 13 (1975), 1–27. MR 0500294
  • [Le2] G. Laumon, Hypergeometric functions, thesis, University of Amsterdam, 1961.
  • Juri I. Manin, Moduli fuchsiani, Ann. Scuola Norm. Sup. Pisa (3) 19 (1965), 113–126 (Italian). MR 0180581
  • Zoghman Mebkhout, Sur le problème de Hilbert-Riemann, Complex analysis, microlocal calculus and relativistic quantum theory (Proc. Internat. Colloq., Centre Phys., Les Houches, 1979) Lecture Notes in Phys., vol. 126, Springer, Berlin-New York, 1980, pp. 90–110 (French). MR 579742
  • [Me2] Z. Mebhkout, Une équivalence de catégories et une autre équivalence de catégories, Compositio Math. 51 (1984), 55-69.
  • [Me3] Z. Mebhkout, Le formalisme des six operations de Grothendieck pour les coefficients de de Rham, Seminaire de Plans-sur-Bex, mars 1984, Travaux en Cours, Hermann, 1986.
  • S. J. Patterson, On the distribution of Kummer sums, J. Reine Angew. Math. 303/304 (1978), 126–143. MR 514676, 10.1515/crll.1978.303-304.126
  • Kenneth A. Ribet, Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math. 98 (1976), no. 3, 751–804. MR 0457455
  • Neantro Saavedra Rivano, Catégories Tannakiennes, Lecture Notes in Mathematics, Vol. 265, Springer-Verlag, Berlin-New York, 1972 (French). MR 0338002
  • Jean-Pierre Serre, Abelian 𝑙-adic representations and elliptic curves, 2nd ed., Advanced Book Classics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. With the collaboration of Willem Kuyk and John Labute. MR 1043865
  • Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 0236190
  • Friedrich Karl Schmidt, Analytische Zahlentheorie in Körpern der Charakteristik 𝑝, Math. Z. 33 (1931), no. 1, 1–32 (German). MR 1545199, 10.1007/BF01174341
  • J.-L. Verdier, Spécialisation de faisceaux et monodromie modérée, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 332–364 (French). MR 737938
  • André Weil, Jacobi sums as “Grössencharaktere”, Trans. Amer. Math. Soc. 73 (1952), 487–495. MR 0051263, 10.1090/S0002-9947-1952-0051263-0
  • Hiroyuki Yoshida, On an analogue of the Sato conjecture, Invent. Math. 19 (1973), 261–277. MR 0337977
  • Yu. G. Zarkhin, Weights of simple Lie algebras in the cohomology of algebraic varieties, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 2, 264–304 (Russian). MR 740792
  • Yu. G. Zarhin, Linear simple Lie algebras and ranks of operators, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 481–495. MR 1106920, 10.1007/978-0-8176-4576-2_12

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1985): 10G05, 11L40, 14F99, 32C38, 22E47

Retrieve articles in all journals with MSC (1985): 10G05, 11L40, 14F99, 32C38, 22E47


Additional Information

DOI: https://doi.org/10.1090/S0273-0979-1990-15922-1