Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

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

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Exponential sums over finite fields and differential equations over the complex numbers: Some interactions
HTML articles powered by AMS MathViewer

by Nicholas M. Katz PDF
Bull. Amer. Math. Soc. 23 (1990), 269-309
References
  • 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, DOI 10.2307/1971424
  • E. Artin, Quadratische Körper im Gebiete der höheren Kongruenzen. I, Math. Z. 19 (1924), no. 1, 153–206 (German). MR 1544651, DOI 10.1007/BF01181074
  • James Ax, Zeroes of polynomials over finite fields, Amer. J. Math. 86 (1964), 255–261. MR 160775, DOI 10.2307/2373163
  • 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, DOI 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, DOI 10.2307/1971424
  • E. Artin, Quadratische Körper im Gebiete der höheren Kongruenzen. I, Math. Z. 19 (1924), no. 1, 153–206 (German). MR 1544651, DOI 10.1007/BF01181074
  • James Ax, Zeroes of polynomials over finite fields, Amer. J. Math. 86 (1964), 255–261. MR 160775, DOI 10.2307/2373163
  • 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, DOI 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 $_nF_{n-1}$, Invent. Math. 95 (1989), no. 2, 325–354. MR 974906, DOI 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, DOI 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 230682, DOI 10.1112/jlms/s1-43.1.57
  • 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, Inc.), New York-London, 1962. MR 0144979
  • Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. 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, DOI 10.1007/BFb0061194
  • P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
  • Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258, DOI 10.1007/BF02684373
  • [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, DOI 10.1007/978-3-540-38955-2
  • 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 387281, DOI 10.1215/S0012-7094-74-04176-3
  • [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 153725, DOI 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, DOI 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, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
  • 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, DOI 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 337959, DOI 10.1007/BF01389714
  • Nicholas M. Katz, On the calculation of some differential Galois groups, Invent. Math. 87 (1987), no. 1, 13–61. MR 862711, DOI 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, DOI 10.1515/9781400882120
  • 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, DOI 10.5802/aif.1069
  • 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, DOI 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, DOI 10.2969/aspm/01710209
  • Nicholas M. Katz, Exponential sums and differential equations, Annals of Mathematics Studies, vol. 124, Princeton University Press, Princeton, NJ, 1990. MR 1081536, DOI 10.1515/9781400882434
  • 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, DOI 10.24033/bsmf.1960
  • 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, DOI 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 240106, DOI 10.2307/2373294
  • Bertram Kostant, A characterization of the classical groups, Duke Math. J. 25 (1958), 107–123. MR 95883
  • 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, DOI 10.1007/BF02698937
  • 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 500294, DOI 10.1007/BF02386195
  • [Le2] G. Laumon, Hypergeometric functions, thesis, University of Amsterdam, 1961.
  • Juri I. Manin, Moduli fuchsiani, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 19 (1965), 113–126 (Italian). MR 180581
  • 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, DOI 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 457455, DOI 10.2307/2373815
  • 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 236190, DOI 10.2307/1970722
  • Friedrich Karl Schmidt, Analytische Zahlentheorie in Körpern der Charakteristik $p$, Math. Z. 33 (1931), no. 1, 1–32 (German). MR 1545199, DOI 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 51263, DOI 10.1090/S0002-9947-1952-0051263-0
  • Hiroyuki Yoshida, On an analogue of the Sato conjecture, Invent. Math. 19 (1973), 261–277. MR 337977, DOI 10.1007/BF01425416
  • 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, DOI 10.1007/978-0-8176-4576-2_{1}2
Similar Articles
Additional Information
  • Journal: Bull. Amer. Math. Soc. 23 (1990), 269-309
  • MSC (1985): Primary 10G05, 11L40, 14F99, 32C38, 22E47
  • DOI: https://doi.org/10.1090/S0273-0979-1990-15922-1
  • MathSciNet review: 1032857