|
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
-
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
(91e:11094), http://dx.doi.org/10.2307/1971424
-
E.
Artin, Quadratische Körper im Gebiete der höheren
Kongruenzen. I, Math. Z. 19 (1924), no. 1,
153–206 (German). MR
1544651, http://dx.doi.org/10.1007/BF01181074
-
James
Ax, Zeroes of polynomials over finite fields, Amer. J. Math.
86 (1964), 255–261. MR 0160775
(28 #3986)
-
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
(89b:14027), http://dx.doi.org/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
(86g:32015)
-
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
(80f:12008)
-
[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, 1974 (French). MR 0384804
(52 #5676)
-
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
(91e:11094), http://dx.doi.org/10.2307/1971424
-
E.
Artin, Quadratische Körper im Gebiete der höheren
Kongruenzen. I, Math. Z. 19 (1924), no. 1,
153–206 (German). MR
1544651, http://dx.doi.org/10.1007/BF01181074
-
James
Ax, Zeroes of polynomials over finite fields, Amer. J. Math.
86 (1964), 255–261. MR 0160775
(28 #3986)
-
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
(89b:14027), http://dx.doi.org/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
(86g:32015)
-
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
(80f:12008)
-
[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, 1974 (French). MR 0384804
(52 #5676)
-
F.
Beukers and G.
Heckman, Monodromy for the hypergeometric function
_{𝑛}𝐹_{𝑛-1}, Invent. Math. 95
(1989), no. 2, 325–354. MR 974906
(90f:11034), http://dx.doi.org/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
(90e:11106), http://dx.doi.org/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
(37 #6242)
-
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
(89g:32014)
-
[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
(83g:17001)
-
[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
(84i:22001)
-
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
(88j:32013)
-
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
(26 #2519)
-
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
(82i:20001)
-
[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
(92d:14002)
-
Pierre
Deligne, Équations différentielles à points
singuliers réguliers, Lecture Notes in Mathematics, Vol. 163,
Springer-Verlag, Berlin, 1970 (French). MR 0417174
(54 #5232)
-
P.
Deligne, Cohomologie étale, Lecture Notes in
Mathematics, Vol. 569, Springer-Verlag, Berlin, 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
(57 #3132)
-
Pierre
Deligne, La conjecture de Weil. I, Inst. Hautes Études
Sci. Publ. Math. 43 (1974), 273–307 (French). MR 0340258
(49 #5013)
-
[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, 1982. MR 654325
(84m:14046)
-
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
(18,113e)
-
B.
Dwork, Bessel functions as 𝑝-adic functions of the
argument, Duke Math. J. 41 (1974), 711–738. MR 0387281
(52 #8124)
-
[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
(92b:14010)
-
[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 (27 #3686), http://dx.doi.org/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 (87h:22012)
-
John
Greene, Hypergeometric functions over finite
fields, Trans. Amer. Math. Soc.
301 (1987), no. 1,
77–101. MR
879564 (88e:11122), http://dx.doi.org/10.1090/S0002-9947-1987-0879564-8
-
A.
Grothendieck, Crystals and the de Rham cohomology of schemes,
Dix Exposés sur la Cohomologie des Schémas, North-Holland,
Amsterdam, 1968, pp. 306–358. MR 0269663
(42 #4558)
-
[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 (81e:10033)
-
P.
Deligne, Cohomologie étale, Lecture Notes in
Mathematics, Vol. 569, Springer-Verlag, Berlin, 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
(57 #3132)
-
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
(89h:11052)
-
Masaki
Kashiwara, The Riemann-Hilbert problem for holonomic systems,
Publ. Res. Inst. Math. Sci. 20 (1984), no. 2,
319–365. MR
743382 (86j:58142), http://dx.doi.org/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
(49 #2728)
-
Nicholas
M. Katz, On the calculation of some differential Galois
groups, Invent. Math. 87 (1987), no. 1,
13–61. MR
862711 (88c:12010), http://dx.doi.org/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
(91a:11028)
-
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
(88a:14032)
-
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
(88i:11053), http://dx.doi.org/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
(92m:11080)
-
Nicholas
M. Katz, Exponential sums and differential equations, Annals
of Mathematics Studies, vol. 124, Princeton University Press,
Princeton, NJ, 1990. MR 1081536
(93a:14009)
-
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
(84h:14014a)
-
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
(82m:10059)
-
Nicholas
M. Katz, Travaux de Laumon, Astérisque
161-162 (1988), Exp. No. 691, 4, 105–132 (1989).
Séminaire Bourbaki, Vol. 1987/88. MR 992205
(90h:14028)
-
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
(87i:14017)
-
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 (88j:12005), http://dx.doi.org/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
(85d:14034)
-
E.
R. Kolchin, Algebraic groups and algebraic dependence, Amer.
J. Math. 90 (1968), 1151–1164. MR 0240106
(39 #1460)
-
Bertram
Kostant, A characterization of the classical groups, Duke
Math. J. 25 (1958), 107–123. MR 0095883
(20 #2381)
-
Serge
Lang, Algebraic number theory, Addison-Wesley Publishing Co.,
Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947
(44 #181)
-
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
(88g:14019)
-
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
(83g:14007)
-
A.
H. M. Levelt, Jordan decomposition for a class of singular
differential operators, Ark. Mat. 13 (1975),
1–27. MR
0500294 (58 #17962)
-
[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
(31 #4815)
-
Zoghman
Mebkhout, Sur le problème de Hilbert-Riemann, (Proc.
Internat. Colloq., Centre Phys., Les Houches, 1979) Lecture Notes in
Phys., vol. 126, Springer, Berlin, 1980, pp. 90–110
(French). MR
579742 (82a:32015)
-
[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
(80m:10031), http://dx.doi.org/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
(56 #15660)
-
Neantro
Saavedra Rivano, Catégories Tannakiennes, Lecture Notes
in Mathematics, Vol. 265, Springer-Verlag, Berlin, 1972 (French). MR 0338002
(49 #2769)
-
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
(91b:11071)
-
Jean-Pierre
Serre and John
Tate, Good reduction of abelian varieties, Ann. of Math. (2)
88 (1968), 492–517. MR 0236190
(38 #4488)
-
Friedrich
Karl Schmidt, Analytische Zahlentheorie in Körpern der
Charakteristik 𝑝, Math. Z. 33 (1931),
no. 1, 1–32 (German). MR
1545199, http://dx.doi.org/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
(86f:32010)
-
André
Weil, Jacobi sums as
“Grössencharaktere”, Trans.
Amer. Math. Soc. 73
(1952), 487–495. MR 0051263
(14,452d), http://dx.doi.org/10.1090/S0002-9947-1952-0051263-0
-
Hiroyuki
Yoshida, On an analogue of the Sato conjecture, Invent. Math.
19 (1973), 261–277. MR 0337977
(49 #2746)
-
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
(85k:14012)
-
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
(92g:17007), http://dx.doi.org/10.1007/978-0-8176-4576-2_12
- [AdSp] A. Adolphson and S. Sperber, Exponential sums and Newton polyhedra; cohomology and estimates, Ann. of Math. 130 (1989), 367-406. MR 1014928
- [Ar] E. Artin, Quadratische Korper in Gebiete der hoheren KongruenzenI and II, Math. Z. 19 (1924), 153-246. MR 1544651
- [Ax] J. Ax, Zeros of polynomials over finite fields, Amer. J. Math. 86 (1964), 255-261. MR 160775
- [Bei] A. A. Beilinson, On the derived category of the category of the category of perverse sheaves, in Yu. I. Manin (ed.), K-Theorey, arithmetic and geometry, Lecture Notes in Math., vol. 1289, Springer-Verlag, Berlin, 1987, pp. 27-41. MR 923133
- [BBD] A. A. Beilinson, I. N. Bernstein and P. Deligne, Faisceaux pervers, entire contents of Analyse et Topologie sur les espaces singuliers I, Conference de Luminy, Astérisque 100 (1982). MR 751966
- [Bel] G. V. Belyi, On Galois extensions of a maximal cyclotomic field, Math. USSR-Izv. 14(2) (1980), 247-256. MR 534593
- [Ber] J. Bernstein, Six lectures on the algebraic theory of D-modules, Xeroxed notes, E.T.H., Zurich, 1983.
- [Bert] P. Berthelot, Cohomologie cristalline des schémas de caractéristique p > 0, Lecture Notes in Math., vol. 407, Springer-Verlag, Berlin, 1974. MR 384804
- [AdSp] A. Adolphson and S. Sperber, Exponential sums and Newton polyhedra; cohomology and estimates, Ann. of Math. 130 (1989), 367-406. MR 1014928
- [Ar] E. Artin, Quadratische Korper in Gebiete der hoheren KongruenzenI and II, Math. Z. 19 (1924), 153-246. MR 1544651
- [Ax] J. Ax, Zeros of polynomials over finite fields, Amer. J. Math. 86 (1964), 255-261. MR 160775
- [Bei] A. A. Beilinson, On the derived category of the category of the category of perverse sheaves, in Yu. I. Manin (ed.), K-Theorey, arithmetic and geometry, Lecture Notes in Math., vol. 1289, Springer-Verlag, Berlin, 1987, pp. 27-41. MR 923133
- [BBD] A. A. Beilinson, I. N. Bernstein and P. Deligne, Faisceaux pervers, entire contents of Analyse et Topologie sur les espaces singuliers I, Conference de Luminy, Astérisque 100 (1982). MR 751966
- [Bel] G. V. Belyi, On Galois extensions of a maximal cyclotomic field, Math. USSR-Izv. 14(2) (1980), 247-256. MR 534593
- [Ber] J. Bernstein, Six lectures on the algebraic theory of D-modules, Xeroxed notes, E.T.H., Zurich, 1983.
- [Bert] P. Berthelot, Cohomologie cristalline des schémas de caractéristique p > 0, Lecture Notes in Math., vol. 407, Springer-Verlag, Berlin, 1974. MR 384804
- [BH] F. Beukers and G. Heckman, Monodromy for the hypergeometric function -1, Invent. Math. 95 (1989), 325-354. MR 974906
- [BBH] F. Beukers, D. Brownawell and G. Heckman, Siegel normality, Ann. of Math. 127(1988), 279-308. MR 932298
- [Bi] B. 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
- [Bor] A. Borel, et al. Algebraic D-Modules, Academic Press, Boston, 1987. MR 882000
- [Bour1 ] N. Bourbaki, Groupes et algebres de Lie, Chapter 1, Diffusion CCLS, Paris, 1971.
- [Bour2] N. Bourbaki, Groupes et algebres de Lie, Chapters 4, 5 and 6, Masson, Paris, 1981. MR 647314
- [Bour3] N. Bourbaki, Groupes et algebres de Lie, Chapters 7 and 8, Diffusion CCLS, Paris, 1975.
- [Bour4] N. Bourbaki, Groupes et algebres de Lie, Chapter 9, Masson, Paris, 1982. MR 682756
- [Br] J.-L. Brylinski, Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Astérisque 140-141 (1986), 3-134. MR 864073
- [CR1] C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Interscience Publ., New York and London, 1962. MR 144979
- [CR2] C. W. Curtis and I. Reiner, Methods of representation theory with applications to finite groups and orders, vol. I, John Wiley and Sons, New York, 1981. 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.
- [De3] P. Deligne, Catégories tannakiennes, Grothendieck Festschrift (to appear). MR 1106898
- [De4] P. Deligne, Equations differentielles à points singuliers reguliers, Lecture Notes in Math., vol. 163, Springer-Verlag, Berlin, 1970. MR 417174
- [De5] P. Deligne, Theoremes de finitude en cohomologique l-adique, in Cohomologie Etale (SGA4½), Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin, 1977, pp. 233-251. MR 463174
- [De6] P. Deligne, La conjecture de WeilI, Publ. Math. I.H.E.S. 48 (1974), 273-308. MR 340258
- [De7] P. Deligne, La conjecture de WeilII, Publ. Math. I.H.E.S. 52 (1981), 313-428.
- [DM] P. Deligne and J. Milne, Tannakian categories, in P. Deligne, J. Milne, A. Ogus and K.-Y. Shih(eds.), Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math., vol. 900, Springer-Verlag, Berlin, 1982, pp. 101-228. MR 654325
- [Deu] Deuring, Die Zetafunktion einer algebraischen Kurve von Geschlechte Eins, Drei Mitteilungen, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II, 1953, pp. 85-94; 1955, pp. 13-43; 1956, pp. 37-76. MR 79611
- [Dw] B. Dwork, Bessel functions as p-adic functions of the argument, Duke Math. J. 41 (1974), 711-738. MR 387281
- [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).
- [Ek] T. Ekedahl, On the adic formalism, Grothendieck Festschrift (to appear). MR 1106899
- [Er] A. Erdelyi, Higher transcendental functions, vol. I (Bateman Manuscript Project), McGraw-Hill, New York, 1953.
- [Ev] L. Evens, A generalization of the transfer map in the cohomology of groups, Trans. Amer. Math. Soc. 108 (1963), 54-65. MR 153725
- [Gel] I. M. Gel' fand, The general theory of hypergeometric functions, Dokl. Akad. Nauk SSSR 288 (1) (1986), 14-18. MR 841131
- [Gre] J. Greene, Hypergeometric functions over finite fields, Transl. Amer. Math. Soc. 301 (1987), 77-101. MR 879564
- [Gro1] A. Grothendieck, Crsytals and the De Rham cohomology of schemes, reprinted in Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968. 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.
- [HBP] D. J. Heath-Brown and S. J. Patterson, The distribution of Kummer sums at prime arguments, J. Reine Angew. Math. 310 (1979), 111-130. MR 546667
- [Ill1] L. Illusie, Appendix to Deligne, P., Theoremes de finitude en cohomologique l-adique, Cohomologie Etale (SGA4½), Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin, 1977, 233-251. MR 463174
- [Ill2] L. Illusie, Deligne's l-adic Fourier transform, Algebraic Geometry. Bowdoin 1985 (S.J. Bloch, ed.), Amer. Math. Soc. Providence, 1987. MR 927979
- [K] M. Kashiwara, The Riemann-Hilbert problem for holonomic systems, RIMS, Kyoto Univ., 1983. MR 743382
- [Ka1] N. Katz, Algebraic solutions of differential equations; p-curvature and the Hodge filtration, Invent. Math. 18 (1972), 1-118. MR 337959
- [Ka2] N. Katz, On the calculation of some differential Galois groups, Invent. Math. 87(1987), 13-61. MR 862711
- [Ka3] N. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Ann. Of Math. Study, vol. 116, Princeton Univ. Press, Princeton, N.J., 1988. MR 955052
- [Ka4] N. Katz, Local to global extensions of representations of fundamental groups, Ann. Inst. Fourier (Grenoble) 36 (4) (1986), 59-106. MR 867916
- [Ka5] N. Katz, On the monodromy groups attached to certain families of exponential sums, Duke Math. J. 54 (1) (1987), 41-56. MR 885774
- [Ka6] N. Katz, Perversity and exponential sums, in Algebraic number theory-in honor of K. Iwasawa, Adv. Studies Pure Math., vol. 17, North-Holland, Amsterdam, 1989, pp. 209-259. MR 1097616
- [Ka7] N. Katz, Exponential sums and differential equations, Ann. of Math. Study 124, Princeton University Press, Princeton, N.J., 1990. MR 1081536
- [Ka8] N. Katz, A conjecture in the arithmetic theory of differential equations, Bull. Soc. Math. France 110 (1982), 203-239. MR 667751
- [Ka9] N. Katz, Sommes exponentielles, rédigé par G. Laumon, Astérisque 79 ( 1980). MR 617009
- [Ka10] N. Katz, Travaux de Laumon, Séminaire Bourbaki 1987-88, Exposé 691, Astérisque (to appear). MR 992205
- [KL] N. Katz and G. Laumon, Transformation de Fourier et majoration de sommes exponentielles, Publ. Math. I.H.E.S. 62 (1986), 361-418. MR 823177
- [KaPi] N. Katz and R. Pink, A note on pseudo-CM representations and differential Galois groups, Duke Math. J. 54 (1) (1987), 57-65. MR 885775
- [Kob] N. Koblitz, The number of points on certain families of hypersurfaces over finite fields, Compositio Math. 48 (1983), 3-23. MR 700577
- [Kol] E. Kolchin, Algebraic groups and algebraic independence, Amer. J. Math. 90(1968), 1151-1164. MR 240106
- [Kos] B. Kostant, A characterization of the classical groups, Duke Math. J. 25 (1958), 107-123. MR 95883
- [La] S. Lang, Algebraic Number Theory, Addison-Wesley, Reading, Mass., 1970. MR 282947
- [Lau1] G. Laumon, Transformation de Fourier, constantes d'équations fonctionelles et conjecture de Weil, Publ. Math. I.H.E.S. 65 (1987), 131-210. MR 908218
- [Lau2] G. Laumon, Semi-continuité du conducteur de Swan (d'après P. Deligne), in Caractéristique d'Euler-Poincaré, Seminaire E.N.S. 1978-79, Astérisque 82-83 (1981), 173-219. MR 629128
- [Lev1] A. H. M. Levelt, Jordan decomposition for a class of singular differential operators, Ark. Mat. 13 (1975), 1-27. MR 500294
- [Le2] G. Laumon, Hypergeometric functions, thesis, University of Amsterdam, 1961.
- [Ma] J. Manin, Moduli fuchsiani, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1965), 13-126. MR 180581
- [Me1 ] Z. Mebhkout, Sur le problème de Hilbert-Riemann, Lecture Notes in Phys., vol. 129, Springer-Verlag, Berlin, 1980, pp. 99-110. 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.
- [Pa] S. J. Patterson, On the distribution of Kummer sums, J. Reine Angew. Math. 303(1978), 126-143. MR 514676
- [Ri] K. Ribet, Galois action on division points of abelian varieties with real multiplication, Amer. J. Math. 98 (1976), 751-805. MR 457455
- [Saa] N. Saaveedra Rivano, Catégories tannakiennes, Lecture Notes in Math., vol. 265, Springer-Verlag, Berlin, 1972. MR 338002
- [Se] J.-P. Serre, Abelian l-adic representations and elliptic curves, Addison-Wesley, Reading, Mass., 1989. MR 1043865
- [SeTa] J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. (2)88(1968), 492-517. MR 236190
- [Sch] F. K. Schmidt, Analytische Zahlentheorie in Korporn der Characteristikp. Math. Z. 33(1931), 1-32. MR 1545199
- [Ver] J.-L. Verdier, Specialization de faisceaux et monodromie moderé, in Analyse et topologie sur les espaces singuliers, Vols. II and III, Astérisque 101-102 (1983), 332-364. MR 737938
- [We] A. Weil, Jacobi sums as Grossencharaktere, Trans. Amer. Math. Soc. 73 (1952), 487-492. MR 51263
- [Yo] H. Yoshida, On an analogue of the Sato conjecture, Invent. Math. 19 ( 1973), 261-277. MR 337977
- [Za1] Yu. G. Zarkhin, Weights of simple Lie algebras in the cohomology of algebraic varieties, Izv. Akad. Nauk. SSSR, Ser. Mat. 48 (1984), 264-304; English translation in Math. USSR-Izv. 24 (1985), 245-281. MR 740792
- [Za2] Yu. G. Zarkhin, Linear simple Lie algebras and ranks of operators, Grothendieck Festschrift (to appear). MR 1106920
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:
http://dx.doi.org/10.1090/S0273-0979-1990-15922-1
PII:
S 0273-0979(1990)15922-1
|