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Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil. (Fourier transformation, constants of the functional equations and Weil conjecture). (French) Zbl 0641.14009
In 1976, P. Deligne introduced a Fourier transformation into algebraic geometry, i.e. an involution for \(\ell\)-adic sheaves on the affine line (more generally on a vector bundle over a scheme of finite type) over a field k of finite characteristic \(p\neq \ell\) (depending on a character \({\mathbb{F}}_ p\to {\mathbb{Q}}_{\ell})\). The properties of this Fourier transformation have been studied by N. M. Katz and the author [Publ. Math., Inst. Hautes Étud. Sci. 62, 145-02 (1985; Zbl 0603.14015)]. In the present paper a local version of Fourier transformation is introduced, defined on \(\ell\)-adic representations of the Galois group of a local field of equal characteristic. At the end of the paper, these Fourier transformations are used to give a new proof of the main result of Deligne’s paper [P. Deligne, Publ. Math., Inst. Haut. Étud. Sci. 52, 137-52 (1980; Zbl 0456.14014) which in turn implies the part of the Weil conjectures which is the analog of the Riemann hypothesis.
The main part of the paper is devoted to Grothendieck’s L-function for a complex K of \(\ell\)-adic sheaves on a curve X over k. This L-function satisfies a functional equation for \(t\to t^{-1}\) in which a constant \(\epsilon\) (X,K) appears. The main result of the paper under review expresses this constant as a product of local constants in the places of the curve. The proof of this product formula reduces to the case \(X={\mathbb{P}}^ 1\) and then uses the Fourier transformation to investigate a one parameter deformation of a certain complex.
The author also explains the relevance of his product formula for Langlands’ conjectures on the corresondence between L-functions and automorphic representations.
Reviewer: F.Herrlich

MSC:
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G15 Finite ground fields in algebraic geometry
11S37 Langlands-Weil conjectures, nonabelian class field theory
14G99 Arithmetic problems in algebraic geometry; Diophantine geometry
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References:
[1] E. Artin andJ. Tate,Class field theory, New York, Benjamin (1967). · Zbl 1179.11040
[2] A. A. Beilinson, I. N. Bernstein etP. Deligne, Faisceaux pervers, dansAnalyse et topologie sur les espaces singuliers (I), Conférence de Luminy, juillet 1981,Astérisque 100 (1982).
[3] A. Borel, Automorphic L-functions, inAutomorphic Forms, Representations and L-functions. Proc. Sympos, Pure Math., vol. 33, part 2, Amer. Math. Soc., Providence, R. I. (1979), 27–61.
[4] A. Borel andH. Jacquet, Automorphic Forms and Automorphic Representations, inAutomorphic Forms, Representations and L-functions, Proc. Sympos. Pure Math., vol. 33, part 1, Amer. Math. Soc., Providence, R. I. (1979), 189–202. · Zbl 0414.22020
[5] J.-L. Brylinski, Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques,Astérisque 140–141 (1986), 3–134.
[6] P. Deligne, Les constantes des équations fonctionnelles des fonctions L, dansModular Functions of One Variable II,Lecture Notes in Math., vol. 349, Springer-Verlag (1973), 55–106.
[7] P. Deligne,Les constantes des équations fonctionnelles des fonctions L, Séminaire à l’I.H.E.S., 1980, notes de L. Illusie.
[8] P. Deligne, La conjecture de Weil I,Publ. Math. I.H.E.S.,43 (1974), 273–307.
[9] P. Deligne, La conjecture de Weil II,Publ. Math. I.H.E.S.,52 (1981), 313–428.
[10] P. Deligne,Lettre à D. Kazhdan, 29 novembre 1976.
[11] P. Deligne,Lettre à J.-P. Serre, février 1974.
[12] P. Deligne,Lettre à J.-L. Verdier, 22 avril 1982.
[13] V. G. Drinfeld, Langlands’ Conjecture for GL(2) over Functional Fields,Proceedings of the International Congress of Mathematicians, Helsinki (1978), 565–574.
[14] V. G. Drinfeld, Two-dimensional -adic Representations of the Fundamental Group of a Curve over a Finite Field and Automorphic Forms on GL(2),Amer. J. of Math.,105 (1983), 85–114. · Zbl 0536.14014 · doi:10.2307/2374382
[15] V. G. Drinfeld, Two-dimensional -adic Representations of the Galois Group of a Global Field of Characteristicp and Automorphic Forms on GL(2) (en russe),Zapiski Nayčnyh Seminarov LOM I,134 (1984), 138–156.
[16] B. Dwork, On the Artin Root Number,Amer. J. of Math. 78 (1956), 444–472. · Zbl 0074.26601 · doi:10.2307/2372524
[17] R. Godement andH. Jacquet, Zeta Functions of Simple Algebras,Lecture Notes in Math., vol. 260, Springer-Verlag (1972). · Zbl 0244.12011
[18] A. Grothendieck, Formule de Lefschetz et rationalité des fonctions L,Séminaire Bourbaki 1964–1965, no 279, dansDix exposés sur la cohomologie des schémas, Amsterdam, North-Holland (1968).
[19] G. Henniart, Les inégalités de Morse (d’après E. Witten),Séminaire Bourbaki 1983–1984, no 617,Astérisque 121–122 (1985), 43–61.
[20] G. Henniart,Lettre à l’auteur, 27 mars 1986.
[21] H. Jacquet andR. P. Langlands, Automorphic Forms on GL(2),Lecture Notes in Math., vol. 114, Springer-Verlag (1970). · Zbl 0236.12010
[22] N. M. Katz, Gauss Sums, Kloosterman Sums and Monodromy Groups, to be published inAnnals of Mathematics Studies, Princeton University Press.
[23] N. M. Katz, Extensions of Representations of Fundamental Groups,Ann. Inst. Fourier, à paraître. · Zbl 0564.14013
[24] N. M. Katz, Wild Ramification and some Problems of “Independence of ”,Amer. J. of Math. 105, (1983), 201–227. · Zbl 0568.14012 · doi:10.2307/2374386
[25] N. M. Katz,Communication personnelle à l’auteur, février 1986.
[26] N. M. Katz etG. Laumon, Transformation de Fourier et majoration de sommes exponentielles,Publ. Math. I.H.E.S.,62 (1985), 361–418. · Zbl 0603.14015
[27] H. Koch, Bemerkungen zur Numerischen Lokalen Langlands-Vermutung,Trudy Matematičeskogo Instituta AN, SSSR,163 (1984), 108–114.
[28] H. Koch, Classification of the Primitive Representations of the Galois Groups of Local Fields,Invent. Math. 40 (1977), 195–216. · Zbl 0376.12003 · doi:10.1007/BF01390345
[29] R. P. Langlands,On the Functional Equation of the Artin L-functions, Yale University (1969?).
[30] R. P. Langlands, Problems in the Theory of Automorphic Forms, inLectures in Modern Analysis and Applications III,Lecture Notes in Math., vol. 170, Springer-Verlag (1970), 18–86.
[31] G. Laumon, Majoration de sommes exponentielles attachées aux hypersurfaces diagonales,Ann. Scient. Ec. Norm. Sup.,16 (1983), 1–58. · Zbl 0518.10043
[32] G. Laumon, Comparaison de caractéristiques d’Euler-Poincaré en cohomologie -adique,C. R. Acad. Sc. Paris,292 (1981), 209–212. · Zbl 0468.14005
[33] G. Laumon, Semi-continuité du conducteur de Swan (d’après P. Deligne), dansCaractéristique d’Euler-Poincaré, Séminaire E.N.S. 1978–1979,Astérisque 82–83 (1981), 173–219.
[34] G. Laumon, Les constantes des équations fonctionnelles des fonctions L sur un corps global de caractéristique Positive,C. R. Acad. Sc. Paris,298 (1984), 181–184. · Zbl 0567.14016
[35] I. Piatetski-Shapiro,Zeta Function of GL(n), Preprint, University of Maryland (1976).
[36] I. Piatetski-Shapiro, Multiplicity One Theorems, inAutomorphic Forms, Representations and L-functions, Proc. Sympos. Pure Math., vol. 33, part 1, Amer. Math. Soc., Providence, R. I. (1979), 209–212.
[37] M. Raynaud, Caractéristiques d’Euler-Poincaré d’un faisceau et cohomologie des variétés abéliennes (d’après Ogg-Shafarévitch et Grothendieck),Séminaire Bourbaki 1964–1965, no 286, dansDix exposés sur la cohomologie des schémas, Amsterdam, North-Holland (1968).
[38] J.-P. Serre,Corps locaux, Paris, Hermann (1968).
[39] J.-P. Serre,Représentations linéaires des groupes finis, Paris, Hermann (1971).
[40] J.-P. Serre,Groupes algébriques et corps de classes, Paris, Hermann (1959). · Zbl 0097.35604
[41] J.-P. Serre, Sur la rationalité des représentations d’Artin,Ann. of Math.,72 (1960), 406–420. · Zbl 0202.32803 · doi:10.2307/1970142
[42] J.-P. Serre, Zeta and L-Functions, inArithmetic Algebraic Geometry, New York, Harper & Row (1965), 82–92.
[43] J.-P. Serre andJ. Tate, Good Reduction of Abelian Varieties,Ann. of Math.,88 (1968), 492–517. · Zbl 0172.46101 · doi:10.2307/1970722
[44] J. Tate, Fourier Analysis in Number Fields and Hecke’s Zeta-Functions, inAlgebraic Number Theory (J. W. S. Cassels andA. Fröhlich), London, Academic Press (1967), 305–347.
[45] J. Tate, Local constants, inProc. Durham Sympos. on Algebraic Number Fields, London, Academic Press (1977), 89–131.
[46] A. Weil, Dirichlet Series and Automorphic Forms,Lecture Notes in Math., vol. 189, Springer-Verlag (1971). · Zbl 0218.10046
[47] A. Weil, L’avenir des mathématiques,Bol. Sao Paulo, 1 (1946), 55–68. · Zbl 0061.00614
[48] E. Witten, Supersymmetry and Morse Theory,J. of Diff. Geometry,17 (1982), 661–692.
[49] Séminaire de Géométrie algébrique du Bois-Marie. Théorie des topos et cohomologie étale des schémas, dirigé parM. Artin, A. Grothendieck etJ.-L. Verdier,Lecture Notes in Math., vol. 269, 270 et 305, Springer-Verlag (1972 et 1973).
[50] Cohomologie étale, parP. Deligne, avec la collaboration deJ.-F. Boutot, A. Grothendieck, L. Illusie etJ.-L. Verdier,Lecture Notes in Math., vol. 569, Springer-Verlag (1977).
[51] Cohomologie -adique et fonctions L, dirigé parA. Grothendieck, avec la collaboration deI. Bucur, C. Houzel, L. Illusie, J.-P. Jouanolou etJ.-P. Serre,Lecture Notes in Math., vol. 589, Springer-Verlag (1977).
[52] Théorie des intersections et théorème de Riemann-Roch, dirigé parP. Berthelot, A. Grothendieck etL. Illusie, avec la collaboration deD. Ferrand, J.-P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud, J.-P. Serre,Lecture Notes in Math., vol. 225, Springer-Verlag (1971).
[53] Groupes de monodromie en géométrie algébrique, dirigé parA. Grothendieck avec la collaboration deM. Raynaud etD. S. Rim pour la partie I et parP. Deligne etN. Katz pour la partie II,Lecture Notes in Math., vol. 288 et 340, Springer-Verlag (1972 et 1973).
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