Skip to main content
Log in

\(D\)-elliptic sheaves and the langlands correspondence

  • Published:
Inventiones mathematicae Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  • [An] Anderson, G.:t-motives. Duke Math. J.53, 457–502 (1986)

    Google Scholar 

  • [Ar-Cl] Arthur, J., Clozel, L.: Base change and the advanced theory of the trace formula. (Ann. Math. Stud. vol. 120) Princeton: Princeton University Press and University of Tokyo Press (1989)

    Google Scholar 

  • [Art] Artin. M.: Versal deformations and algebraic stacks. Invent. Math.27, 165–189 (1974)

    Google Scholar 

  • [Be-Ze 1] Bernstein, I.N., Zelevinskii, A.V.: Representations of the group GL(n, F) whereF is a non-archimedean local field. Russ. Math. Surv.31(3), 1–68 (1976)

    Google Scholar 

  • [Be-Ze 2] Bernstein, I.N. Zelevinskii, A.V.: Induced representations of reductivep-adic groups. I. Ann. Sci. Ec. Norm. Supér., IV. Sér.10, 441–472 (1977)

    Google Scholar 

  • [Ca] Cartier, P.: Representations of ℘-adic groups: A survey. In: Borel, A., Casselman, W. (eds.) Corvallis conference on Automorphic forms, Representations andL-functions. (Proc. Symp. Pure Math. vol. XXXIII, part 1, pp. 111–156) Providence, RI: Am. Math. Soc. 1979

    Google Scholar 

  • [Cu-Re] Curtis, C., Reiner, I.: Methods of representation theory: with applications to finite groups and orders, vol. 1. New York: Wiley 1981

    Google Scholar 

  • [De 1] Deligne, P.: Les constantes des équations fonctionelles des fonctionsL. In: Deligne, P., Kuyk, W. (eds.) Modular functions of one variable II. Antwerpen conference 1972. (Lect. Notes Math., 349, pp. 501–597) Berlin Heidelberg New York: Springer 1973

    Google Scholar 

  • [De 2] Deligne, P.: La conjecture de Weil II. Publ. Math., Inst. Hautes Étud. Sci.52, 137–252 (1980)

    Google Scholar 

  • [De 3] Deligne, P.: Formes modulaires et représentations de GL(2). In: Deligne, P., Kuyk, W. (eds.) Modular functions of one variable II. Antwerpen conference 1972. (Lect. Notes Math., vol. 349, pp. 55–105) Berlin Heidelberg New York: Springer 1973

    Google Scholar 

  • [De-Hu] Deligne, P., Husemoller, D.: Survey of Drinfel'd modules. In: Ribet, K.A. (ed.) Current trends in arithmetical algebraic geometry. (Contemp. Math. vol. 67, pp. 25–91) Providence, RI: Am. Math. Soc. 1987

    Google Scholar 

  • [De-Ka-Vi] Deligne, P., Kazhdan, D., Vigneras, M.-F.: Représentations des algèbres centrales simplesp-adiques. In: Bernstein, I.N., Deligne, P., Kazhdan, D., Vigneras, M.-F. (eds.) Représentations des groupes réductifs sur un corps local, pp. 33–118. Paris: Hermann 1984

    Google Scholar 

  • [De-Mu] Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. Math., Inst. Hautes Étud. Sci.36, 75–110 (1969)

    Google Scholar 

  • [De-Ra] Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. In: Deligne, P., Kuyk, W. (eds.) Modular functions of one variable II. Antwerpen conference 1972. (Lect. Notes Math., vol. 349, pp. 143–316) Berlin Heidelberg New York: Springer 1973

    Google Scholar 

  • [Dr 1] Drinfel'd, V.G.: Elliptic modules. Math. USSR, Sb.23, 561–592 (1974)

    Google Scholar 

  • [Dr 2] Drinfel'd, V.G.: Elliptic modules. II. Math. USSR, Sb.31, 159–170 (1977)

    Google Scholar 

  • [Dr 3] Drinfel'd, V.G.: Commutative subrings of certain noncommutative rings. Funct. Anal. Appl.11, 9–12 (1977)

    Google Scholar 

  • [Dr 4] Drinfel'd, V.G.: Letter to H. Carayol (January 12th, 1980)

  • [Dr 5] Drinfel'd, V.G.: Letter to U. Stuhler. (September 17th, 1985)

  • [Dr 6] Drinfel'd, V.G.: Varieties of modules ofF-sheaves. Funct. Anal. Appl.21 107–122 (1987)

    Google Scholar 

  • [Dr 7] Drinfel'd, V.G.: The proof of Petersson's conjecture for GL(2) over a global field of characteristicp. Funct. Anal. Appl.22, 28–43 (1988)

    Google Scholar 

  • [Dr 8] Drinfel'd, V.G.: Cohomology of compactified manifolds of modules ofF-sheaves of rank 2. J. Soy. Math.46, (no. 1), 1789–1821 (1989)

    Google Scholar 

  • [Fl-Ka] Flicker, Y., Kazhdan, D.: Geometric Ramanujan conjecture and Drinfeld reciprocity law. In: Aubert, K., Bombieri, E., Goldfeld, D. (eds.) Number theory, trace formulas and discrete groups, pp. 201–218. New York London: Academic Press 1989

    Google Scholar 

  • [Go-Ja] Godement, R., Jacquet, H.: Zeta functions of simple algebras. (Lect. Notes Math., 260) Berlin Heidelberg New York: Springer 1972

    Google Scholar 

  • [Gr] Grayson, D.R.: Finite generation ofK-groups of a curve over a finite field (after D. Quillen). In: Dennis, R.K. (ed.). Algebraic K-Theory. Proceedings, 1980, Part I. (Lect. Notes Math., vol. 966, pp. 69–90) Berlin Heidelberg New York: Springer 1982

    Google Scholar 

  • [Gro] Grothendieck, A.: Techniques de construction et théorèmes d'existence en géométrie algébrique IV: Les schémas de Hilbert. Sémin. Bourbaki, exposé 221 (1960/61)

  • [Ha-Na] Harder, G., Narasimhan, M.S.: On the cohomology of moduli spaces of vector bundles on curves. Math. Ann.212, 215–248 (1975)

    Google Scholar 

  • [H-C] Harish-Chandra: A submersion principle and its applications. In: Collected papers, vol. IV, pp. 439–446. Berlin Heidelberg New York: Springer 1984

    Google Scholar 

  • [He 1] Henniart, G.: On the local Langlands conjecture for GL(n): The cyclic case. Ann. Math.123, 145–203 (1986)

    Google Scholar 

  • [He 2] Henniart, G.: La conjecture de Langlands locale numérique pour GL(n), Ann. Sci. Éc. Norm. Supér., IV Sér21, 497–544 (1988)

    Google Scholar 

  • [He 3] Henniart, G.: La conjecture de Langlands locale pour GL(3). Mém. Soc. Math. Fr. Nouv. Sér.11/12, 497–544 (1984)

    Google Scholar 

  • [He 4] Henniart, G.: Le point sur la conjecture de Langlands pour GL(N) sur un corps local. In: Goldstein, C. (ed.). Séminaire de théorie des nombres de Paris 1983–1984. (Prog. Math., vol. 59, pp. 115–131) Boston Basel Stŭttgart: Birkhäuser 1985

    Google Scholar 

  • [He 5] Henniart, G.: Appendix to this paper

  • [Her] Herstein, I.N.: Noncommutative rings. Carus Math. Monogr.15 (1968)

  • [Il] Illusie, L.: Complexe cotangent et déformations. I, II. (Lect. Notes Math., vols. 239, 283) Berlin Heidelberg New York: Springer 1971, 1973

    Google Scholar 

  • [Ja-Pi-Sh 1] Jacquet, H., Piatetski-Shapiro, I., Shalika, J.: Automorphic forms on GL3. I, II. Ann. Math.109, 169–258 (1979)

    Google Scholar 

  • [Ja-Pi-Sh 2] Jacquet, H., Piatetski-Shapiro, I., Shalika, J.: Rankin-Selberg convolutions. Am. J. Math.105, 367–483 (1983)

    Google Scholar 

  • [Ja-Sh 1] Jacquet, H., Shalika, J.: Conducteur des représentations du groupe linéaire. Math. Ann.256, 199–214, (1979)

    Google Scholar 

  • [Ja-Sh 2] Jacquet, H., Shalika, J.: On Euler products and the classification of automorphic representations. I. Am. J. Math.103, 499–558 (1981); II. Am. J. Math.103, 777–815 (1981)

    Google Scholar 

  • [Ka] Kazhdan, D.: An introduction to Drinfeld's “shtuka”. In: Borel, A., Casselman W. (eds.) Corvallis conference on Automorphic forms, Representations andL-functions. (Proc. Symp. Pure Math. vol. XXXIII, part 2, pp. 347–356) Providence, RI: Am. Math. Soc. 1979

    Google Scholar 

  • [Ko] Koch, H.: Eisensteinsche Polynomfolgen und Arithmetik in Divisionsalgebren über lokalen Körpern. Math. Nachr.104, 229–251 (1981)

    Google Scholar 

  • [Kot 1] Kottwitz, R.: Sign changes in harmonic analysis on reductive groups. Trans. Am. Math. Soc.278, 289–297 (1983)

    Google Scholar 

  • [Kot 2] Kottwitz, R.: Tamagawa numbers. Ann. Math.127, 629–646 (1988)

    Google Scholar 

  • [Kot 3] Kottwitz, R.: On the λ-adic representations associated to some simple Shimura varieties, Invent. math.108, 653–665 (1992)

    Google Scholar 

  • [La] Langlands, R.P.: Euler Products. Yale University Press 1971

  • [Lan] Langton, S.: Valuative criteria for families of vector bundles on algebraic varieties. Ann. Math.101, 215–248 (1975)

    Google Scholar 

  • [Lau 1] Laumon, G.: Transformation de Fourier, constantes d'équations fonctionelles et conjecture de Weil. Publ. Math., Inst. Hautes Étud. Sci.65, 131–210 (1987)

    Google Scholar 

  • [Lau 2] Laumon, G.: Cohomology with compact supports of Drinfel'd modular varieties. Part I. Prépublications (91-01), pp. 1–201. Orsay: Université de Paris-Sud 1991

    Google Scholar 

  • [Mo] Morris, L.: Eisenstein series for reductive groups over global function fields. I. The cusp form case. Can. J. Math.34, 91–168 (1982)

    Google Scholar 

  • [Mo-Wa] Moeglin, C., Waldspurger, J.-L.: Le spectre résiduel de GL(n), Ann. Sci. Éc. Norm. Supér., IV. Sér.22, 605–674 (1989)

    Google Scholar 

  • [Mu] Mustafin, G.: Nonarchimedian uniformization. Math. USSR, Sb.34, 187–214 (1978)

    Google Scholar 

  • [Ok-Sch-Sp] Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces. Boston Basel Stuttgart: Birkhäuser 1980

    Google Scholar 

  • [Ra] Rapoport, M.: On the bad reduction of Shimura varieties. In: Clozel, L., Milne, J. (eds.) Automorphic forms, Shimura varieties andL-functions, vol. II. (Perspect. Math., vol. 11, pp. 253–321) Boston: Academic Press 1990

    Google Scholar 

  • [Re] Reiner, I.: Maximal orders. New York London: Academic Press 1975

    Google Scholar 

  • [Ro] Rogawski, J.: Representations of GL(n) and division algebras over ap-adic field, Duke Math. J.50, 161–196 (1983)

    Google Scholar 

  • [Se] Seshadri, C.S.: Fibrés vectoriels sur les courbes algébriques (redigée par J.-M. Drezet). Astérisque96 (1982)

  • [Sh] Shalika, J.: The multiplicity one theorem for GL(n). Ann. Math.100, 171–193 (1974)

    Google Scholar 

  • [Sha] Shahidi, F.: On certainL-functions. Am. J. Math.103, 297–355 (1981)

    Google Scholar 

  • [St] Stuhler, U.:P-adic homogeneous spaces and moduli problems. Math. Z.192, 491–540 (1986)

    Google Scholar 

  • [Ta] Tadič, M.: On the classification of irreducible unitary representations of GL(n) and the conjectures of Bernstein and Zelevinsky (non-archimedian case). Ann. Sci. Éc. Norm. Supér., IV. Sér.19, 335–382 (1986)

    Google Scholar 

  • [Tat] Tate, J.: Local constants. In: Fröhlich, A. (ed.) Algebraic number fields, Durham conference, pp. 89–131 New York London: Academic Press 1977

    Google Scholar 

  • [Wa] Wang, S.: On the commutator group of a simple algebra. Am. J. Math.72, 323–334 (1950)

    Google Scholar 

  • [We]) Weil, A.: Basic number theory. Berlin Heidelberg New York: Springer 1967

    Google Scholar 

  • [SGA 4, vol. III] Artin, M., Grothendieck, A., Verdier, J.-L.: Théorie des topos et cohomologie étale des schémas. (Lect. Notes Math., vol. 305) Berlin Heidelberg New York: Springer 1973

    Google Scholar 

  • [SGA5] Grothendieck, A.: Cohomologiel-adique et fonctionsL. (Lect. Notes Math., vol. 589) Berlin Heidelberg New York: Springer 1977

    Google Scholar 

  • [SGA 7 II] Deligne, P., Katz, N.: Groupes de monodromie en géométrie algébrique. (Lect. Notes Math., vol. 340) Berlin Heidelberg New York: Springer 1973

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Oblatum 11-XI-1991 & 16-XII-1992

Rights and permissions

Reprints and permissions

About this article

Cite this article

Laumon, G., Rapoport, M. & Stuhler, U. \(D\)-elliptic sheaves and the langlands correspondence. Invent Math 113, 217–338 (1993). https://doi.org/10.1007/BF01244308

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01244308

Navigation