Representations of a Weil Group

  • Yuval Z. Flicker
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


Let \(F = {\mathbb{F}}_{q}(C)\) be the field of functions on a smooth projective absolutely irreducible curve C over \({\mathbb{F}}_{q}\), \(\mathbb{A}\) its ring of adèles, \(\overline{F}\) a separable algebraic closure of F, G = GL(r), and a fixed place of F, as in Chap. 2. This section concerns the higher reciprocity law, which parametrizes the cuspidal \(G(\mathbb{A})\)-modules whose component at is cuspidal, by irreducible continuous constructible r-dimensional -adic (p) representations of the Weil group \(W(\overline{F}/F)\), or irreducible rank r smooth -adic sheaves on SpecF which extend to smooth sheaves on an open subscheme of the smooth projective curve whose function field is F, whose restriction to the local Weil group \(W({\overline{F}}_{\infty }/{F}_{\infty })\) at is irreducible. This law is reduced to Theorem 11.1, which depends on Deligne’s conjecture (Theorem 6.8). This reduction uses the Converse Theorem 13.1, and properties of ε-factors attached to Galois representations due to Deligne (SLN 349:501–597, 1973) and Laumon (Publ Math IHES 65:131–210, 1987). We explain the result twice. A preliminary exposition in the classical language of representations of the Weil group, then in the equivalent language of smooth -adic sheaves, used e.g. in (Deligne and Flicker, Counting local systems with principal unipotent local monodromy. flicker.1/df.pdf). Note that in this chapter we denote a Galois representation by ρ, as σ is used to denote an element of a Galois group.


  1. A1.
    Arthur, J.: A trace formula for reductive groups I. Duke Math. J. 45, 911–952 (1978)Google Scholar
  2. A2.
    Arthur, J.: On a family of distributions obtained from orbits. Can. J. Math. 38, 179–214 (1986)Google Scholar
  3. A3.
    Arthur, J.: The local behaviour of weighted orbital integrals. Duke Math. J. 56, 223–293 (1988)Google Scholar
  4. AM.
    Atiyah, M., Macdonald, I.: Introduction to Commutative Algebra. Addison-Wesley, Reading (1969)Google Scholar
  5. B.
    Bernstein, J.: P-invariant distributions on \(GL(N)\). Lecture Notes in Mathematics 1041, 50–102. Springer, New York (1984)Google Scholar
  6. BD.
    Bernstein, J., rédigé par Deligne, P.: Le “centre” de Bernstein, dans Représentations des groupes réductifs sur un corps local. Hermann, Paris (1984)Google Scholar
  7. BDK.
    Bernstein, J., Deligne, P., Kazhdan, D.: Trace Paley-Wiener theorem. J. Anal. Math. 47, 180–192 (1986)Google Scholar
  8. BZ.
    Bernstein, J., Zelevinski, A.: Representations of the group \(GL(n,F)\) where F is a nonarchimedean local field. Uspekhi Mat. Nauk 31, 5–70 (1976). (Russian Math. Surveys 31, 1–68, 1976)Google Scholar
  9. Bo.
    Borel, A.: Admissible representations of a semisimple group over a local field with vectors fixed under an Iwahori subgroup. Invent. Math. 35, 233–259 (1976)Google Scholar
  10. BJ.
    Borel, A., Jacquet, H.: Automorphic forms and automorphic representations. Proc. Sympos. Pure Math. 33, I, 111–155 (1979)Google Scholar
  11. BN.
    Bourbaki, N.: Commutative Algebra. Hermann, Paris (1972)Google Scholar
  12. C.
    Casselman, W.: Characters and jacquet modules. Math. Ann. 230, 101–105 (1977)Google Scholar
  13. CPS.
    Cogdell, J., Piatetski-Shapiro, I.: Converse theorems for \(GL(n)\). Publ. Math. Inst. Hautes Études Sci. 79, 157–214 (1994)Google Scholar
  14. De1.
    Deligne, P.: Formes modulaires et représentations de \(GL(2)\). In: Deligne, P., Kuyk, W. (eds.) Modular Functions of One Variable II. Antwerpen Conference 1972, Springer Lecture Notes, vol. 349, pp. 55–105. Springer, New York (1973)Google Scholar
  15. De2.
    Deligne, P.: Les constantes des équations fonctionnelles des fonctions L. Lecture Notes in Mathematics 349, 501–597. Springer, New York (1973). Scholar
  16. De3.
    Deligne, P.: La conjecture de Weil : II. Publ. Math. IHES 52, 137–252 (1980)Google Scholar
  17. DF.
    Deligne, P., Flicker, Y.: Counting local systems with principal unipotent local monodromy. Annals of Math. (2013). flicker.1/df.pdfGoogle Scholar
  18. DH.
    Deligne, P., Husemoller, D.: Survey of drinfeld modules. Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985), pp. 25–91. Contemporary Mathematics, vol. 67, American Mathematical Society, Providence (1987)Google Scholar
  19. D1.
    Drinfeld, V.: Elliptic modules. Mat. Sbornik 94 (136) (1974)(4)= Math. USSR Sbornik 23 (1974), 561–592.Google Scholar
  20. D2.
    Drinfeld, V.: Elliptic modules. II. Mat. Sbornik 102 (144) (1977)(2)= Math. USSR Sbornik 31 (1977), 159–170.Google Scholar
  21. F1.
    Flicker, Y.: The trace formula and base change for \(GL(3)\). In: Lecture Notes in Mathematics, vol. 927. Springer, New York (1982)Google Scholar
  22. F2.
    Flicker, Y.: Rigidity for automorphic forms. J. Anal. Math. 49, 135–202 (1987)Google Scholar
  23. F3.
    Flicker, Y.: Regular trace formula and base change lifting. Am. J. Math. 110, 739–764 (1988)Google Scholar
  24. F4.
    Flicker, Y.: Base change trace identity for U(3). J. Anal. Math. 52, 39–52 (1989)Google Scholar
  25. F5.
    Flicker, Y.: Regular trace formula and base change for \(GL(n)\). Ann. Inst. Fourier 40, 1–36 (1990)Google Scholar
  26. F6.
    Flicker, Y.: Transfer of orbital integrals and division algebras. J. Ramanujan Math. Soc. 5, 107–121 (1990)Google Scholar
  27. F7.
    Flicker, Y.: The tame algebra. J. Lie Theor. 21, 469–489 (2011)Google Scholar
  28. FK1.
    Flicker, Y., Kazhdan, D.: Metaplectic correspondence. Publ. Math. IHES 64, 53–110 (1987)Google Scholar
  29. FK2.
    Flicker, Y., Kazhdan, D.: A simple trace formula. J. Anal. Math. 50, 189–200 (1988)Google Scholar
  30. FK3.
    Flicker, Y., Kazhdan, D.: Geometric Ramanujan conjecture and Drinfeld reciprocity law. In: Number Theory, Trace Formulas and Discrete subgroups. In: Proceedings of Selberg Symposium, Oslo, June 1987, pp. 201–218. Academic Press, Boston (1989)Google Scholar
  31. Fu.
    Fujiwara, K.: Rigid geometry, Lefschetz-Verdier trace formula and Deligne’s conjecture. Invent. math. 127, 489–533 (1997)Google Scholar
  32. GK.
    Gelfand, I., Kazhdan, D.: On representations of the group \(GL(n,K)\), where K is a local field. In: Lie Groups and Their Representations, pp. 95–118. Wiley, London (1975)Google Scholar
  33. H.
    Henniart, G.: Caractérisation de la correspondance de Langlands locale par les facteurs ε de paires. Invent. Math. 113, 339–350 (1993)Google Scholar
  34. JPS1.
    Jacquet, H., Piatetskii-Shapiro, I., Shalika, J.: Conducteur des représentations du groupe linéaire. Math. Ann. 256, 199–214 (1981)Google Scholar
  35. JPS.
    Jacquet, H., Piatetski-Shapiro, I., Shalika, J.: Rankin-Selberg convolutions. Am. J. Math. 104, 367–464 (1982)Google Scholar
  36. JS.
    Jacquet, H., Shalika, J.: On Euler products and the classification of automorphic forms II. Am. J. Math. 103, 777–815 (1981)Google Scholar
  37. K1.
    Kazhdan, D.: Cuspidal geometry of p-adic groups. J. Anal. Math. 47, 1–36 (1986)Google Scholar
  38. K2.
    Kazhdan, D.: Representations of groups over close local fields. J. Anal. Math. 47, 175–179 (1986)Google Scholar
  39. Ko.
    Koblitz, N.: p-adic numbers, p-adic analysis, and zeta functions, 2nd edn., GTM, vol. 58. Springer, New York (1984)Google Scholar
  40. Lf1.
    Lafforgue, L.: Chtoucas de Drinfeld et conjecture de Ramanujan-Petersson. Asterisque 243, ii+329 (1997)Google Scholar
  41. Lf2.
    Lafforgue, L.: Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math. 147, 1–241 (2002)Google Scholar
  42. Lm1.
    Laumon, G.: Transformation de Fourier, constantes d’équations fonctionelles et conjecture de Weil. Publ. Math. IHES 65, 131–210 (1987)Google Scholar
  43. Lm2.
    Laumon, G.: Cohomology of Drinfeld Modular Varieties, volumes I et II. Cambridge University Press, Cambridge (1996)Google Scholar
  44. LRS.
    Laumon, G., Rapoport, M., Stuhler, U.: \(\mathcal{D}\)-elliptic sheaves and the Langlands correspondence. Invent. math. 113, 217–338 (1993)Google Scholar
  45. Mi.
    Milne, J.: Étale cohomology, Princeton Mathematical Series, vol. 33. Princeton University Press, Princeton (1980)Google Scholar
  46. P.
    Pink, R.: On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne. Ann. Math. 135, 483–525 (1992)Google Scholar
  47. S1.
    Serre, J.P.: Zeta and L-functions. In: Schilling, O.F.G. (ed.) Arithmetic Algebraic Geometry. Proc. Conf. Purdue University, 1963. Harper and Row, New York (1965)Google Scholar
  48. S2.
    Serre, J.P.: Abelian ℓ-adic Representations and Elliptic Curves. Benjamin, New-York (1968)Google Scholar
  49. Sh.
    Shintani, T.: On an explicit formula for class 1 “Whittaker functions” on \({GL}_{n}\) over p-adic fields. Proc. Japan Acad. 52, 180–182 (1976)Google Scholar
  50. Sp.
    Shpiz, E.: Thesis. Harvard University, Cambridge (1990)Google Scholar
  51. Ta.
    Tate, J.: p-divisible groups. In: Proceedings of Conference on Local Fields, NUFFIC Summer School, Driebergen, Springer (1967)Google Scholar
  52. V.
    Varshavsky, Y.: Lefschetz-Verdier trace formula and a generalization of a theorem of Fujiwara. Geom. Funct. Anal. 17, 271–319 (2007)Google Scholar
  53. W.
    Waterhouse, W.: Introduction to affine group schemes, GTM 66. Springer, New York (1979)Google Scholar
  54. Z.
    Zelevinski, A.: Induced representations of reductive p-adic groups II. On irreducible representations of \(GL(n)\). Ann. Scient. Ec. Norm. Sup. 13, 165–210 (1980)Google Scholar
  55. Zi.
    Zink, Th: The Lefschetz trace formula for an open algebraic surface. In: Automorphic Forms, Shimura Varieties, and L-Functions, vol. II (Ann Arbor, MI, 1988), pp. 337–376. Perspectives of Mathematics, vol. 11, Academic Press, Boston (1990)Google Scholar
  56. EGA.
    Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique. Springer, Berlin (1971)Google Scholar
  57. SGA1.
    Grothendieck, A.: Revêtements étales et groupe fondamental. In: Lecture Notes in Mathematics, vol. 224. Springer, New York (1971)Google Scholar
  58. SGA4.
    Artin, M., Grothendieck, A., Verdier, J.-L.: Théorie des topos et cohomologie étale des schémas. In: Lecture Notes in Mathematics, vol. 269, 270, 305. Springer, New York (1972–1973)Google Scholar
  59. SGA4 1/2.
    Deligne, P.: Cohomologie étale. In: Lecture Notes in Mathematics, vol. 569. Springer, New York (1977)Google Scholar
  60. SGA5.
    Grothendieck, A.: Cohomologie ℓ-adique et fonctions L. In: Lecture Notes in Mathematics, vol. 589. Springer, New York (1977)Google Scholar

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© Yuval Z. Flicker 2013

Authors and Affiliations

  • Yuval Z. Flicker
    • 1
  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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