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Covering Schemes

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

Abstract

The group \(GL(r, {\mathbb{A}}_{f})\) acts (by Prop.4.15) on the moduli scheme \({M}_{r} = Spec{A}_{r} =\mathop{ \lim }_\longleftarrow {M}_{r,I}\) constructed in Theorem 4.10. The central group F × acts trivially. In this section we construct a covering scheme \({\widetilde{M}}_{r}\) of M r for which the action of \(GL(r, {\mathbb{A}}_{f})\) extends nontrivially to an action of \((GL(r, {\mathbb{A}}_{f}) \times{D}_{\infty }^{\times })/{F}^{\times }\), where D is a division algebra of rank r over F .

References

  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). http://www.springerlink.com/content/t5v71453lj557n02/fulltext.pdf
  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). http://www.math.osu.edu/ flicker.1/df.pdf
  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

Copyright information

© Yuval Z. Flicker 2013

Authors and Affiliations

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

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