Skip to main content
SpringerLink
Log in

Hilbert modular forms and the Ramanujan conjecture

  • Chapter
Noncommutative Geometry and Number Theory

Part of the book series: Aspects of Mathematics ((ASMA))

Abstract

This paper completes the proof, at all finite places, of the Ramanujan Conjecture for motivic holomorphic Hilbert modular forms which belong to the discrete series at the infinite places. In addition, the Weight-Monodromy Conjecture of Deligne is proven for the Shimura varieties attached to GL(2) and its inner forms, and the conjecture of Langlands, often today called the local-global compatibility, is established at all places for these varieties. This latter conjecture gives, for a finite place v of the field of definition F, an automorphic description of the action of decomposition group Γv of the Galois group Gal \( (\bar F/F) \) on the l-adic cohomology of the the variety, at least if l is distinct from the residue characteristic of v. In particular, the Hasse-Weil zeta functions of these varieties are computed at all places.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arthur, J.; Clozel, L., Simple algebras, base change, and the advanced theory of the trace formula. Annals of Mathematics Studies, 120. Princeton University Press, Princeton, NJ, 1989.

    Google Scholar 

  2. Blasius, D.; Rogawski, J., Motives for Hilbert modular forms. Invent. Math. 114 (1993), no. 1, 55–87.

    Article  MATH  MathSciNet  Google Scholar 

  3. Blasius, Don; Rogawski, Jonathan D., Zeta functions of Shimura varieties. Motives (Seattle, WA, 1991), 525–571, Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI, 1994.

    Google Scholar 

  4. Brylinski, J.-L.; Labesse, J.-P., Cohomologie d’intersection et fonctions L de certaines variétés de Shimura. Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 3, 361–412.

    MATH  MathSciNet  Google Scholar 

  5. Carayol, Henri, Sur les représentations l-adiques associées aux formes modulaires de Hilbert. Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 409–468.

    MATH  MathSciNet  Google Scholar 

  6. Clozel, Laurent, Motifs et formes automorphes: applications du principe de fonctorialité. Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), 77–159, Perspect. Math., 10, Academic Press, Boston, MA, 1990.

    Google Scholar 

  7. De Jong, A. J., Smoothness, semi-stability and alterations. Inst. Hautes études Sci. Publ. Math. No. 83 (1996), 51–93.

    Article  MATH  Google Scholar 

  8. De Shalit, E., The p-adic monodromy-weight conjecture for p-adically uniformized varieties. Comp. Math. 141 (2005), 101–120.

    Article  MATH  Google Scholar 

  9. Deligne, P., Théorie de Hodge I. Actes ICM Nice, t.I, pp. 425–430, Gauthier-Villars, 1970.

    Google Scholar 

  10. Deligne, P., Travaux de Shimura. Séminaire Bourbaki 1970-71, Exposé 389, Lecture Notes in Mathematics, Vol. 244. Springer-Verlag, Berlin-New York, 1971.

    Google Scholar 

  11. Deligne, P., Les constantes des équations fonctionnelles des fonctions L. Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 501–597. Lecture Notes in Mathematics, Vol. 349, Springer-Verlag, Berlin, 1973.

    Chapter  Google Scholar 

  12. Deligne, Pierre, La conjecture de Weil II. Inst. Hautes Études Sci. Publ. Math. No. 52 (1980), 137–252.

    Article  MATH  MathSciNet  Google Scholar 

  13. Deligne, P., Milne, J., Tannakian Categories. Hodge cycles, motives, and Shimura varieties. Lecture Notes in Mathematics, Vol. 900. Springer-Verlag, Berlin-New York, 1982.

    Google Scholar 

  14. Deligne, P.; Milne, J.; Ogus, A.; Shih, K., Hodge cycles, motives, and Shimura varieties. Lecture Notes in Mathematics, Vol. 900. Springer-Verlag, Berlin-New York, 1982.

    MATH  Google Scholar 

  15. Harris, M.; Taylor, R., The Geometry and Cohomology of Some Simple Shimura Varieties, Annals of Mathematics Studies, 151. Princeton University Press, Princeton, NJ, 2001.

    Google Scholar 

  16. Grothendieck, A., Groupes de monodromie en géométrie algébrique, I. Lecture Notes in Mathematics, Vol. 288, Springer Verlag, Berlin-New York, 1972.

    Google Scholar 

  17. Illusie, Luc, Autour du théoréme de monodromie locale. Périodes p-adiques (Bures-sur-Yvette, 1988). Astérisque No. 223 (1994), 9–57.

    MathSciNet  Google Scholar 

  18. Ito, T., Weight-monodromy conjecture for certain 3-folds in mixed characteristic, math.NT/0212109, 2002.

    Google Scholar 

  19. Ito, T., Weight-monodromy conjecture for p-adically uniformized varieties, MPI 2003-6, math.NT/0301202, 2003.

    Google Scholar 

  20. Jacquet, H.; Langlands, R. P., Automorphic forms on GL(2). Lecture Notes in Mathematics, Vol. 114. Springer-Verlag, Berlin-New York, 1970.

    MATH  Google Scholar 

  21. Jarvis, Frazer, On Galois representations associated to Hilbert modular forms. J. Reine Angew. Math. 491 (1997), 199–216.

    MATH  MathSciNet  Google Scholar 

  22. Kudla, Stephen S., The local Langlands correspondence: the non-Archimedean case. Motives (Seattle, WA, 1991), 365–391, Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI, 1994.

    Google Scholar 

  23. Langlands, R. P., On the zeta functions of some simple Shimura varieties. Canad. J. Math. 31 (1979), no. 6, 1121–1216.

    MATH  MathSciNet  Google Scholar 

  24. Langlands, R.P., Base change for GL(2). Annals of Mathematics Studies, 96. Princeton University Press, Princeton, N.J.,1980.

    Google Scholar 

  25. Ohta, Masami, On l-adic representations attached to automorphic forms. Japan. J. Math. (N.S.) 8 (1982), no. 1, 1–47.

    MATH  MathSciNet  Google Scholar 

  26. Rapoport, M.; Zink, Th., Ueber die lokale Zetafunktion von Shimuravarietaeten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik. Invent. Math. 68 (1982), no. 1, 21–101.

    Article  MATH  MathSciNet  Google Scholar 

  27. Rapoport, M., On the bad reduction of Shimura varieties. Automorphic forms, Shimura varieties, and L-functions, Vol. II (Ann Arbor, MI, 1988), 253–321, Perspect. Math., 11, Academic Press, Boston, MA, 1990.

    Google Scholar 

  28. Reimann, Harry, The semi-simple zeta function of quaternionic Shimura varieties. Lecture Notes in Mathematics, Vol.1657. Springer-Verlag, Berlin, 1997. viii+143 pp.

    MATH  Google Scholar 

  29. Reimann, Harry, On the zeta function of quaternionic Shimura varieties. Math. Ann. 317 (2000), no. 1, 41–55.

    Article  MATH  MathSciNet  Google Scholar 

  30. Reimann, Harry; Zink, Thomas, Der Dieudonnmodul einer polarisierten abelschen Mannigfaltigkeit vom CM-Typ. Ann. of Math. (2) 128 (1988), no. 3, 461–482.

    Article  MathSciNet  Google Scholar 

  31. Rogawski, J. D.; Tunnell, J. B., On Artin L-functions associated to Hilbert modular forms of weight one. Invent. Math. 74 (1983), no. 1, 1–42.

    Article  MATH  MathSciNet  Google Scholar 

  32. Rohrlich, David E., Elliptic curves and the Weil-Deligne group. Elliptic curves and related topics, 125–157, CRM Proc. Lecture Notes, 4, Amer. Math. Soc., Providence, RI, 1994.

    Google Scholar 

  33. Saito, Takeshi, Weight-monodromy conjecture for l-adic representations associated to modular forms. A supplement to: “Modular forms and p-adic Hodge theory” [Invent. Math. 129 (1997), no. 3, 607–620], in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 427–431, NATO Sci. Ser. C Math. Phys. Sci., 548, Kluwer Acad. Publ., Dordrecht, 2000.

    Google Scholar 

  34. Serre, Jean-Pierre, Abelian l-adic Representations and Elliptic Curves. Addison-Wesley, Redwood City, 1989.

    Google Scholar 

  35. Serre, Jean-Pierre; Tate, John, Good reduction of abelian varieties. Ann. of Math. (2) 88 (1968) 492–517.

    Article  MathSciNet  Google Scholar 

  36. Shahidi, Freydoon, On the Ramanujan conjecture and finiteness of poles for certain L-functions. Ann. of Math. (2) 127 (1988), no. 3, 547–584.

    Article  MathSciNet  Google Scholar 

  37. Shimura, Goro, Construction of class fields and zeta functions of algebraic curves. Ann. of Math. (2) 85 (1967), 58–159.

    Article  MathSciNet  Google Scholar 

  38. Tadić, Marko, Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case). Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 335–382.

    Google Scholar 

  39. Tate, J., Number theoretic background. Automorphic forms, representations and L-functions (Corvallis, Oregon, 1977), 3–26, Proc. Sympos. Pure Math.,XXXIII, Part 2, Amer. Math. Soc., Providence, R.I., 1979.

    Google Scholar 

  40. Taylor, R., On Galois representations associated to Hilbert modular forms. Invent. Math. 98 (1989), no. 2, 265–280.

    Article  MATH  MathSciNet  Google Scholar 

  41. Taylor, R., On Galois representations associated to Hilbert modular forms. II. Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), 185–191, Ser. Number Theory, I, Internat. Press, Cambridge, MA, 1995.

    Google Scholar 

  42. Taylor, R.; Yoshida, Teruyoshi, Compatibility of local and global Langlands correspondences preprint, math.NT/0412357, December 2004.

    Google Scholar 

  43. Yoshida, Hiroyuki, On a conjecture of Shimura concerning periods of Hilbert modular forms. Am. J. Math. 117 (1995), no. 4, 1019–1038.

    Article  MATH  Google Scholar 

  44. Wiles, A., On ordinary λ-adic representations associated to modular forms. Invent. Math. 94 (1988), no. 3, 529–573.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Authors
  1. Don Blasius
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, The Johns Hopkins University, 3400 North Charles Street, Baltimore, MD, 21218, USA

    Caterina Consani

  2. Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111, Bonn

    Matilde Marcolli

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Friedr. Vieweg & Sohn Verlag ∣ GWV Fachverlage GmbH, Wiesbaden

About this chapter

Cite this chapter

Blasius, D. (2006). Hilbert modular forms and the Ramanujan conjecture. In: Consani, C., Marcolli, M. (eds) Noncommutative Geometry and Number Theory. Aspects of Mathematics. Vieweg. https://doi.org/10.1007/978-3-8348-0352-8_2

Download citation

Publish with us

Policies and ethics

Search

Navigation