Mathematische Annalen

, Volume 286, Issue 1–3, pp 481–509 | Cite as

Mixed Hodge structures and automorphic forms for Siegel modular varieties of degree two

  • Takayuki Oda
  • Joachim Schwermer
Article

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Borel, A.: Stable real cohomology of arithmetic groups. II. In: J. Hano etal. (ed.) Manifolds and Lie groups. Progress in Maths Vol. 14, pp. 21–55. Boston Basel Stuttgart: Birkhäuser 1981CrossRefGoogle Scholar
  2. 2.
    Borel, A., Garland H.: Laplacian and the discrete spectrum of an arithmertic group. Am. J. Math.105, 309–335 (1983)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Borel, A., Serre, J.-P.: Corners and arithmetic groups. Comment. Math. Helv.48, 436–491 (1973)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups and representations of reductive groups. Ann. Math. Stud. 94, Princeton: University Press (1980)MATHGoogle Scholar
  5. 5.
    Baily, W.: On Satake's compactification ofV n. Am. J. Math.80, 348–364 (1958).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Clzel, L.: On limit multiplicities of discrete series representations in the space of automorphic forms. Invent. Math.83, 265–284 (1986).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Deligne, P.: Théorie de Hodge. II. Publ. Math. Inst. Hautes Etud. Sci.40, 5–58 (1971).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Deligne, P.: Théorie de Hodge, III. Publ. Math. Inst. Hautes Etud. Sci.44, 5–78 (1974)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Faltings, G.: Arithmetic theory of Siegel modular forms. In: Number theory ed. D.V. Chudnovsky et al. (Lecture Notes Mathematics, Vol.1240, pp. 101–108). Berlin Heidelberg New York: Springer 1987CrossRefGoogle Scholar
  10. 10.
    Freitag, E.: Thetareihen mit harmonischen Koeffizienten zur Siegelschen Modulgruppe. Math. Ann.254, 27–51 (1980)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Freitag, E.: Holomorphe Differentialformen zu Kongruenzgruppen der Siegelschen Modulgruppe Zweiten Grades. Math. Ann.216, 155–164 (1975)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fulton, W.: Intersection theory. Ergeb. d. Mathematik u. ihrer Grenzgebiete 3. Folge Bd. 2. Berlin Heidelberg New York: Springer 1984Google Scholar
  13. 13.
    Harish-Chandra: Discrete series for semisimple Lie groups. II. Acata Math.116, 1–111 (1966)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Harish-Chandra: Automorphic forms on semisimple Lie groups. (Lecture Notes Mathematics, Vol. 62). Berlin Heidelberg New York: Springer 1968MATHGoogle Scholar
  15. 15.
    Harris, M.: Automorphic forms of\(\bar \partial\)-cohomology type as coherent cohomology classes. J. Differ. Geom. (to appear)Google Scholar
  16. 16.
    Hecke, E.: Zur Theorie der elliptischen Modulfunktionen. Math. Ann.97, 210–246 (1926) (Mathematische Werke, 428–460)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Howe, R., Piatetski-Shapiro, I.: Some examples of automorphic forms onSp(4). Duke Math. J.50, 55–106 (1983)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Igusa, J.: A desingularization problem in theory of Siegel modular functions. Math. Ann.168, 228–260 (1967)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kashiwara, M., Vergne, M.: On the Segal-Shale-Weil representation and harmonic polynomials. Invent. Math.44, 1–47 (1978)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Labesse, J.-P., Schwermer, J.: On liftings and cusp cohomology of arithmetic groups. Invent. Math.83, 383–401 (1986)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lee, R., Schwermer, J.: Cohomology of arithmetic subgroups ofSL 3at infinity. J. Reine Angew. Math.330, 100–131 (1982)MathSciNetMATHGoogle Scholar
  22. 22.
    Lee, R., Weintraub, S.: Cohomology ofSp 4(ℤ) and related groups and spaces. Topology24, 391–410 (1985)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Oda, T.: Theta series of definite quadratic forms and Siegel modular forms. Unpublished manuscript 1979Google Scholar
  24. 14.
    Oda, T.: Hodge structures of Shimura varieties attached to the unit groups of quaternion algebras. In: Galois groups and their representations. Adv. Studies in Pure Math., Vol. 2, pp. 15–36, Tokyo Amsterdam: Kinokuniya, Elsevier 1983Google Scholar
  25. 25.
    Oda, T.: Distinguished cycles and Shimura varieties. In: Automorphic forms of several variables. Progress in Maths Vol. 4, pp. 298–332 Boston Basel Stuttgart: Birkhäuser 1984Google Scholar
  26. 26.
    Oda, T.: Hodge structures attached to geometric automorphic forms. In: Automorphic forms and number theory. Adv. Studies in Pure Maths. Vol. 7, pp. 223–276. Tokyo Amsterdam: Kinokuniya Elsevier 1985Google Scholar
  27. 27.
    Piatetski-Shapiro, I.: On the Saito-Kurokawa lifting. Invent. Math.71, 309–338 (1983).MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Piatetski-Shapiro, I.: A remark on my paper “On the Saito-Kurokawa lifting”. Invent. Math.76, 75–76 (1984)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Saito, M.: Representations unitaires des groupes symplectiques. J. Math. Soc. Japan24, 232–251 (1972)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Schwermer, J.: On arithmetic quotients of the Siegel upper half space of degree two. Compos. Math.58, 233–258 (1986)MathSciNetMATHGoogle Scholar
  31. 31.
    Shioda, T.: On elliptic modular surfaces. J. Math. Soc. Japan,24, 20–59 (1972)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Soudry, D.: The CAP representations ofGSp(4,A).. Reine Angew. Math.383, 87–108 (1988)MathSciNetMATHGoogle Scholar
  33. 33.
    Srinivasan, B.: The characters of the finite symplectic groupSp(4,q). Trans. Am. Math. Soc.131, 488–525 (1963)Google Scholar
  34. 34.
    Steenbrink, J.: Mixed Hodge structures and singularities. Preprint 1987Google Scholar
  35. 35.
    Vogan, D.A., Jr.: Representations of real reductive Lie groups. Progress in maths. Vol. 15. Boston Basel Stuttgart: Birkhäuser 1981MATHGoogle Scholar
  36. 36.
    Vogan, D.A., Jr., Zuckerman, G.: Unitary representations with non-zero cohomology. Compos. Math.53, 51–90 (1984)MathSciNetMATHGoogle Scholar
  37. 37.
    Wallach, N.: On the constant term of a square integrable automorphic form. In: Operator algebras and group representations. II. Monographs and Studies in Maths.18, pp. 227–237. London: Pitman 1984Google Scholar
  38. 38.
    Weil, A.: Sur certains groupes d'opérateurs unitaires. Acta. Math.111, 143–211 (1964) (œuvres scientifiques, III, 1–71)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Weissauer, R. Differentialformen zu Untegrupen der Siegelchen Modulgruppe zweiten Grades. Presprint 1987Google Scholar
  40. 40.
    Yamazaki, T.: On Siegel modular forms of degree two. Am. J. Maths98, 39–53 (1976)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Zucker, S.: Satake compactifications. Comment. Math. Helf.58, 312–343 (1983)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Takayuki Oda
    • 1
  • Joachim Schwermer
    • 2
  1. 1.Department of Mathematics, Faculty of ScienceNiigata UniversityNiigataJapan
  2. 2.Mathematisch-Geographische FakultätKatholische University EichsättEichstättFederal Republic of Germany

Personalised recommendations