Inventiones mathematicae

, Volume 60, Issue 3, pp 193–248 | Cite as

Siegel's modular forms and the arithmetic of quadratic forms

  • Hiroyuki Yoshida


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  1. 1.
    Andrianov, A.N.: Dirichlet series with Euler products in the theory of Siegel modular forms of genus 2, Trudy Math. Inst. Steklov112, 73–94 (1971)Google Scholar
  2. 2.
    Andrianov, A.N.: Euler products corresponding to Siegel modular forms of genus 2, Uspekhi Math. Nauk29, 43–110 (1974)Google Scholar
  3. 3.
    Andrianov, A.N.: Modular descent or on Saito-Kurokawa conjecture, Inventiones Math.53, 267–280 (1979)Google Scholar
  4. 4.
    Andrianov, A.N., Maloletkin, G.N.: Behavior of thete series of degreeN under modular substitutions, Izv. Akad. Nauk SSSR39, 227–241 (1975)Google Scholar
  5. 5.
    Eichler, M.: Über die Idealklassenzahl total definiter Quaternionenalgebren, Math. Zeitschr.43, 102–109 (1938)Google Scholar
  6. 6.
    Eichler, M.: Zur Zahlentheorie der Quaternionen-Algebren, J. Reine Angew. Math.195, 127–151 (1955)Google Scholar
  7. 7.
    Eichler, M.: The basis problem for modular forms and the traces of Hecke operators, Lecture notes in math.320, pp. 75–151. Berlin Heidelberg New York: Springer-Verlag 1973Google Scholar
  8. 8.
    Gelbart, S.: Weil's representation and the spectrum of the metaplectic group, Lecture notes in math. 530. Berlin Heidelberg New York: Springer-Verlag 1976Google Scholar
  9. 9.
    Hecke, E.: Mathematische Werke, Zweite Auflage, Vandenhoeck, 1970Google Scholar
  10. 10.
    Howe, R.: θ-series and invariant theory, Proc. of symposia in pure mathematics,XXXIII (vol. 1), 275–286 (1979).Google Scholar
  11. 11.
    Ihara, Y.: On certain arithmetical Dirichlet series. J. Math. Soc. Japan,16, 214–225 (1964)Google Scholar
  12. 12.
    Jacquet, H., Langlands, R.P.: Automorphic forms onGL(2). Lecture notes in math. 114. Berlin Heidelberg New York: Springer-Verlag 1970Google Scholar
  13. 13.
    Kudla, S.: Theta functions and Hilbert modular forms, Nagoya Math. J.69, 97–106 (1978)Google Scholar
  14. 14.
    Kurokawa, N.: Examples of eigenvalues of Hecke operators on Siegel cusp forms of degree two, Inventiones Math.49, 149–165 (1978)Google Scholar
  15. 15.
    Maaß, H.: Die Primzahlen in der Theorie der Siegelschen Modulfunktionen. Math. Ann.124, 87–122 (1951)Google Scholar
  16. 16.
    Matsuda, I.: Dirichlet series corresponding to Siegel modular forms of degree 2, levelN. Sci. Pap. Coll. Gen. Educ. Univ. Tokyo28, 21–49 (1978)Google Scholar
  17. 17.
    Niwa, S.: Modular forms of half integral weight and the integral of certain theta functions. Nagoya Math. J.56, 147–161 (1974)Google Scholar
  18. 18.
    Oda, T.: On modular forms associated with indefinite quadratic forms of signature (2,n−2). Math. Ann.231, 97–144 (1977)Google Scholar
  19. 19.
    Pizer, A.: Type numbers of Eichler orders. J. Reine Angew. Math.264, 76–102 (1973)Google Scholar
  20. 20.
    Rallis, S.: On a relation between\(\widetilde{SL}_2 \) cusp forms and cusp forms on tube domain associated to orthogonal groups. Proc. of symposia in pure mathematics,XXXIII (vol. 1), 297–314 (1979)Google Scholar
  21. 21.
    Satake, I.: Theory of spherical functions on reductive algebraic groups overp-adic fields. Publ. Math. IHES18, 229–293 (1963)Google Scholar
  22. 22.
    Serre, J.P.: Facteurs locaux des fonctions zêta des variétés algébriques. Séminaire Delage-Pisot-Poitou, No 19, 1969/70Google Scholar
  23. 23.
    Shimizu, H.: Theta series and automorphic forms onGL 2. J. Math. Soc. Japan,24, 638–683 (1972)Google Scholar
  24. 24.
    Shimura, G.: On modular correspondences forSp(N, Z) and their congruence relations. Proc. of the Nat. Acad. of Sciences49, 824–828 (1963)Google Scholar
  25. 25.
    Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Iwanami-Shoten and Princeton University Press 1971Google Scholar
  26. 26.
    Shimura, G.: On modular forms of half integral weight. Ann. of Math.97, 440–481 (1973)Google Scholar
  27. 27.
    Shintani, T.: On construction of holomorphic cusp forms of half integral weight. Nagoya Math. J.58, 83–126 (1975)Google Scholar
  28. 28.
    Siegel, C.L.: Gesammelte Abhandlungen. Berlin Heidelberg New York: Springer-Verlag 1966Google Scholar
  29. 29.
    Steinberg, R.: Lectures on Chevalley groups. Yale Univ. lecture note, 1967Google Scholar
  30. 30.
    Tamagawa, T.: On class numbers of quadratic forms, Proc. of Japan-U.S. Seminar on modern methods in number theory (1971) (in Japanese)Google Scholar
  31. 31.
    Weil, A.: Sur certains groupes d'opérateur unitaires. Acta Math.111, 143–211 (1964)Google Scholar
  32. 32.
    Weil, A.: Sur la formule de Siegel dans la théorie des groupes classique, ibid.113, 1–87 (1965)Google Scholar
  33. 33.
    Weil, A.: Dirichlet series and automorphic forms. Lecture notes in math. 189. Berlin Heidelberg New York: Springer-Verlag 1971Google Scholar
  34. 34.
    Yoshida, H.: Weil's representations of the symplectic groups over finite fields. J. Math. Soc. Japan31, 399–426 (1979)Google Scholar
  35. 35.
    Yoshida, H.: On an explicit construction of Siegel modular forms of genus 2. Proc. of Japan Acad. October 1979Google Scholar

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© Springer-Verlag 1980

Authors and Affiliations

  • Hiroyuki Yoshida
    • 1
  1. 1.SFB, Theoretische MathematikUniversität BonnBonn 1Germany

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