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On the arithmetic of Siegel-Hilbert cuspforms: Petersson inner products and Fourier coefficients

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References

  • [Ba] Baily, W.L.: Arithmetic Hilbert modular forms. Rev. Mat. Iberoam.1, 85–119 (1985)

    Google Scholar 

  • [B] Borel, A.: Introduction to automorphic forms. (Proc. Symp. Pure Math., vol.IX, pp. 199–210 Providence, R.I.:Am. Math. Soc. 1966

    Google Scholar 

  • [BT] Borel, A., Tits, J.: Groupes reductifs. Publ. Math., Inst. Hautes Étud. Sci.27, 55–151 (1965)

    Google Scholar 

  • [BrT] Bruhat, F., Tits, J.: BN-paires de type affine et donnees radicielles. C.R. Acad. Sci. Paris, Ser. A263, 598–601 (1966); Groupes simples residuellement deployes sur un corps local. Ibid C.R. Acad. Sci. Paris Ser. A,263, 766–768 (1966); Groupes algebriques simple sur un corps local. Ibid C.R. Acad. Sci. Paris Ser. A,263, 822–825 (1966); Groupes algebriques simple sur un corps local: cohomologie galoisienne, decompositions d'Iwasawa et de Cartan. Ibid. C.R. Acad. Sci. Paris Ser. A,263, 867–869 (1966)

    Google Scholar 

  • [C] Cartier, P.: Representations of reductivep-adic groups: a survey. (Proc. Symp. Pure Math., vol. 33, part 1, pp. 111–156) Providence, R.I.: Am. Math. Soc. 1979

    Google Scholar 

  • [G1] Garrett, P.B.: Arithmetic properties of Fourier-Jacobi expansions of automorphic forms in several variables. Am. J. Math.103, 1103–1134 (1981)

    Google Scholar 

  • [G4] Garrett, P.B.: Pullbacks of Eisenstein series; applications. In: Satake, I., Morita, Y. (eds.) Automorphic Forms of Several Variables, pp. 114–137, Boston: Birkhäuser 1984

    Google Scholar 

  • [GPR] Gelbart, S., Piatetski-Shapiro, I.I., Rallis, S.: Explicit constructions of automorphicL-functions. (Lect. Notes Math., vol. 1254) Berlin Heidelberg New York: Springer 1987

    Google Scholar 

  • [Go1] Godement, R.: Fonctions holomorphes de carre sommable dans le demi-plan de Siegel. Ou l'on generalise un integrale etudiee par C.L. Siegel, et generalization de la fonction Θ. In. Semin. H. Cartan, Paris: Ec. Norm. Supér. 1957–58

    Google Scholar 

  • [Go2] Godement, R.: Series d'Eisenstein. In: Semin. H. Cartan. Paris: Éc. Norm. Supér. 1957–58

    Google Scholar 

  • [Go3] Godement, R.: Domaines fondamentaux des groupes arithmetiques. Semin. Bourbak:257 (1962/63)

  • [H1] Harris, M.: Maass operators and Eisenstein series. Math. Ann.258, 135–144 (1981)

    Google Scholar 

  • [H2] Harris, M.: Special values of zeta functions attached to Siegel modular forms. Ann. Sci. Éc. Norm. Super.14, 77–120 (1981)

    Google Scholar 

  • [H3] Harris, M.: Eisenstein series on Shimura varieties. Ann. Math.119, 59–94 (1984)

    Google Scholar 

  • [K1] Klingen, H.: Über die Werte der Dedekindschen Zetafunktionen. Math. Ann.145, 265–272 (1962)

    Google Scholar 

  • [K2] Klingen, H.: Über den arithmetischen Charakter der Fourierkoeffizienten von Modulformen. Math. Ann.147, 176–188 (1962)

    Google Scholar 

  • [Kn] Kneser, M.: Starke Approximation in algebraischen Gruppen. J. Reine Angew. Math.218, 190–203 (1965)

    Google Scholar 

  • [M] Milne, J.S.: Canonical models of (mixed) Shimura varieties and automorphic vector bundles. In: Clozel, L., Milne, J.S. (eds.) Automorphic Forms, Shimura Varieties, andL-functions, pp. 283–414. Boston: Academic Press 1990

    Google Scholar 

  • [S] Satake, I.: Theory of spherical functions on reductive algebraic groups overp-adic fields. Publ. Math., Inst. Hautes Étud. Sci.18, 1–69 (1963)

    Google Scholar 

  • [Sh1] Shimura, G.: On canonical models of arithmetic quotients of bounded symmetric domains I, II. Ann. Math.91, 144–22 (1970);92, 528–549 (1970)

    Google Scholar 

  • [Sh2] Shimura, G.: On some properties of modular forms in one and several variables. Ann. Math.102, 491–515 (1975)

    Google Scholar 

  • [Sh3] Shimura, G.: On Fourier cofficients of modular forms of several variables. Nachr. Akad. Wiss. Gött.17, 261–268 (1975)

    Google Scholar 

  • [Sh4] Shimura, G.: The special values of the zeta functions associated with Hilbert modular forms. Duke J. Math.50, 637–679 (1978)

    Google Scholar 

  • [Sh5] Shimura, G.: On Eisenstein series. Duke J. Math.50, 417–476 (1983)

    Google Scholar 

  • [Sh6] Shimura, G.: Algebraic relations between critical values of zeta functions and inner products. Am. J. Math.105, 253–285 (1983)

    Google Scholar 

  • [Si] Siegel, C.L.: Einfuhrung in die Theorie der Modulfunktionenn-ten Grades. Math. Ann.116, 617–657 (1939)

    Google Scholar 

  • [T] Tits, J.: Reductive groups over local fields. (Proc. Symp. Pure Math., vol. 33 part I, pp. 29–70) Providence, RI: Am. Math. Soc. 1979

    Google Scholar 

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Oblatum 18-IX-1990 & VII-1991

This work was partially supported by a grant from the National Science Foundation

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Garrett, P.B. On the arithmetic of Siegel-Hilbert cuspforms: Petersson inner products and Fourier coefficients. Invent Math 107, 453–481 (1992). https://doi.org/10.1007/BF01231899

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