Special values of zeta functions attached to Siegel modular forms

Harris, Michael

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 14 (1981) no. 1, pp. 77-120

[1] A. N. Andrianov, Symmetric Squares of Zeta-Functions of Siegel's Modular Forms of Genus 2 (Trudy. Mat. Inst. Steklov, Vol. 142, 1976, pp. 22-45). | Zbl | MR

[2] A. N. Andrianov and V. L. Kalinin, On Analytic Properties of the Standard Zeta-Functions of Siegel Modular Forms (Mat. Sbornik, Vol. 106, 1978, pp. 323-339). | Zbl | MR

[3] W. L. Baily Jr., On the Theory of θ-Functions, the Moduli of Abelian Varieties, and the Moduli of Curves (Ann. of Math., Vol. 75, 1962, pp. 342-381). | Zbl

[4] W. L. Baily Jr. and A. Borel, Compactification of Arithmetic Quotients of Bounded Symmetric Domains (Ann. of Math., Vol. 84, 1966, pp. 442-528). | Zbl | MR

[5] A. Borel, Introduction to Automorphic Forms, in Proc. Symp. Pure Math., Vol. IX, Amer. Math. Soc., Providence, 1966, pp. 199-210. | Zbl | MR

[6] P. Deligne, Travaux de Shimura (Sém. Bourbaki, Vol. 23, (1970/1971), No. 389). | Zbl | Numdam

[7] P. Deligne, Valeurs de fonctions L et périodes d'intégrales, in Proc. Symp. Pure Math., Vol. XXXIII, Amer. Math. Soc., Providence, 1979, Part 2, pp. 313-346. | Zbl | MR

[8] J. Dixmier, Algèbres enveloppantes, Gauthier-Villars, Paris, 1974. | Zbl | MR

[9] E. Freitag, Holomorphe Differentialformen zu Kongruenzgruppen der Siegelschen Modulgruppe (Inv. Math., Vol. 30, 1975, pp. 181-196). | Zbl | MR

[10] V. A. Gritsenko, Analytic Continuation of Symmetric Squares (Mat. Sbornik, Vol. 107, 1978, pp. 323-346). | Zbl | MR

[11] A. Grothendieck, On the De Rham Cohomology of Algebraic Varieties (Publ. Math. I.H.E.S., Vol. 29, 1966). | Zbl | MR | Numdam

[12] Harish-Chandra, Representations of Semi-Simple Lie groups IV, V, VI (Am. J. Math., Vol. 77-78, 1955-1956). | Zbl | MR

[13] Harish-Chandra, Automorphic Forms on Semi-Simple Lie Groups (Lecture Notes in Math., 62, 1968). | Zbl | MR

[14] M. Harris, A Note on Three Lemmas of Shimura (Duke Math., J., Vol. 46, 1979, pp. 871-879). | Zbl | MR

[15] H. P. Jakobsen, Intertwining Differential Operators for Mp (n, R) and SU(n, n) (Trans. A.M.S., Vol. 246, 1978, pp. 311-337). | Zbl | MR

[16] H. P. Jakobsen and M. Vergne, Restrictions and Expansions of Holomorphic Representations (J. Fun. Anal., Vol. 34, 1979, pp. 29-53). | Zbl | MR

[17] N. M. Katz, Algebraic Solutions of Differential Equations (p-Curvature and the Hodge Filtration) (Inv. Math., Vol. 18, 1972, pp. 1-118). | Zbl | MR

[18] N. M. Katz, p-Adic Properties of Modular Schemes and Modular Forms, in Modular Functions of One Variable III (Lecture Notes in Math., No. 350, 1973, pp. 63-190). | Zbl | MR

[19] N. M. Katz, p-Adic L-Functions for CM Fields (Inv. Math., Vol. 49, 1978, pp. 199-297). | Zbl | MR

[20] N. M. Katz and T. Oda, On the Differentiation of De Rham Cohomology Classes with Respect to Parameters (J. Math. Kyoto Univ., Vol. 8, 1968, pp. 199-213). | Zbl | MR

[21] R. P. Langlands, Euler Products, New Haven, Yale University Press, 1971. | Zbl

[22] H. Maaβ, Die Differentialgleichungen in der Theorie der Siegelschen Modulfunktionen (Math. Ann., Vol. 126, 1953, pp. 44-68). | Zbl | MR

[23] H. Maaβ, Siegel's Modular Forms and Dirichlet Series (Lecture Notes in Math., No. 216, 1971). | Zbl | MR

[24] Yu. I. Manin and A. A. Panchishkin, Twists of Hecke Series and Their Values at Integer Points (Mat. Sbornik, Vol. 104, 1977, pp. 617-651). | Zbl

[25] D. Mumford, Geometric Invariant Theory, Springer-Verlag, Berlin, 1965. | Zbl | MR

[26] D. Mumford, Abelian Varieties, Oxford University Press, London, 1970. | Zbl | MR

[27] H. L. Resnikoff, Automorphic Forms of Singular Weight are Singular Forms (Math. Ann., Vol. 215, 1975, pp. 173-193). | Zbl | MR

[28] H. Rossi and M. Vergne, Analytic Continuation of the Holomorphic Discrete Series, (Acta. Math., Vol. 136, 1976, pp. 1-59). | Zbl | MR

[29] I. Satake, Theory of Spherical Functions on Reductive Algebraic Groups Over p-Adic Fields (Publ. Math. I.H.E.S., Vol. 18, 1963, pp. 1-69). | Zbl | MR | Numdam

[30] W. Schmid, Die Randwerte holomorpher Funktionen auf Hermitesch symmetrischen Raumen (Inv. Math., Vol. 9, 1969, pp. 61-80). | Zbl | MR

[31] Séminaire H. Cartan 1957/1958, Fonctions Automorphes, Paris, Secrétariat Mathématique, 1958. | Zbl

[32] G. Shimura, On Some Arithmetic Properties of Modular Forms of One and Several Variables (Ann. of Math., Vol. 102, 1975, pp. 491-515). | Zbl | MR

[33] G. Shimura, On the Fourier Coefficients of Modular Forms of Several Variables (Göttingen Nachr. Akad. Wiss., 1975, pp. 261-268). | Zbl | MR

[34] G. Shimura, The Special Values of the Zeta Functions Associated with Cusp Forms, (Comm. Pure Appl. Math., Vol. 29, 1976, pp. 783-804). | Zbl | MR

[35] G. Shimura, On the Periods of Modular Forms (Math. Ann., Vol. 229, 1977, pp. 211-221). | Zbl | MR

[36] G. Shimura, On the Derivatives of Theta Functions and Modular Forms (Duke Math. J., Vol. 44, 1977, pp. 365-387). | Zbl | MR

[37] G. Shimura, The Special Values of the Zeta Functions Associated with Hilbert modular forms, (Duke Math. J., Vol. 45, 1978, pp. 637-679). | Zbl | MR

[38] G. Shimura, On Certain Reciprocity-Laws for Theta Functions and Modular Forms (Acta. Math., Vol. 141, 1978, pp. 35-71). | Zbl | MR

[39] T. Shintani, On automorphic Forms on Unitary Groups of Order 3, preprint.

[40] C. L. Siegel, Einführung in die Theorie der Modulfunktionen n-ten Grades (Math. Ann., Vol. 116, 1939, pp. 617-657). | Zbl | MR | JFM

[41] C. L. Siegel, Symplectic Geometry (Am. J. Math., Vol. 65, 1943, pp. 1-86). | Zbl | MR

[42] N. Wallach, Harmonic Analysis on Homogeneous Spaces, Marcel Dekker, Inc., New York, 1973. | Zbl | MR

[42a] N. Wallach, Representations of Reductive Lie groups, in Proc. Symp. Pure Math., Vol. XXXIII, Amer. Math. Soc., Providence, 1979, Part 1, pp. 71-86). | Zbl | MR

[43] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Springer-Verlag, Berlin, 1972. | Zbl

[44] A. Weil, Introduction à l'étude des variétés kählériennes, nouvelle édition corrigée, Hermann, Paris, 1958. | Zbl

[45] H. Weyl, The Classical Groups, Princeton University Press, 1946.

[46] D. Zagier, Modular Forms Whose Fourier Coefficients Involve Zeta-Functions of Quadratic Fields, in Modular Functions of One Variable VI (Lecture Notes in Math., No. 627, 1977, pp. 105-170). | Zbl | MR

[47] J. Sturm, Special Values of Zeta Functions, and Eisenstein Series of Half Integral Weight (Am. J. Math., Vol. 102, 1980, pp. 219-240). | Zbl | MR

[48] D. Mumford, An Analytic Construction of Degenerating Abelian Varieties Over Complete Local Rings (Compositio Math., Vol. 24, 1972, pp. 239-272). | Zbl | MR | Numdam

[49] A. N. Andrianov, On Factorization of Hecke Polynomials for the Symplectic Group of Genus n, (Math. U.S.S.R. Sbornik, Vol. 33, 1977, pp. 343-373). | Zbl

[50] A. N. Andrianov, Euler Expansions of Theta-Transforms of Siegel Modular Forms of Degree n (Math. U.S.S.R. Sbornik, Vol. 34, 1978, pp. 259-300). | Zbl

[51] M. Karel, Eisenstein Series and Fields of Definition (Compositio Math., Vol. 37, 1978, pp. 121-169). | Zbl | MR | Numdam

[52] M. Harris and H. P. Jakobsen, Singular Modular Forms and Singular Holomorphic Representations, in preparation.

[53] J. L. Brylinski, "l-motifs" et formes automorphes, preprint.

[54] A. N. Andrianov, The Multiplicative Arithmetic of Siegel Modular Forms (Russian Math. Surveys, Vol. 34, 1979, pp. 75-148). | Zbl | MR

[55] M. Harris, The Rationality of Holomorphic Eisenstein Series [Inv. Math. (in press)]. | Zbl