On Dirichlet Series and Petersson Products for Siegel Modular Forms

Böcherer, Siegfried; Chiera, Francesco Ludovico

Annales de l'Institut Fourier, Tome 58 (2008) no. 3, pp. 801-824.

On démontre que la série de Dirichlet à la Rankin-Selberg associée à toute paire de formes modulaires de Siegel (non nécessairement paraboliques) de degré $n$ et poids $k\ge n/2$ admet un prolongement méromorphe à $ℂ$. En outre, on montre que le produit de Petersson de toute paire de formes modulaires de carré-intégrable et de poids $k\ge n/2$ a une expression en termes du résidu en $s=k$ de la série de Dirichlet associée. Ces résultats sont bien connus pour les formes paraboliques. La méthode que nous adoptons généralise celle qui a été introduite par Maass (dans le cas $n=2$) et se base sur l’utilisation de certains opérateurs différentiels invariants.

We prove that the Dirichlet series of Rankin–Selberg type associated with any pair of (not necessarily cuspidal) Siegel modular forms of degree $n$ and weight $k\ge n/2$ has meromorphic continuation to $ℂ$. Moreover, we show that the Petersson product of any pair of square–integrable modular forms of weight $k\ge n/2$ may be expressed in terms of the residue at $s=k$ of the associated Dirichlet series.

Classification : 11F46, 11F60, 11F66
Keywords: Rankin-Selberg method, Petersson product, non-cuspidal modular forms, invariant differential operators.
Mot clés : méthode de Rankin et Selberg, produit de Petersson, formes modulaires non paraboliques, opérateurs différentielles invariants

[1] Böcherer, S. Über die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen., Math. Z., Volume 183 (1983), pp. 21-46 | DOI | EuDML | MR | Zbl

[2] Böcherer, S.; Chiera, F.L. Petersson products of singular and almost singular theta series., Manuscr. Math., Volume 115 (2004) no. 3, pp. 281-297 | DOI | MR | Zbl

[3] Böcherer, S.; Raghavan, S. On Fourier coefficients of Siegel modular forms., J. Reine Angew. Math., Volume 384 (1988), pp. 80-101 | DOI | EuDML | MR | Zbl

[4] Chiera, F.L. On Petersson products of not necessarily cuspidal modular forms., J. Number Theory, Volume 122 (2007) no. 1, pp. 13-24 | DOI | MR | Zbl

[5] Courtieu, Michel; Panchishkin, A. Non-Archimedean $L$-functions and arithmetical Siegel modular forms. 2nd, augmented ed., Lecture Notes in Mathematics 1471. Berlin: Springer. viii, 196 p., 2004 | MR | Zbl

[6] Deitmar, Anton; Krieg, Aloys Theta correspondence for Eisenstein series., Math. Z., Volume 208 (1991) no. 2, pp. 273-288 | DOI | EuDML | MR | Zbl

[7] Feit, Paul Poles and residues of Eisenstein series for symplectic and unitary groups., Mem. Am. Math. Soc., Volume 346 (1986), pp. 89 p. | MR | Zbl

[8] Freitag, E. Siegelsche Modulfunktionen., Grundlehren der mathematischen Wissenschaften, 254. Berlin-Heidelberg-New York: Springer-Verlag. X, 341 S. DM 168.00; $ 67,20, 1983 | MR | Zbl

[9] Harish-Chandra Discrete series for semisimple Lie groups. II: Explicit determination of the characters., Acta Math., Volume 116 (1966), pp. 1-111 | DOI | MR | Zbl

[10] Harris, Michael The rationality of holomorphic Eisenstein series., Invent. Math., Volume 63 (1981), pp. 305-310 | DOI | MR | Zbl

[11] Harris, Michael; Jakobsen, Hans Plesner Singular holomorphic representations and singular modular forms., Math. Ann., Volume 259 (1982), pp. 227-244 | DOI | MR | Zbl

[12] Kalinin, V.L. Eisenstein series on the symplectic group., Math. USSR, Sb., Volume 32 (1978), pp. 449-476 | DOI | MR | Zbl

[13] Kalinin, V.L. Analytic properties of the convolution of Siegel modular forms of genus n., Math. USSR, Sb., Volume 48 (1984), pp. 193-200 | DOI | MR | Zbl

[14] Kitaoka, Y. Lectures on Siegel modular forms and representation by quadratic forms., Lectures on Mathematics and Physics. Mathematics, 77. Tata Institute of Fundamental Research, Bombay. Berlin etc.: Springer-Verlag. V, 227 p., 1986 | MR | Zbl

[15] Klingen, Helmut Introductory lectures on Siegel modular forms., Cambridge Studies in Advanced Mathematics, 20. Cambridge: Cambridge University Press. x, 162 p., 1990 | MR | Zbl

[16] Kohnen, W. A simple remark on eigenvalues of Hecke operators on Siegel modular forms., Abh. Math. Semin. Univ. Hamb., Volume 57 (1987), pp. 33-36 | DOI | MR | Zbl

[17] Kohnen, W.; Skoruppa, N.-P. A certain Dirichlet series attached to Siegel modular forms of degree two., Invent. Math., Volume 95 (1989) no. 3, pp. 541-558 | DOI | MR | Zbl

[18] Lang, Serge Introduction to modular forms. (With two appendices, by D. B. Zagier and by W. Feit)., Grundlehren der mathematischen Wissenschaften, 222. Berlin-Heidelberg-New York: Springer-Verlag. IX, 261 p. with 9 figs., 1976 | MR | Zbl

[19] Lieman, Daniel B. The $GL\left(3\right)$ Rankin-Selberg convolution for functions not of rapid decay., Duke Math. J., Volume 69 (1993) no. 1, pp. 219-242 | DOI | MR | Zbl

[20] Maass, Hans Siegel’s modular forms and Dirichlet series. Course given at the University of Maryland, 1969-1970., Lecture Notes in Mathematics. 216. Berlin-Heidelberg-New York: Springer-Verlag. 328 p. DM 20.00; $ 5.50, 1971 | Zbl

[21] Maass, Hans Dirichletsche Reihen und Modulformen zweiten Grades. (Dirichlet series and modular forms of second degree)., Acta Arith., Volume 24 (1973), pp. 225-238 | MR | Zbl

[22] Mizumoto, Shin-ichiro Eisenstein series for Siegel modular groups., Math. Ann., Volume 297 (1993) no. 4, pp. 581-625 | DOI | MR | Zbl

[23] Mizuno, Y. The Rankin-Selberg convolution for Cohen’s Eisenstein series of half integral weight., Abh. Math. Semin. Univ. Hamb., Volume 75 (2005), pp. 1-20 | DOI | Zbl

[24] Petersson, Hans Über die Berechnung der Skalarprodukte ganzer Modulformen., Comment. Math. Helv., Volume 22 (1949), pp. 168-199 | DOI | MR | Zbl

[25] Rankin, R.A. Contributions to the theory of Ramanujan’s function $\tau \left(n\right)$ and similar arithmetical functions. II. The order of the Fourier coefficients of integral modular forms., Proc. Camb. Philos. Soc., Volume 35 (1939), pp. 357-372 | DOI | Zbl

[26] Satake, I. Caractérisation de l’espace des Spitzenformen, Séminaire Henri Cartan; 10 no.1 (1957/1958). Fonctions Automorphes. Exposé 9 bis. Secrétariat Math., Paris, 1958., 1958 | Numdam

[27] Sato, Mikio; Shintani, Takuro On zeta functions associated with prehomogeneous vector spaces., Ann. of Math., Volume 100 (1974) no. 2, pp. 131-170 | DOI | MR | Zbl

[28] Shimura, Goro The special values of the zeta functions associated with cusp forms., Commun. Pure Appl. Math., Volume 29 (1976), pp. 783-804 | DOI | MR | Zbl

[29] Shimura, Goro Invariant differential operators on Hermitian symmetric spaces., Ann. of Math., Volume 132 (1990) no. 2, pp. 237-272 | DOI | MR | Zbl

[30] Shimura, Goro Differential operators, holomorphic projection, and singular forms., Duke Math. J., Volume 76 (1994) no. 1, pp. 141-173 | DOI | MR | Zbl

[31] Weissauer, Rainer Vektorwertige Siegelsche Modulformen kleinen Gewichtes., J. Reine Angew. Math., Volume 343 (1983), pp. 184-202 | DOI | MR | Zbl

[32] Yamazaki, Tadashi Rankin-Selberg method for Siegel cusp forms., Nagoya Math. J., Volume 120 (1990), pp. 35-49 | MR | Zbl

[33] Zagier, Don The Rankin-Selberg method for automorphic functions which are not of rapid decay., J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 28 (1981), pp. 415-437 | MR | Zbl