On Dirichlet Series and Petersson Products for Siegel Modular Forms
Annales de l'Institut Fourier, Volume 58 (2008) no. 3, pp. 801-824.

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 kn/2 has meromorphic continuation to . Moreover, we show that the Petersson product of any pair of square–integrable modular forms of weight kn/2 may be expressed in terms of the residue at s=k of the associated Dirichlet series.

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 kn/2 admet un prolongement méromorphe à . En outre, on montre que le produit de Petersson de toute paire de formes modulaires de carré-intégrable et de poids kn/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.

DOI: 10.5802/aif.2370
Classification: 11F46,  11F60,  11F66
Keywords: Rankin-Selberg method, Petersson product, non-cuspidal modular forms, invariant differential operators.
Böcherer, Siegfried 1; Chiera, Francesco Ludovico 2

1 Universität Mannheim Fakultät für Mathematik und Informatik A5, 68131 Mannheim(Germany)
2 Università “La Sapienza” di Roma Dipartimento di Matematica P. le A. Moro 2 00185 Rome (Italy)
@article{AIF_2008__58_3_801_0,
     author = {B\"ocherer, Siegfried and Chiera, Francesco Ludovico},
     title = {On {Dirichlet} {Series} and {Petersson} {Products} for {Siegel} {Modular} {Forms}},
     journal = {Annales de l'Institut Fourier},
     pages = {801--824},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {58},
     number = {3},
     year = {2008},
     doi = {10.5802/aif.2370},
     mrnumber = {2427511},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2370/}
}
TY  - JOUR
AU  - Böcherer, Siegfried
AU  - Chiera, Francesco Ludovico
TI  - On Dirichlet Series and Petersson Products for Siegel Modular Forms
JO  - Annales de l'Institut Fourier
PY  - 2008
DA  - 2008///
SP  - 801
EP  - 824
VL  - 58
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2370/
UR  - https://www.ams.org/mathscinet-getitem?mr=2427511
UR  - https://doi.org/10.5802/aif.2370
DO  - 10.5802/aif.2370
LA  - en
ID  - AIF_2008__58_3_801_0
ER  - 
%0 Journal Article
%A Böcherer, Siegfried
%A Chiera, Francesco Ludovico
%T On Dirichlet Series and Petersson Products for Siegel Modular Forms
%J Annales de l'Institut Fourier
%D 2008
%P 801-824
%V 58
%N 3
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2370
%R 10.5802/aif.2370
%G en
%F AIF_2008__58_3_801_0
Böcherer, Siegfried; Chiera, Francesco Ludovico. On Dirichlet Series and Petersson Products for Siegel Modular Forms. Annales de l'Institut Fourier, Volume 58 (2008) no. 3, pp. 801-824. doi : 10.5802/aif.2370. http://archive.numdam.org/articles/10.5802/aif.2370/

[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(3) 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 τ(n) 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

Cited by Sources: