Surjectivity of Siegel Φ-operator for square free level and small weight
Annales de l'Institut Fourier, Volume 62 (2012) no. 1, pp. 121-144.

We show the surjectivity of the (global) Siegel Φ-operator for modular forms for certain congruence subgroups of Sp(2,) and weight k=4, where the standard techniques (Poincaré series or Klingen-Eisenstein series) are no longer available. Our main tools are theta series and genus versions of basis problems.

Nous démontrons la surjectivité de l’opérateur Φ de Siegel pour des formes modulaires pour certains groupes de congruence de Sp(2,) et de poids 4, où les techniques standards (séries de Poincaré ou séries de Klingen-Eisenstein) ne marchent pas. Nous utilisons des séries thêta et le problème de base pour plusieurs genres.

DOI: 10.5802/aif.2702
Classification: 11F46, 11F27
Keywords: Siegel modular form, $\Phi $-operator, Theta series
Mot clés : formes modulaires de Siegel, l’opérateur $\Phi $, séries de thêta

Böcherer, Siegfried 1; Ibukiyama, Tomoyoshi 2

1 Kunzenhof 4B 79117 Freiburg (Germany)
2 Osaka University Graduate School of Science Department of Mathematics Machikaneyama 1-1, Toyonaka Osaka, 560-0043 (Japan)
@article{AIF_2012__62_1_121_0,
     author = {B\"ocherer, Siegfried and Ibukiyama, Tomoyoshi},
     title = {Surjectivity of {Siegel} $\Phi $-operator for square free level and small weight},
     journal = {Annales de l'Institut Fourier},
     pages = {121--144},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {1},
     year = {2012},
     doi = {10.5802/aif.2702},
     mrnumber = {2986268},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2702/}
}
TY  - JOUR
AU  - Böcherer, Siegfried
AU  - Ibukiyama, Tomoyoshi
TI  - Surjectivity of Siegel $\Phi $-operator for square free level and small weight
JO  - Annales de l'Institut Fourier
PY  - 2012
SP  - 121
EP  - 144
VL  - 62
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2702/
DO  - 10.5802/aif.2702
LA  - en
ID  - AIF_2012__62_1_121_0
ER  - 
%0 Journal Article
%A Böcherer, Siegfried
%A Ibukiyama, Tomoyoshi
%T Surjectivity of Siegel $\Phi $-operator for square free level and small weight
%J Annales de l'Institut Fourier
%D 2012
%P 121-144
%V 62
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2702/
%R 10.5802/aif.2702
%G en
%F AIF_2012__62_1_121_0
Böcherer, Siegfried; Ibukiyama, Tomoyoshi. Surjectivity of Siegel $\Phi $-operator for square free level and small weight. Annales de l'Institut Fourier, Volume 62 (2012) no. 1, pp. 121-144. doi : 10.5802/aif.2702. https://aif.centre-mersenne.org/articles/10.5802/aif.2702/

[1] Arakawa, Tsuneo Vector-valued Siegel’s modular forms of degree two and the associated Andrianov L-functions, Manuscripta Math., Volume 44 (1983) no. 1-3, pp. 155-185 | DOI | MR | Zbl

[2] Böcherer, Siegfried On Eisenstein series of degree two for squarefree levels and the genus version of the basis problem. I, Automorphic forms and zeta functions, World Sci. Publ., Hackensack, NJ, 2006, pp. 43-70 (Proceedings of the conference in memory of T.Arakawa, Ed. S. Böcherer, T. Ibukiyama, M. Kaneko, F. Sato) | MR

[3] Böcherer, Siegfried The genus version of the basis problem II: The case of oldforms (2009) (Preprint)

[4] Böcherer, Siegfried; Furusawa, Masaaki; Schulze-Pillot, Rainer On the global Gross-Prasad conjecture for Yoshida liftings, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 105-130 | MR

[5] Böcherer, Siegfried; Hironaka, Yumiko; Sato, Fumihiro Linear independence of local densities of quadratic forms and its application to the theory of Siegel modular forms, Quadratic forms—algebra, arithmetic, and geometry (Contemp. Math.), Volume 493, Amer. Math. Soc., Providence, RI, 2009, pp. 51-82 | MR

[6] Böcherer, Siegfried; Schulze-Pillot, Rainer Siegel modular forms and theta series attached to quaternion algebras, Nagoya Math. J., Volume 121 (1991), pp. 35-96 | MR | Zbl

[7] Garrett, Paul B. Pullbacks of Eisenstein series; applications, Automorphic forms of several variables (Katata, 1983) (Progr. Math.), Volume 46, Birkhäuser Boston, Boston, MA, 1984, pp. 114-137 | MR | Zbl

[8] Garrett, Paul B. Integral representations of Eisenstein series and L-functions, Number theory, trace formulas and discrete groups (Oslo, 1987), Academic Press, Boston, MA, 1989, pp. 241-264 | MR | Zbl

[9] Ibukiyama, Tomoyoshi On some alternating sum of dimensions of Siegel cusp forms of general degree and cusp configurations, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 40 (1993) no. 2, pp. 245-283 | MR | Zbl

[10] Ibukiyama, Tomoyoshi; Wakatsuki, Satoshi Siegel modular forms of small weight and the Witt operator, Quadratic forms—algebra, arithmetic, and geometry (Contemp. Math.), Volume 493, Amer. Math. Soc., Providence, RI, 2009, pp. 189-209 | MR | Zbl

[11] Katsurada, Hidenori; Schulze-Pillot, Rainer Genus theta series, Hecke operators and the basis problem for Eisenstein series, Automorphic forms and zeta functions, World Sci. Publ., Hackensack, NJ, 2006, pp. 234-261 (Proceedings of the conference in memory of T.Arakawa. Ed. S. Böcherer, T. Ibukiyama, M. Kaneko, F. Sato) | MR | Zbl

[12] Klein, M. Verschwindungssätze für Hermitesche sowie Siegelsche Modulformen zu Γ 0 n (N) sowie Γ 1 n (N), Saarbrücken (2004) (Ph. D. Thesis)

[13] Miyake, Toshitsune Modular forms, Springer-Verlag, Berlin, 1989 (Translated from the Japanese by Yoshitaka Maeda) | MR | Zbl

[14] Poor, C.; Yuen, D. S. Dimensions of cusp forms for Γ 0 (p) in degree two and small weights, Abh. Math. Sem. Univ. Hamburg, Volume 77 (2007), pp. 59-80 | DOI | MR | Zbl

[15] Satake, I. Compactification de espaces quotients de Siegel II, Séminaire Cartan, E. N. S., 1957/58, pp. 1-10 (Exposé 13) | EuDML

[16] Satake, I. L’opérateur Φ, Séminaire Cartan, E. N. S., 1957/58, pp. 1-18 (Exposé 14) | EuDML

[17] Satake, I. Surjectivité globale de opérateur Φ, Séminaire Cartan, E. N. S., 1957/58, pp. 1-17 (Exposé 16) | EuDML

[18] Shimura, Goro Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971 (Kanô Memorial Lectures, No. 1) | MR | Zbl

[19] Siegel, Carl Ludwig Über die analytische Theorie der quadratischen Formen, Ann. of Math. (2), Volume 36 (1935) no. 3, pp. 527-606 | DOI | MR | Zbl

[20] Waldspurger, J.-L. Engendrement par des séries thêta de certains espaces de formes modulaires, Invent. Math., Volume 50 (1978/79) no. 2, pp. 135-168 | DOI | EuDML | MR | Zbl

Cited by Sources: