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Siegfried Böcherer; Tomoyoshi Ibukiyama
Surjectivity of Siegel $\Phi $-operator for square free level and small weight
(Surjectivité de l’opérateur $\Phi $ de Siegel pour des niveaux sans facteur carré et pour petit poids)
Annales de l'institut Fourier, 62 no. 1 (2012), p. 121-144, doi: 10.5802/aif.2702
Article PDF | Reviews MR 2986268 | Zbl pre06064515
Class. Math.: 11F46, 11F27
Keywords: Siegel modular form, $\Phi $-operator, Theta series

Résumé - Abstract

We show the surjectivity of the (global) Siegel $\Phi $-operator for modular forms for certain congruence subgroups of $\mathrm{Sp}(2,\mathbb{Z})$ 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.

Bibliography

[1] Tsuneo Arakawa, “Vector-valued Siegel’s modular forms of degree two and the associated Andrianov $L$-functions”, Manuscripta Math. 44 (1983) no. 1-3, p. 155-185 Article |  MR 709851 |  Zbl 0517.10024
[2] Siegfried Böcherer, 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, p. 43–70  MR 2208209
[3] Siegfried Böcherer, “The genus version of the basis problem II: The case of oldforms”, Preprint, 2009
[4] Siegfried Böcherer, Masaaki Furusawa & Rainer Schulze-Pillot, On the global Gross-Prasad conjecture for Yoshida liftings, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, 2004, p. 105–130  MR 2058606
[5] Siegfried Böcherer, Yumiko Hironaka & Fumihiro Sato, 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. 493, Amer. Math. Soc., 2009, p. 51–82  MR 2537093
[6] Siegfried Böcherer & Rainer Schulze-Pillot, “Siegel modular forms and theta series attached to quaternion algebras”, Nagoya Math. J. 121 (1991), p. 35-96  MR 1096467 |  Zbl 0726.11030
[7] Paul B. Garrett, Pullbacks of Eisenstein series; applications, Automorphic forms of several variables (Katata, 1983), Progr. Math. 46, Birkhäuser Boston, 1984, p. 114–137  MR 763012 |  Zbl 0544.10023
[8] Paul B. Garrett, Integral representations of Eisenstein series and $L$-functions, Number theory, trace formulas and discrete groups (Oslo, 1987), Academic Press, 1989, p. 241–264  MR 993320 |  Zbl 0671.10024
[9] Tomoyoshi Ibukiyama, “On some alternating sum of dimensions of Siegel cusp forms of general degree and cusp configurations”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 40 (1993) no. 2, p. 245-283  MR 1255043 |  Zbl 0830.11019
[10] Tomoyoshi Ibukiyama & Satoshi Wakatsuki, Siegel modular forms of small weight and the Witt operator, Quadratic forms—algebra, arithmetic, and geometry, Contemp. Math. 493, Amer. Math. Soc., 2009, p. 189–209  MR 2537101
[11] Hidenori Katsurada & Rainer Schulze-Pillot, Genus theta series, Hecke operators and the basis problem for Eisenstein series, Automorphic forms and zeta functions, World Sci. Publ., Hackensack, NJ, 2006, p. 234–261  MR 2208777
[12] M. Klein, Verschwindungssätze für Hermitesche sowie Siegelsche Modulformen zu $\Gamma ^n_0(N)$ sowie $\Gamma _1^n(N)$, Ph. D. Thesis, Saarbrücken, (available from Schulze-Pillot’s homepage), 2004
[13] Toshitsune Miyake, Modular forms, Springer-Verlag, Berlin, 1989, Translated from the Japanese by Yoshitaka Maeda  MR 1021004 |  Zbl 0701.11014
[14] C. Poor & D. S. Yuen, “Dimensions of cusp forms for $\Gamma _0(p)$ in degree two and small weights”, Abh. Math. Sem. Univ. Hamburg 77 (2007), p. 59-80 Article |  MR 2379329
[15] I. Satake, Compactification de espaces quotients de Siegel II, Séminaire Cartan, E. N. S., 1957/58, p. 1–10
[16] I. Satake, L’opérateur $\Phi $, Séminaire Cartan, E. N. S., 1957/58, p. 1–18
[17] I. Satake, Surjectivité globale de opérateur $\Phi $, Séminaire Cartan, E. N. S., 1957/58, p. 1–17
[18] Goro Shimura, 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 314766 |  Zbl 0221.10029
[19] Carl Ludwig Siegel, “Über die analytische Theorie der quadratischen Formen”, Ann. of Math. (2) 36 (1935) no. 3, p. 527-606 Article |  MR 1503238 |  Zbl 0012.19703
[20] J.-L. Waldspurger, “Engendrement par des séries thêta de certains espaces de formes modulaires”, Invent. Math. 50 (1978/79) no. 2, p. 135-168 Article |  MR 517775 |  Zbl 0393.10025
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