Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-25T03:32:29.430Z Has data issue: false hasContentIssue false
  • English
  • Français

ON CHARACTERIZATION OF SIEGEL CUSP FORMS OF DEGREE 2 BY THE HECKE BOUND

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  27 May 2014

Yoshinori Mizuno
Yoshinori Mizuno*
Affiliation:
Faculty and School of Engineering, The University of Tokushima, 2-1, Minami-josanjima-cho, Tokushima, 770-8506, Japan email mizuno@pm.tokushima-u.ac.jp
Get access

Abstract

We give a new proof of a recent result of Kohnen–Martin on a characterization of degree 2 Siegel cusp forms by the growth of their Fourier coefficients. Our main tools are Koecher–Maass series, Imai’s converse theorem and the theory of singular modular forms.

MSC classification

Primary: 11F30: Fourier coefficients of automorphic forms
Secondary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Type
Research Article
Information
Mathematika , Volume 61 , Issue 1 , January 2015 , pp. 89 - 100
Copyright
Copyright © University College London 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andrianov, A. N., Quadratic Forms and Hecke Operators (Grundlehren der Mathematischen Wissenschaften 286), Springer (Berlin, 1987).Google Scholar
Arakawa, T., Makino, I. and Sato, F., Converse theorem for not necessarily cuspidal Siegel modular forms of degree 2 and Saito–Kurokawa lifting. Comment. Math. Univ. St. Pauli 50(2) 2001, 197–234.Google Scholar
Böcherer, S. and Das, S., Characterization of Siegel cusp forms by the growth of their Fourier coefficients. Math. Ann. 359(1–2) 2014, 169–188.CrossRefGoogle Scholar
Duke, W. and Imamoḡlu, Ö., A converse theorem and the Saito–Kurokawa lift. Int. Math. Res. Not. 7 1996, 347–355.CrossRefGoogle Scholar
Ibukiyama, T., A survey on the new proof of Saito–Kurokawa lifting after Duke and Imamoḡlu (in Japanese). Report of the fifth summer school of number theory “Introduction to Siegel Modular Forms”, 1997, 134–176; available at http://www.math.sci.osaka-u.ac.jp/∼ibukiyam/research.html.Google Scholar
Ibukiyama, T., Saito–Kurokawa liftings of level N and practical construction of Jacobi forms. Kyoto J. Math. 52 2012, 141–178.Google Scholar
Imai, K., Generalization of Hecke’s correspondence to Siegel modular forms. Amer. J. Math. 102(5) 1980, 903–936.Google Scholar
Iwaniec, H., Spectral Methods of Automorphic Forms (Graduate Studies in Mathematics 53), 2nd edn, American Mathematical Society and Revista Matemática Iberoamericana (Providence, RI and Madrid, 2002).Google Scholar
Kitaoka, Y., Two theorems on the class number of positive definite quadratic forms. Nagoya Math. J. 51 1973, 79–89.Google Scholar
Klingen, H., Introductory Lectures on Siegel Modular Forms (Cambridge Studies in Advanced Mathematics 20), Cambridge University Press (Cambridge, 1990).Google Scholar
Kohnen, W. and Martin, Y., A characterization of degree 2 Siegel cusp forms by the growth of their Fourier coefficients. Forum Math. (to appear); doi:10.1515/forum-2011-0142.Google Scholar
Krieg, A., Modular Forms on Half-spaces of Quaternions (Lecture Notes in Mathematics 1143), Springer (Berlin, 1985).Google Scholar
Maass, H., Siegel’s Modular Forms and Dirichlet Series (Lecture Notes in Mathematics 216), Springer (Berlin, New York, 1971).CrossRefGoogle Scholar
Manickam, M., Ramakrishnan, B. and Vasudevan, T. C., On Saito–Kurokawa descent for congruence subgroups. Manuscripta Math. 81 1993, 161–182.Google Scholar
Matthes, R. and Mizuno, Y., Spectral theory on 3-dimensional hyperbolic space and Hermitian modular forms. Forum Math. (to appear); doi:10.1515/forum-2011-0113.Google Scholar
Rademacher, H., On the Phragmen–Lindelöf theorem and some applications. Math. Z. 72 1959–1960, 192–204.Google Scholar
Shintani, T., On zeta-functions associated with the vector space of quadratic forms. J. Fac. Sci. Univ. Tokyo, Sect. IA, Math. 22 1975, 25–65.Google Scholar