Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A note on eigenvalues of Hecke operators on Siegel modular forms of degree two


Author: Winfried Kohnen
Journal: Proc. Amer. Math. Soc. 113 (1991), 639-642
MSC: Primary 11F60; Secondary 11F46, 11F66
MathSciNet review: 1068125
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ F$ be a cuspidal Hecke eigenform of even weight $ k$ on $ {\operatorname{Sp} _4}(\mathbb{Z})$ with associated eigenvalues $ {\lambda _m}(m \in \mathbb{N})$. Under the assumption that its first Fourier-Jacobi coefficient does not vanish it is proved that the abscissa of convergence of the Dirichlet series $ {\sum _{m \geq 1}}\left\vert {{\lambda _m}} \right\vert{m^{ - s}}$ is less than or equal to $ k$.


References [Enhancements On Off] (What's this?)

  • [1] A. N. Andrianov, Euler products that correspond to Siegel’s modular forms of genus 2, Uspehi Mat. Nauk 29 (1974), no. 3 (177), 43–110 (Russian). MR 0432552
  • [2] Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735
  • [3] S. A. Evdokimov, Characterization of the Maass space of Siegel modular cusp forms of genus 2, Mat. Sb. (N.S.) 112(154) (1980), no. 1(5), 133–142, 144 (Russian). MR 575936
  • [4] E. Freitag, Siegelsche Modulfunktionen, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 254, Springer-Verlag, Berlin, 1983 (German). MR 871067
  • [5] V. A. Gritsenko, The action of modular operators on the Fourier-Jacobi coefficients of modular forms, Math. USSR-Sb. 47 (1984), 237-267.
  • [6] Yoshiyuki Kitaoka, Fourier coefficients of Siegel cusp forms of degree two, Nagoya Math. J. 93 (1984), 149–171. MR 738922
  • [7] Helmut Klingen, Über Kernfunktionen für Jacobiformen und Siegelsche Modulformen, Math. Ann. 285 (1989), no. 3, 405–416 (German). MR 1019710, 10.1007/BF01455065
  • [8] W. Kohnen and N.-P. Skoruppa, A certain Dirichlet series attached to Siegel modular forms of degree two, Invent. Math. 95 (1989), no. 3, 541–558. MR 979364, 10.1007/BF01393889
  • [9] Aloys Krieg, Das Vertauschungsgesetz zwischen Hecke-Operatoren und dem Siegelschen 𝜑-Operator, Arch. Math. (Basel) 46 (1986), no. 4, 323–329 (German). MR 847098, 10.1007/BF01200463
  • [10] Takayuki Oda, On the poles of Andrianov 𝐿-functions, Math. Ann. 256 (1981), no. 3, 323–340. MR 626953, 10.1007/BF01679701
  • [11] Nils-Peter Skoruppa, Developments in the theory of Jacobi forms, Automorphic functions and their applications (Khabarovsk, 1988) Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk, 1990, pp. 167–185. MR 1096975
  • [12] -, Computations of Siegel modular forms of genus two, preprint, MPI Bonn, 1990.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11F60, 11F46, 11F66

Retrieve articles in all journals with MSC: 11F60, 11F46, 11F66


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1068125-2
Article copyright: © Copyright 1991 American Mathematical Society