# On Siegel modular forms

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On Siegel modular forms COMPOSITIO MATHEMATICA WINFRIEDKOHNEN OnSiegelmodular forms Compositio Mathematica, tome 103, no 2 (1996), p. 219-226. <http://www.numdam.org/item?id=CM_1996__103_2_219_0> © Foundation Compositio Mathematica, 1996, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 219Compositio Mathematica 103: 219-226, 1996. © 1996 Kluwer Academic Publishers. Printed in the Netherlands. On Siegel modular forms WINFRIED KOHNEN Universitiit Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany Received 18 April 1994: accepted in final form 30 August 1994 1. Introduction and statement of result Let F be a holomorphic cusp form of integral weight on the Siegel modular group r = Sp (Z) of genus g and denote by a (T) (T a positive definite symmetric even integral g, g)-matrix) its Fourier coefficients. If g = 1 and k &#x3E; 2, then by Deligne’s theorem (previously the Ramanujan- Petersson conjecture) one has and since by [10] this bound is best possible. For arbitrary g &#x3E; 2 our knowledge of how to obtain good bounds for the coefficients a(T) in terms of det(T) is still extremely limited. For g &#x3E;, 2 and k &#x3E; 9 + 1 Bôcherer and the author in [4] proved that where The bound (1) for arbitrary g seems to be the best one known so far. Note, however, that for g - oo it is still of the same order of magnitude as Hecke’s bound In the present paper we shall prove THEOREM. Suppose that 4/g. Then there exists r, = k ( g ) E N with thefollowin

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