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On Siegel modular forms. (English) Zbl 0858.11028
So far a lot of work has been done in order to generalize Deligne’s theorem, i.e. the former Ramanujan-Petersson conjecture, to modular forms of several variables. In this context the author proves the following result:
Given \(g \equiv 0 \pmod 4\) there exists \(\kappa (g)\in\mathbb{N}\) such that for all \(n\in\mathbb{N}\) there exist \(k\in\{N,N+1, \dots, N + \kappa (g)-1\}\) and a Siegel cusp form \(F\neq 0\) of weight \(k\) and genus \(g\), whose Fourier coefficients satisfy \[ a(T) \ll_{\varepsilon,F} (\text{det} T)^{k/2-1/2 + \varepsilon} (\varepsilon > 0). \] \(F\) is constructed from theta series with harmonic forms.
Reviewer: A.Krieg (Aachen)
MSC:
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F30 Fourier coefficients of automorphic forms
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References:
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