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Estimates for Fourier coefficients of Siegel cusp forms. (English) Zbl 0787.11017
Let \(F\) be a cusp form of integral weight \(k\) on the Siegel modular group \(\text{Sp}_ g(\mathbb Z)\) of genus \(g\) and denote by \(a(T)\) (\(T\) a positive definite symmetric half-integral \((g,g)\)-matrix) its Fourier coefficients. Set \[ m_{g-1}(T):= \min\{U'TU|_{g-1}\mid U\in \text{GL}_ g(\mathbb Z)\} \] where \(U'TU|_{g-1}\) denotes the determinant of the leading \((g-1)\)-rowed submatrix of \(U'TU\). Define \(\alpha_ g \in \mathbb R_ +\) by \(\alpha_ g^{-1}:= 4(g-1) + 4\left[{g-1\over 2}\right] + {2\over g+2}\). The main result of the paper is:
Theorem. Let \(g \geq 2\) and suppose that \(k > g + 1\). Then \[ a(T) \ll_{\varepsilon,F}(m_{g-1}(T))^{1/2-\alpha_ g+\varepsilon}(\det\;T)^{(k-1 )/2+\varepsilon}\quad (\varepsilon > 0). \] Since by reduction theory \(m_{g-1}(T)\ll (\text{det }T)^{1- 1/g}\), one deduces:
Corollary. Under the above assumptions we have \[ a(T)\ll_{\varepsilon,F}(\det\;T)^{k/2-1/2g-(1-1/g)\alpha_ g + \varepsilon}\quad (\varepsilon > 0). \] If \(g = 2\), the above results were proved previously in [W. Kohnen, Estimates for Fourier coefficients of Siegel cusp forms of degree two. I, Compos. Math. 87, 231-240 (1993; Zbl 0783.11023); II, Nagoya Math. J. 128, 171–176 (1992; Zbl 0792.11013)] using the theory of Jacobi forms. The proof in the general case follows essentially the same pattern as that in the case \(g=2\).
Note that the above estimates improve upon the estimates obtained previously in [S. Böcherer and S. Raghavan, J. Reine Angew. Math. 384, 80–101 (1988; Zbl 0636.10022)] and [S. Raghavan and R. Weissauer, Number Theory and Dynamical Systems, Lond. Math. Soc. Lect. Note Ser. 134, 87–102 (1989; Zbl 0686.10019)].

MSC:
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F30 Fourier coefficients of automorphic forms
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