Fourier coefficients of Siegel cusp forms of genus n

Abstract

Let F(Z) be a cusp form of integral weight k relative to the Siegel modular group Spn(Z) and let f(N) be its Fourier coefficient with index N. Making use of Rankin's convolution, one proves the estimate $$f(\mathcal{N}) = O(\left| \mathcal{N} \right|^{\tfrac{k}{2} - \tfrac{1}{2}\delta (n)} ),$$ (1) where $$\delta (n) = \frac{{n + 1}}{{\left( {n + 1} \right)\left( {2n + \tfrac{{1 + ( - 1)^n }}{2}} \right) + 1}}.$$ Previously, for n ≥ 2 one has known Raghavan's estimate $$f(\mathcal{N}) = O(\left| \mathcal{N} \right|^{\tfrac{k}{2}} )$$ In the case n=2, Kitaoka has obtained a result, sharper than (1), namely: $$f(\mathcal{N}) = O(\left| \mathcal{N} \right|^{\tfrac{k}{2} - \tfrac{1}{4} + \varepsilon } ).$$ (2) At the end of the paper one investigates specially the case n=2. It is shown that in some cases the result (2) can be improved to, apparently, unimprovable estimates if one assumes some analogues of the Petersson conjecture. These results lead to a conjecture regarding the optimal estimates of f(N), n=2.