On Fourier coefficients for Siegel cusp forms of degree~$n$

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 6, Tome 144 (1985), pp. 155-166

Résumé

Let $F(Z)$ be a cusp form of integral weight $k$ relative to the Siegel modular group $Sp_n(\mathbb{Z})$ and let $f(N)$ be its Fourier coefficient with index $N$. Making use of Rankin's convolution, one proves the estimate $$ f(N)=O\Bigl(|N|^{\frac k2-\frac17\delta(n)}\Bigr), \qquad (1) $$ where $$ \delta(n)=\frac{n+1}{(n+1)\Bigl(zn+\frac{1+(-1)^n}{2}\Bigr)+1} $$ Previously, for $n\geq2$ one has known Raghavan's estimate $$ f(N)=O(|N|^{\frac k2}) $$ In the case $n=2$, Kitaoka has obtained a result, sharper than (1), namely: $$ f(N)=O\Bigl(|N|^{\frac k2-\frac14+\varepsilon}\Bigr) \qquad (2) $$ 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 $f(N)$, $n=2$.