Non-vanishing of Fourier coefficients of Siegel cusp forms

Y. Martin

Every Siegel cusp form $F$ of weight $k$ over a Hecke congruence subgroup $\Gamma _0^n(N)$ of the symplectic group $Sp_n(\mathbb {Z}) \subseteq \mathbb {Z}^{2n, 2n}$ has a Fourier series representation

\[ F(Z) = \sum _{T \in J_n} c(T) \exp {(2\pi i \ trace (TZ))}. \]

Here $Z$ belongs to the Siegel space of degree $n$ and $J_n$ denotes the set of positive-definite, half-integral, symmetric, $n$ by $n$ matrices.

In this talk we discuss the existence of simple looking matrices $T$ such that $c_F(T) \neq 0$ whenever $F \neq 0$. In this manner one gets proper, uncomplicated subsets $\Delta $ of $J_n$ such that

\[ F = G \ \text{if, and only if}, \ c_F(T) = c_G(T) \ \text{for all $T \in \Delta $}. \]

In particular, for $n=3$ we exhibit a set $\Delta $ parametrized by triples of primes numbers which characterize all Siegel cusp forms in $S_k^3(N, \chi )$ whenever $N$ is odd and conductor of $\chi = N$.