Fundamental Fourier coefficients of Siegel modular forms of higher degrees and levels

Abstract: We prove the following statement about any Siegel modular form $F$ of degree $n$ and arbitrary odd level $N$ on the group $\Gamma_1^{(n)}(N)$. Let $A(F,T)$ denote the Fourier coefficients of $F$ and write $T=(T(i,j))$. Suppose that $F$ has a non-zero Fourier coefficient $A(F,T_0)$ such that $(T_0(n,n),N)=1$. Then there exist infinitely many odd and square-free (and thus fundamental) integers $m$ such that $m=\mathrm{discriminant}(T)$ and $A(F,T)\neq 0$. In the case of odd degrees, we prove a stronger result by replacing odd and square-free with odd and prime. We also prove quantitative results towards this. As a consequence, we can show in particular that the statement of the main result in arXiv:2408.03442 about the algebraicity of certain critical values of the spinor $L$-functions of holomorphic newforms (in the ambit of Deligne's conjectures) on congruence subgroups of $\mathrm{GSp}(3)$ is unconditional.