Family of Siegel groups (awaiting review)

Siegel modular forms of degree $g$ transform under the action of a discrete subgroup $\Gamma$ of $\mathrm{Sp}_{2g}(\Q)$. We focus on certain discrete subgroups $\Gamma_F(N)$ belonging to a family $F$, indexed by $N \in \Z_{\geq 0}$, as follows:

• Siegel (parabolic) subgroup: the subgroup of $\mathrm{Sp}_{2g}(\Z)$ whose lower $g \times g$ block is zero modulo $N$.
• principal congruence subgroup: the kernel of $\mathrm{Sp}_{2g}(\Z) \to \mathrm{Sp}_{2g}(\Z/N\Z)$.
• paramodular subgroup : corresponds to $(1,\dots,1,N)$-polarized abelian $g$-folds.