An Isomorphism Between Moduli Spaces of Abelian Varieties

In a previous paper we showed that for every polarization on an abelian variety there is a dual polarization on the dual abelian variety. In this note we extend this notion of duality to families of polarized abelian varieties. As a main consequence we obtain an involution on the set of moduli spaces of polarized abelian varieties of dimension $g$. In particular, the moduli spaces ${\mathcal{A}}_{(d_1,\mathrm{.}\mathrm{.}\mathrm{.},d_g)}$ and $\mathcal A_{(d_1,\frac{d_1d_g}{d_{g-1}},…,\frac{d_1d_g}{d_2},d_g)}$ are canonically isomorphic.