Let $A_g$ be the moduli space of principally polarised abelian varieties of dimension $g$ over the complex numbers. (EDIT: I mean the coarse moduli space.) Is this smooth?

Since $A_g$ is the quotient of Siegel upper half space by $\mathrm{Sp}_{2g}(\mathbb{Z})$ and this group has torsion elements, it seems likely that the answer is no. On the other hand $\mathrm{SL}_2(\mathbb{Z})$ already has torsion elements and $A_1$ is smooth.

Any attempt I make to search for information on this question only leads to information about the smoothness of the boundaries in compactifications of $A_g$.