Is the moduli space of ppAVs smooth? 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 $A_g$.
 A: The answer is no, it is not smooth for any $g \geq 2$. For $g \geq 3$ the singular locus is precisely the locus of PPAVs with automorphism group greater than $\pm \mathrm{id}$, as proven in Oort, Frans: "Singularities of coarse moduli schemes". For $g=2$ there is IIRC a unique singular point which is in $M_2$ (the open subvariety of $A_2$ of Jacobians), I think this is in Igusa's paper "Arithmetic variety of moduli for genus two".
A: There is an ambiguity in the question, which lies of course in the definition of moduli space,
as the functor defining $A_g$ is not representable in the category of schemes. One solution, which seems the one considered implicitly by the OP (as suggested by the claim that for $g=1$ the moduli space is $\mathbb A^1$), and by Dan Petersen in his answer, is to define $A_g$ as the coarse moduli space of PPAV of genus $g$. And in this case, indeed, $A_g$ is not smooth.
An other solution, which has many advantaged, is to consider $A^g$ not as a scheme but as an
algebraic stack. In many respect this is the right thing to do, and in this case
then $A^g$ is smooth as an algebraic stack. A proof for this is in the book
of Faltings-Chai, Degenerations of abelian varieties.  
(PS: I believe that the OP is well-aware of this distinction coarse/fine moduli space, but since it was mentioned in the question nor in the first answer, it seemed important re recall it for other readers).
