  We consider some classical maps from the theory of abelian varieties and
their moduli spaces and prove their definability, on restricted domains, in the
o-minimal structure $\Rae$. In particular, we prove that the embedding of
moduli space of principally polarized ableian varierty, $Sp(2g,\Z)\backslash
\CH_g$, is definable in $\Rae$, when restricted to Siegel's fundamental set
$\fF_g$. We also prove the definability, on appropriate domains, of embeddings
of families of abelian varieties into projective space.
