Definability of restricted theta functions and families of abelian varieties

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 $\text{Rae}$. In particular, we prove that the embedding of moduli space of principally polarized ableian varierty, $Sp(2g,\Z)\backslash \CH_g$,isdefinablein$\Rae$,whenrestrictedtoSiege{l}^{\mathrm{\prime }}sfundamentalset$\fF_g$. We also prove the definability, on appropriate domains, of embeddings of families of abelian varieties into projective space.