Definability of restricted theta functions and families of abelian varieties

Abstract

We consider some classical maps from the theory of abelian varieties and their moduli spaces, and we prove their definability on restricted domains in the o-minimal structure ${\mathbb{R}}_{\mathrm{an,exp}}$. In particular, we prove that the projective embedding of the moduli space of the principally polarized abelian variety $Sp(2g,\mathbb{Z})\backslash {\mathbb{H}}_g$ is definable in ${\mathbb{R}}_{\mathrm{an,exp}}$ when restricted to Siegel’s fundamental set ${\mathfrak{F}}_g$. We also prove the definability on appropriate domains of embeddings of families of abelian varieties into projective spaces.