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 Ran,exp. In particular, we prove that the projective embedding of the moduli space of the principally polarized abelian variety Sp(2g,Z)\Hg is definable in Ran,exp when restricted to Siegel’s fundamental set Fg. We also prove the definability on appropriate domains of embeddings of families of abelian varieties into projective spaces.