Definability of restricted theta functions and families of abelian varieties

TL;DR

This paper proves that certain classical maps related to abelian varieties and their moduli spaces are definable within the o-minimal structure , including embeddings of moduli spaces and families of abelian varieties.

Contribution

It establishes the definability of key maps in the theory of abelian varieties within the o-minimal structure , extending the understanding of their logical and geometric properties.

Findings

Embedding of moduli space of principally polarized abelian varieties is definable in  on Siegel's fundamental set.

Definability of embeddings of families of abelian varieties into projective space.

Results apply to classical maps in the theory of abelian varieties and their moduli.

Abstract

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.