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 \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

Ax-Schanuel for Shimura varieties

We prove the Ax-Schanuel theorem for a general (pure) Shimura variety

Hyperbolic Ax-Lindemann theorem in the cocompact case

We prove an analogue of the classical Ax-Lindemann theorem in the context of compact Shimura varieties. Our work is motivated by J. Pila's strategy for proving the Andr\'e-Oort conjecture unconditionallyComment: To appear in Duke Mathematical Journa