Ax-Lindemann for $\mathcal{A}_g$

Abstract

We prove the Ax-Lindemann theorem for the coarse moduli space $\mathcal{A}_{g}$ of principally polarized abelian varieties of dimension $g\ge 1$. We affirm the André-Oort conjecture unconditionally for $\mathcal{A}_g$ for $g\le 6$, and under GRH for all $g$.