Ax-Lindemann for \mathcal{A}_g

TL;DR

This paper proves the Ax-Lindemann theorem for the moduli space of principally polarized abelian varieties and confirms the André-Oort conjecture for dimensions up to 6, advancing understanding in algebraic geometry and number theory.

Contribution

It establishes the Ax-Lindemann theorem for alg and unconditionally verifies the André-Oort conjecture for alg with g , extending previous results.

Findings

Proved Ax-Lindemann theorem for alg.

Confirmed André-Oort conjecture for alg with g .

Enhanced understanding of special points and subvarieties in moduli spaces.

Abstract

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