A proof of the Andre-Oort conjecture for A_g

We give a proof of the Andr'e-Oort conjecture for ${\mathcal{A}}_g$ - the moduli space of principally polarized abelian varieties. In particular, we show that a recently proven `averaged’ version of the Colmez conjecture yields lower bounds for Galois orbits of CM points. The Andr'e-Oort conjecture then follows from previous work of Pila and the author.