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.
Comment: The previous version (v2) claiming to prove the general case has a serious error, as was kindly pointed out by Klingler,Ullmo and Yafaev. As such, the article is being reverted to claiming only the case for A_g