he Andre-Oort conjecture for Ag, under GRH, after Pila and Tsimerman

In their arxiv preprint of June 2012 Jonathan Pila and Jacob Tsimerman prove that the Andre-Oort conjecture for the moduli spaces of principally polarised abelian varieties is a consequence of the generalised Riemann hypothesis for number fields. I will discuss their proof. Some essential ingredients are the so-called Ax-Lindemann theorem, proved in the same article, the "counting theorems'' by Pila and Wilkie, and a lower bound for the number of points in Galois orbits of CM-points by Tsimerman. Unfortunately I have not yet studied more recent work by Klingler, Ulmmo and Yafaev on the Ax-Lindemann conjecture for general Shimura varieties, and the generalisation to mixed Shimura varieties by Ziyang Gao.