Abstract: the theory of complex multiplication is more than a century old; its origins date back to Klein, Hilbert, Kummer, Weber, Deuring and many others. It has been instrumental in the development of class field theory and algebraic number theory. Yet, more than a century later we find new theorems that are truly surprising.
I will start with this historical perspective and try to position some of these new developments in the light of the André-Oort conjecture - a conjecture in the area of Shimura varieties that was recently resolved by Tsimerman, building on ideas of Edixhoven, Pila, Wilkie and Zannier. The resolution rests on the averaged Colmez conjecture, a conjecture that addresses the arithmetic complexity of abelian varieties with complex multiplication, which was proved by Andreatta-Howard-Madapusi Pera and the speaker, and, independently, by Yuan-Zhang.