The averaged Colmez conjecture was proved by Xinyi Yuan and Shou-Wu Zhang and independently by Andreatta, Goren, Howard and Madapusi-Pera. In 2019-2020, Gal Biniyamini, Harry Schmidt and Andrei Yafaev, building on previous work and ideas of Harry Schmidt on torsion points in tori and abelian varieties and Gal Biniyamini's point counting results, have formulated a conjecture on bounds of heights of special points and deduced from its validity the bounds for the Galois degrees of special points needed for the proof of the full André-Oort conjecture. In September 2021, Jonathan Pila, Ananth Shankar, and Jacob Tsimerman claimed in a paper (featuring an appendix written by Hélène Esnault and Michael Groechenig) a proof of the Biniyamini-Schmidt-Yafaev height conjecture, thus completing the proof of the André-Oort conjecture using the Pila-Zannier strategy. Coleman–Oort conjecture A related conjecture that has two forms, equivalent if the André–Oort conjecture is assumed, is the Coleman–Oort conjecture. Robert Coleman conjectured that for sufficiently large g, there are only finitely many smooth projective curves C of genus g, such that the Jacobian variety J(C) is an abelian variety of CM-type. Oort then conjectured that the Torelli locus – of the moduli space of abelian varieties of dimension g – has for sufficiently large g no special subvariety of dimension > 0 that intersects the image of the Torelli mapping in a dense open subset. Generalizations Manin-Mumford and André–Oort conjectures can be generalized in many directions, for example by relaxing the properties of points being `special' (and considering the so-called `unlikely locus' instead) or looking at more general ambient varieties: abelian or semi-abelian schemes, mixed Shimura varieties etc.... These generalizations are colloquially known as the Zilber–Pink conjectures because problems of this type were proposed by Richard Pink and Boris Zilber. Most of these questions are open and are a subject of current active research. See also Zilber–Pink conjecture References Further reading Diophantine geometry Conjectures