GAUS AG: The André-Oort conjecture (Talk 12)

Jacob Tsimerman (University of Toronto)

Abstract: We explain how to prove upper bounds for heights of special points on arbitrary Shimura Varieties, completing the proof after the work of Binyamini-Schmidt-Yafaev. The proof in fact reduces to the case of A_g, and thus requires a comparison of height functions between distinct Shimura varieties. To facilitate this, we introduce a  canonical height function corresponding to automorphic vector bundles on it and explain their functorial properties. This is achieved using relative p-adic Hodge theory, combining the de-Rham and crystalline formalisms. This also reduces the question of bounding these canonical heights in  0-dimensional settings. Finally, we explain how a trick of Deligne allows us to conclude this case using functoriality and the Abelian case.