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Heights on arbitrary Shimura Varieties and Tori
June 30 at 16:30 – 17:30 CEST
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.