Departmental Colloquium: Canonical heights and the Andre-Oort conjecture, by Jacob Tsimerman
September 17, 2021
3:00 PM - 3:50 PM
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Jacob Tsimerman (University of Toronto): Canonical heights and the Andre-Oort conjecture
(Joint with Jonathan Pila and Ananth Shankar) Shimura varieties are objects which are at the heart of arithmetic geometry, yet they start life as purely complex analytic objects. Many Shimura varieties are moduli spaces---mostly relating to abelian varieties---and this makes their arithmetic much more accessible. However, for the ones that have no moduli interpretation, we only see shadows of these structures as Galois representations, variations of Hodge structures, modular forms, etc. We explain how recent advances---particularly in relative p-adic Hodge theory---allow us to make precise arithmetic statements about Shimura varieties that were previously inaccessible. Primarily, we explain how to create a canonical height for arbitrary Shimura varieties, analogous to the Faltings height for the Siegel modular variety.
As our chief application, we explain how to use this theory to complete the proof of the Andre--Oort conjecture. While this conjecture can be made purely in the analytic world---describing the Zariski distribution of special points---its proof involves much algebraic and analytic number theory relying heavily on point counting results of Pila-Wilkie and recent improvements of Binyamini. We will try to present an overview of all the ingredients that go into the proof.
Sep 28, 2021
Sep 28, 2021