Basic notions: Gal Binyamini (Weizmann) - Point counting: old and new

Thu, 30/12/202116:00-17:15
Ross 70
Link to live recording:





"Point counting" refers to a body of results establishing asymptotic upper bounds on the number of rational or algebraic points in various types of sets, as a function of height and degree. The proofs usually involve auxiliary polynomial type constructions similar to the ones used in classical “transcendence methods”. I will explain the ideas in the most basic case and formulate some more general results and conjectures.

Point counting has become popular over the last 10-15 years thanks to an abundance of applications in arithmetic geometry and related areas. I will try to give a bird's-eye view of the progress in this area by outlining the (very recent) proof of the Andre-Oort conjecture for general Shimura varieties, which employs three distinct applications of point counting at different stages of the argument (as opposed to two applications in previous incarnations of this approach). However, much of the argument can be presented in the more elementary context of the Manin-Mumford conjecture, and the talks will be accessible without advanced background.