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Title:聽 Quadratic Chabauty for modular curves聽

Abstract:聽By Faltings' theorem, the set of rational points on a curve of genus 2 or more is finite. We give a survey of the quadratic Chabauty method, which is used to determine a finite superset of the set of rational points for certain curves of genus 2 or more. In particular, we discuss what aspects of quadratic Chabauty can be made practical for certain modular curves and highlight several examples, including the non-split Cartan modular curve of level 27. This is based on joint work with Alexander Betts, Netan Dogra, Daniel Hast, Aashraya Jha, Steffen Mueller, Jan Tuitman, and Jan Vonk.