Belyi's Theorem

Will Sawin , Princeton University
Lewis Library 121

Belyi's Theorem provides a surprising and fairly recently-proved connection between the analytic and arithmetic theory of Riemann surfaces/algebraic curves. It states that a Riemann surface/algebraic curve has a holomorphic map to CP1 that is unramified except over 3 points if and only if it has a projective embedding where it is defined by polynomial equations with coefficients in a number field. I will explain what that means, provide a proof, and discuss consequences, such as the fact that the deep and mysterious absolute Galois group of Q acts faithfully on things you can draw on a blackboard.