Algebraic Curves and Belyi's theorem by Anand Deopurkar

Chair for the Talk: Nitin Nitsure, TIFR.

Abstract: Belyi's theorem states that a smooth projective curve $X$ over the complex numbers is defined over $\overline{\mathbb Q}$ if and only if it can be expressed as a finite cover of the projective line ramified over at most three points. As applications, we get that every such curve can be expressed as a compactification of the upper half plane by a finite index subgroup of the modular group. I will prove Belyi's theorem and discuss some of these applications.