Special values of modular functions and S-units

The values of modular functions at complex multiplication points are classically used to construct explicit abelian extensions of quadratic imaginary fields. The factorization of (the algebraic norm of) these numbers is also a problem that has been extensively studied, at least since the beginnings of the 1900s. More recently, particular attention has been given to the question of when these values are algebraic units, and more generally how often these special values are S-units when S is a fixed finite set of prime numbers. In this talk I will present a brief overview of known results and open related problems, and I will explain joint work with Ricardo Menares (PUC Chile) and Juan Rivera-Letelier (U. of Rochester) on the finiteness of special values of modular functions that are S-units.