Obstructions for rational points on curves

Felipe Voloch

This series of four lectures will discuss the various obstructions to the existence of rational points on curves over global fields, such as the Brauer-Manin obstruction, the finite descent obstruction and the obstructions coming from Grothendieck's section conjecture and the Chabauty method as well as weakenings and strengthenings of these various obstructions. We will discuss the relationship among these conjectures and some of the progress that has been made. We will look at the function field case, where more is known. We will discuss the connection with modularity, where some progress can be made sometimes contingent on some familiar conjectures.