Useful geometry and modular forms

Algebraic geometry has become the standard language for many number theorists in recent decades. In this talk, we will define modular forms and related objects in the language of modern geometers, thereby giving a geometric motivation for their study. We will ask some naive questions from a purely geometric point of view about these objects, and try to answer them using standard geometric techniques. If time permits, we will discuss some rather deep consequences in number theory of our geometric excursion, and mention open problems in geometry whose solution would have profound consequences in number theory.