Modular forms are functions on the upper half plane which satisfy certain transformation laws given by the action of certain arithmetic groups. Though often introduced as a complex analytic object, modular forms have a tendency to appear in seemingly unrelated areas of mathematics. How many ways can an integer be written as a sum of four squares? Modular forms can be used to solve this. What is the densest arrangement of non-overlapping spheres in ? Modular forms can be used to solve this. The coefficients of the Fourier expansion of some important modular function encode the dimensions of the irreducible representations of the monster group (moonshine!!!). In the context of geometry, modular forms give pluricanonical forms on some moduli space. The goal of this talk is to introduce modular forms, marvel at some of their interesting properties, and then explain why I care about modular forms as an algebraic geometer.
Modular Forms and the Baily-Borel Theorem