The notion of ``generalized modular conditions'' was introduced by physicists to treat rational conformal field theory. It is an article of faith among physicists that these conditions, along with certain arithmetic restrictions, force a function to be a classical modular form. In joint work with G. Mason of UCSC, we develop a theory of generalized modular forms, and show that this belief is not too far afield. We also show that there exist generalized modular forms that are not classical modular forms.
This talk will discuss these results after a gentle introduction to the basics of modular forms on finite index subgroups of SL(2,Z).