Modular forms and p-curvature

Because of applications primarily to geometry and physics,
there has been increasing attention paid to modular forms with say
rational Fourier coefficients but denominators which get arbitrarily
large. By an old conjecture of Atkin and Swinnerton-Dyer, modular forms
of noncongruence subgroups of the modular group must have
this property (unless they're also modular forms for a congruence
subgroup, of course). Despite work by many people, there has been
relatively  little progress on this conjecture. In my talk I'll explain
a new approach to this area.