Title: Geometry and arithmetic of noncongruence subgroups of SL(2,Z)
Abstract: Famously, SL(2,Z) does not have the congruence subgroup property. This means that it admits finite index subgroups which do not contain the kernel of the reduction map to SL(2,Z/n) for any n. Given the outstanding success of the congruence theory over the last century, it is natural to wonder if noncongruence subgroups may eventually aspire to the same success. In this talk I'll explain how a couple of exceptional isomorphisms in "low degree" leads to an understanding of noncongruence subgroups as capturing the geometry and arithmetic of punctured elliptic curves. While elliptic curves are abelian varieties, punctured elliptic curves are "anabelian" varieties in the sense of Grothendieck. A key observation is that noncongruence modular curves are moduli spaces for branched (possibly nonabelian) covers of elliptic curves, at most branched above the origin. This leads to a host of new questions, in topics ranging from discrete groups, Teichmuller dynamics, representation theory, algebraic geometry, and of course number theory. As time allows I'll describe some results and open problems.