Abstract

While the theory of congruence subgroups of SL(2,Z) has played a central role in modern number theory, the theory of noncongruence subgroups is still relatively underdeveloped. In this talk we will explain how noncongruence modular curves can be understood as moduli spaces of elliptic curves equipped with a generalized notion of level structure, which are parametrized by finite groups G. A central problem that arises is that of determining the connected components of Hurwitz spaces of G-covers of elliptic curves. This is also related to the group theoretic problem of understanding Nielsen equivalence classes of generating pairs of G. In both problems, the particular regime in which we are interested remains quite mysterious, and it is not even clear what to expect. Thankfully, in the case of G = SL(2,p), using results of Bourgain, Gamburd, and Sarnak, this problem can be solved. This leads to a number of interesting corollaries, including the Diophantine application that the Markoff equation x^2 + y^2 + z^2 - xyz = 0 satisfies "strong approximation".