Arithmetic algebraization and arithmetic groups

Abstract:

It was already known to Klein that the modular group SL_2(Z) has
extraordinarily many finite index subgroups, most of which are not defined
by congruence conditions on the matrix entries. The Unbounded Denominators
conjecture, solved in a recent joint work with Frank Calegari and Yunqing
Tang, is an amendment for the failure of the Congruence Subgroup Property
for SL_2(Z): a Z[[q]] condition on the Fourier expansions of the modular
forms living on a given finite index subgroup of SL_2(Z) turns out to be a
necessary and sufficient condition for the subgroup to be congruence.