Polynomial representation growth and the congruence subgroup problem

Let Γ be an S-arithmetic group in a semisimple group. We show that if Γ has the congruence subgroup property then the number of isomorphism classes of irreducible complex n-dimensional characters of Γ is polynomially bounded. In characteristic zero, the converse is also true. We conjecture that the converse also holds in positive characteristic, and we prove some partial results in this direction.