COUNTING POINTS OF SCHEMES OVER FINITE RINGS AND COUNTING REPRESENTATIONS OF ARITHMETIC LATTICES

We relate the singularities of a scheme X to the asymptotics of the number of points of X over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More precisely, if Gamma is an arithmetic lattice whose Q-rank is greater than 1, then let r(n) (Gamma) be the number of irreducible n-dimensional representations of Gamma up to isomorphism. We prove that there is a constant C (in fact, any C > 40 suffices) such that r(n) (Gamma) = O(n(C)) for every such Gamma. This answers a question of Larsen and Lubotzky.