Roots µ mod m of polynomial congruences F(X) ≃ 0 mod m are fundamental objects in number theory.
Statistical information about the normalized roots µ(m) ∈ R / Z for m up to large x is used, for example, to prove that n^2 + 1 is infinitely often a product of at most two primes.
Originally, this statistical information for quadratic roots was obtained from algebraic geometry (the Weil bound) and later from the spectral theory of automorphic forms.
Recently in joint work with Jens Marklof and Zonglin Li, we studied quadratic roots using techniques from homogeneous dynamics on SL(2, R) that were similar to techniques developed to study Farey fractions.
In this talk, I present work in progress on extending these techniques to roots of cubic congruences and SL(3,R)..