FINE-SCALE DISTRIBUTION OF ROOTS OF QUADRATIC CONGRUENCES

We establish limit laws for the distribution in small intervals of the roots of the quadratic congruence mu(2) = D mod m, with D > 0 square-free and D (sic) 1 mod 4. This is achieved by translating the problem to convergence of certain geodesic ran-dom line processes in the hyperbolic plane. This geometric interpretation allows us in particular to derive an explicit expression for the pair correlation density of the roots.