Geometry and distribution of roots of ${\mu }^2\equiv D\mathrm{mod}m$ with $D\equiv 1\mathrm{mod}4$

Abstract

We extend the geometric interpretation of the roots of the quadratic congruence ${\mu }^2\equiv D(\mathrm{mod}m)$ as the tops of certain geodesics in the hyperbolic plane to the case when $D$ is square-free and $D\equiv 1(\mathrm{mod}4)$. This allows us to establish limit laws for the fine-scale statistics of the roots using results by Marklof and the second author.

33 pages, 6 figures