Zonglin Li

Geometry and distribution of roots of $\mu^2 \equiv D \bmod m$ with $D \equiv 1 \bmod 4$

Marklof and Welsh established limit laws for the distribution in small intervals of the roots of the quadratic congruence $\mu^2 \equiv D \bmod m$, where $D>0$ is a square free integer and $D \equiv 1 \bmod 4$. In this talk, we will investigate the remaining case when $D \equiv 1 \bmod 4$. As not all ideals in $\mathbb{Z}[\sqrt{D}]$ are invertible, we need to consider geodesics associated with two different rings, $\mathbb{Z}[\sqrt{D}]$ and $\mathbb{Z}[\frac{1+\sqrt{D}}{2}]$. We will establish the connection between the roots $\frac{\mu}{m}$ and tops of geodesics in the Poincare upper half plane.