Zonglin Li: Fine-scale statistics of roots of quadratic congruences

Abstract: Let $D>1$ be a square-free integer. Marklof and Welsh established limit laws for the fine-scale distribution of roots of the quadratic congruence $\mu^2\equiv D \pmod m$ with $D \not\equiv 1 \pmod 4$. This is achieved by considering the convergence of certain geodesic random line processes in the hyperbolic plane. Strikingly, they derived an explicit formula for the pair correlation density of the roots. In this talk, I will introduce their work and our recent result without the restriction $D \not\equiv 1 \pmod 4$. This is joint work with Matthew Welsh.