Geometry and distribution of roots of $\mu ^2\equiv D\ {\rm mod\ } m$ with $D \equiv 1\ {\rm mod\ } 4$
Volume 215 / 2024
Acta Arithmetica 215 (2024), 193-228
MSC: Primary 11K31; Secondary 37D40, 37A44
DOI: 10.4064/aa230523-8-3
Published online: 10 July 2024
Abstract
We extend the geometric interpretation of the roots of the quadratic congruence $\mu ^2 \equiv D$ (mod $m$) as the tops of certain geodesics in the hyperbolic plane to the case when the integer $D \gt 1$ is square-free and $D\equiv 1$ (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.
