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

Zonglin Li; Matthew Welsh

doi:10.4064/aa230523-8-3

Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

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

Volume 215 / 2024

Zonglin Li, Matthew Welsh 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.

Authors

Zonglin LiSchool of Mathematics
University of Bristol
Bristol BS8 1UG, UK
e-mail
Matthew WelshDepartment of Mathematics
Michigan State University
East Lansing, MI 48824, USA
e-mail

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Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

Acta Arithmetica

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

Volume 215 / 2024

Abstract

Authors

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