Geometry and distribution of roots of μ2≡D mod m with D≡1 mod 4

Zonglin Li; Matthew C Welsh

doi:10.4064/aa230523-8-3

Geometry and distribution of roots of μ²≡D mod m with D≡1 mod 4

Zonglin Li, Matthew C Welsh

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

We extend the geometric interpretation of the roots of the quadratic congruence μ²≡D (mod m) as the tops of certain geodesics in the hyperbolic plane to the case when the integer D>1 is square-free and D≡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.

Original language	English
Pages (from-to)	193-228
Number of pages	36
Journal	Acta Arithmetica
Volume	215
Issue number	3
DOIs	https://doi.org/10.4064/aa230523-8-3
Publication status	Published - 10 Jul 2024

Bibliographical note

Publisher Copyright:
© Instytut Matematyczny PAN, 2024.

Keywords

Pair correlations
Quadratic congruences
Geodesics

Access to Document

10.4064/aa230523-8-3

Full-text PDF (accepted manuscript)
This is the accepted author manuscript (AAM) of the article which has been made Open Access under the University of Bristol's Scholarly Works Policy. The final published version (Version of Record) can be found on the publisher's website. The copyright of any third-party content, such as images, remains with the copyright holder.
Accepted author manuscript, 782 KBLicence: CC BY

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Persistent link

Wave Transport in low-density matter, Siegel theta functions, and homogeneous flows
Marklof, J. (Principal Investigator)
28/10/19 → 27/04/23
Project: Research

Cite this

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N2 - We extend the geometric interpretation of the roots of the quadratic congruence μ²≡D (mod m) as the tops of certain geodesics in the hyperbolic plane to the case when the integer D>1 is square-free and D≡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.

AB - We extend the geometric interpretation of the roots of the quadratic congruence μ²≡D (mod m) as the tops of certain geodesics in the hyperbolic plane to the case when the integer D>1 is square-free and D≡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.

KW - Pair correlations

KW - Quadratic congruences

KW - Geodesics

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M3 - Article (Academic Journal)

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JO - Acta Arithmetica

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