Density and Defect Distribution of Reduced Ideals Missed by the Reduction Algorithm in Real Quadratic Fields

Schoof asked how many reduced Arakelov divisors cannot arise from the reduction algorithm. Tian and Tran subsequently characterized the image of this algorithm in real quadratic fields as precisely the $1$-reduced ideals. We give a quantitative answer to Schoof's question in the real-quadratic setting by determining the asymptotic proportion of reduced ideals missed by the algorithm. A key arithmetic input is the criterion that a reduced ideal with associated reduced form $(A,b,-r)$ fails to be $1$-reduced if and only if $A-r

Authors

Publication Details

Journal
arXiv (Cornell University)
Published
2026-10-27
DOI
https://doi.org/10.48550/arxiv.2610.36800
Primary Topic
Algebraic Geometry and Number Theory
Type
preprint
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preprint

Density and Defect Distribution of Reduced Ideals Missed by the Reduction Algorithm in Real Quadratic Fields

Tarō Hayashi, Mengzhe Guo, Ryota Sueyosh
arXiv (Cornell University)
Algebraic Geometry and Number Theory
preprint

Density and Defect Distribution of Reduced Ideals Missed by the Reduction Algorithm in Real Quadratic Fields

Tarō Hayashi, Mengzhe Guo, Ryota Sueyosh
preprint en

Abstract

Schoof asked how many reduced Arakelov divisors cannot arise from the reduction algorithm. Tian and Tran subsequently characterized the image of this algorithm in real quadratic fields as precisely the $1$-reduced ideals. We give a quantitative answer to Schoof's question in the real-quadratic setting by determining the asymptotic proportion of reduced ideals missed by the algorithm. A key arithmetic input is the criterion that a reduced ideal with associated reduced form $(A,b,-r)$ fails to be $1$-reduced if and only if $A-r

arXiv (Cornell University)
Algebraic Geometry and Number Theory
DOIOpen Access PDFOpenAlex
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Density and Defect Distribution of Reduced Ideals Missed by the Reduction Algorithm in Real Quadratic Fields — Tarō Hayashi, Mengzhe Guo, et al. · arXiv (Cornell University) (2026) | TGRS Research Map | TGRS