二元二次型和类数的地形图

IF 0.8 3区 数学 Q2 MATHEMATICS
Mathematika Pub Date : 2025-09-01 DOI:10.1112/mtk.70042
Cormac O'Sullivan
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引用次数: 0

摘要

在这项工作中,我们比以前更详细地研究了J.H. Conway的积分二元二次型的地形。这些是平面上的树,其区域用整数标记,遵循一个简单的模式。每个地形图可以显示单一形式的值,或者表示等价的形式类。利用拓扑图和复数的一个新的连分式,给出了正则等价类表示的形式约化的新处理。这允许对任何正的、负的、平方的或非平方的判别式进行统一的约简。地形几何还提供了新的类数公式,以及关于三个平方和的高斯结果的简短证明。对类数的Hurwitz级数的推广给出了在地形图的区域或边缘上求和的某些无穷级数的评价。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Topographs for binary quadratic forms and class numbers

Topographs for binary quadratic forms and class numbers

Topographs for binary quadratic forms and class numbers

In this work, we study in greater detail than before, J.H. Conway's topographs for integral binary quadratic forms. These are trees in the plane with regions labeled by integers following a simple pattern. Each topograph can display the values of a single form, or represent an equivalence class of forms. We give a new treatment of reduction of forms to canonical equivalence class representatives by employing topographs and a novel continued fraction for complex numbers. This allows uniform reduction for any positive, negative, square, or nonsquare discriminant. Topograph geometry also provides new class number formulas, and short proofs of results of Gauss relating to sums of three squares. Generalizations of the series of Hurwitz for class numbers give evaluations of certain infinite series, summed over the regions or edges of a topograph.

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来源期刊
Mathematika
Mathematika MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
1.40
自引率
0.00%
发文量
60
审稿时长
>12 weeks
期刊介绍: Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.
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