Topographs for binary quadratic forms and class numbers

TL;DR

This paper enhances the understanding of Conway's topographs for integral binary quadratic forms, introducing new reduction methods, class number formulas, and applications to classical number theory problems.

Contribution

It provides a novel approach to form reduction using topographs and a new continued fraction for complex numbers, enabling uniform reduction across all discriminants.

Findings

New class number formulas derived from topograph geometry

A novel continued fraction for complex numbers used in reduction

Simplified proofs of Gauss's results on sums of three squares

Abstract

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 non-square 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.