Exploring Quadratic Form Composition on Conway's Topograph

Undergraduate Honors Thesis

Citations

Citeable URL: https://scholar.colorado.edu/concern/undergraduate_honors_theses/wm117q09d

Abstract

In Disquistones Arithmeticae, Carl Friedrich Gauss constructs a composition law for binary quadratic forms. Gauss’s Composition law defines a group structure to the set of equivalence classes of primitive binary quadratic forms with the same discriminant. These group structures correspond to ideal classes of imaginary quadratic fields. In 1997, John Conway developed what is known as Conway’s Topograph: a method for visualizing binary quadratic forms. Binary forms depicted on the topograph form “lakes,” “wells”, and “rivers” which are unique to each equivalence class of a primitive form. For example, a well found in the primitive form will be found in all quadratics in the same equivalence class as the primitive form. Since there exists a bijection between the ideal class groups and binary quadratic forms and a bijection between quadratic forms and Conway’s Topograph, we wish to explore the bijection between the ideal class groups and Conway’s Topograph

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