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A. Muhammed Uludağ; Ayberk Zeytin; Merve Durmuş
Binary quadratic forms as dessins
Journal de théorie des nombres de Bordeaux, 29 no. 2 (2017), p. 445-469, doi: 10.5802/jtnb.987
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Class. Math.: 11H55, 05C10
Keywords: binary quadratic forms, dessins d’enfants, bipartite ribbon graphs, çarks, ambiguous forms, reciprocal forms, Markoff number

Résumé - Abstract

We show that the class of every primitive indefinite binary quadratic form is naturally represented by an infinite graph (named çark) with a unique cycle embedded on a conformal annulus. This cycle is called the spine of the çark. Every choice of an edge of a fixed çark specifies an indefinite binary quadratic form in the class represented by the çark. Reduced forms in the class represented by a çark correspond to some distinguished edges on its spine. Gauss reduction is the process of moving the edge in the direction of the spine of the çark. Ambiguous and reciprocal classes are represented by çarks with symmetries. Periodic çarks represent classes of non-primitive forms.

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