Binary quadratic forms as dessins

Uludağ, A. Muhammed; Zeytin, Ayberk; Durmuş, Merve

Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 2, pp. 445-469.

Nous montrons que la classe de toute forme quadratique binaire indéterminée et primitive est représentée de façon naturelle par un graphe infini (appellé çark) avec un unique cycle, plongé dans une couronne conforme. Ce cycle est appelé le rachis du çark. Le choix d’un arc d’un çark donné spécifie une forme quadratique binaire indéterminée dans la classe représentée par le çark. Les formes réduites dans la classe représentée par un çark correspondent à certains arcs distingués sur son rachis. La réduction de Gauss est le processus de déplacement de l’arc vers la direction du rachis du çark. Les classes ambiguës et réciproques sont représentées par des çarks ayant une symétrie. Les çarks périodiques représentent les classes des formes non-primitives.

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.

Classification : 11H55, 05C10
Mots clés : binary quadratic forms, dessins d’enfants, bipartite ribbon graphs, çarks, ambiguous forms, reciprocal forms, Markoff number

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