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Binary quadratic forms as dessins. (English. French summary) Zbl 1420.11096

Summary: 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.

MSC:

11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11G32 Arithmetic aspects of dessins d’enfants, Belyĭ theory
05C10 Planar graphs; geometric and topological aspects of graph theory
05E15 Combinatorial aspects of groups and algebras (MSC2010)

Software:

PARI/GP; OEIS; InfoMod
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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