Binary quadratic forms as dessins

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ULUDAĞ A. M., ZEYTİN A., Durmus M.

JOURNAL DE THEORIE DES NOMBRES DE BORDEAUX, vol.29, no.2, pp.445-469, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 29 Issue: 2
  • Publication Date: 2017
  • Doi Number: 10.5802/jtnb.987
  • Journal Name: JOURNAL DE THEORIE DES NOMBRES DE BORDEAUX
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.445-469
  • Keywords: binary quadratic forms, dessins d'enfants, bipartite ribbon graphs, carks, ambiguous forms, reciprocal forms, Markoff number, MODULAR GROUP
  • Galatasaray University Affiliated: Yes

Abstract

We show that the class of every primitive indefinite binary quadratic form is naturally represented by an infinite graph (named cark) with a unique cycle embedded on a conformal annulus. This cycle is called the spine of the cark. Every choice of an edge of a fixed cark specifies an indefinite binary quadratic form in the class represented by the cark. Reduced forms in the class represented by a cark 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 cark. Ambiguous and reciprocal classes are represented by carks with symmetries. Periodic carks represent classes of non-primitive forms.