Binary quadratic forms as dessins (1508.01677v1)

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