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arXiv · 1508.01677

Binary quadratic forms as dessins

A. Muhammed Uludağ·Ayberk Zeytin·Merve Durmuş
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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.

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A. Muhammed Uludağ, Ayberk Zeytin, Merve Durmuş. 2015-08-07. Binary quadratic forms as dessins. https://arxiv.org/abs/1508.01677

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