Binary quadratic forms and Eichler orders. (English) Zbl 1079.11022

Let \(\mathcal O\) be an Eichler order in an indefinite quaternion algebra \(H\). So there is a standard embedding \(H\hookrightarrow\text{M}_2(\mathbb R)\), and each element \(\left(\begin{smallmatrix} a & b\\ c & d\end{smallmatrix}\right)\) of \(H\) can be associated with a quadratic form \(cx^2+(d-a)xy-by^2\). Therefore, the order \(\mathcal O\) gives rise to a class \(\mathcal H(\mathcal O)\) of binary quadratic forms. With respect to the Fuchsian group \(\Gamma(\mathcal O) :=\{ \alpha\in\mathcal O\mid \text{nr}(\alpha)>0\}\subset\text{SL}_2(\mathbb R)\), the author classifies primitive forms in \(\mathcal H(\mathcal O)\), generalizing the classical theory over \(\text{SL}_2(\mathbb R)\). Furthermore, the \(\Gamma(\mathcal O)\)-reduced forms are descibed in terms of fundamental domains.


11E16 General binary quadratic forms
11G18 Arithmetic aspects of modular and Shimura varieties
11R52 Quaternion and other division algebras: arithmetic, zeta functions
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