## 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.

### MSC:

 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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### References:

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