## Fundamental domains for Shimura curves.(English)Zbl 1044.11052

Let $$A$$ be an indefinite quaternion algebra over $$\mathbb Q$$ with discriminant $$D$$, and let $$\mathcal O$$ be a maximal order in $$A$$. An Eichler order of index $$N$$ in $$\mathcal O$$ defines a discrete subgroup $$\Gamma^D_0 (N)$$ of $$\text{PSL}_2 (\mathbb R)$$, and the quotient $$X^D_0 (N) = \Gamma^D_0 (N) \backslash \mathcal H$$ of the Poincaré upper half plane $$\mathcal H$$ by $$\Gamma^D_0 (N)$$ is a Shimura curve. In this paper the authors describe a process for determining a fundamental domain for $$X^D_0 (N)$$ in $$\mathcal H$$ and show that such a fundamental domain realizes a finite presentation of the quaternion unit group modulo its center. They also provide explicit examples for the curves $$X^6_0 (1)$$, $$X^{15}_0 (1)$$, and $$X^{35}_0 (1)$$.

### MSC:

 11G18 Arithmetic aspects of modular and Shimura varieties 14G35 Modular and Shimura varieties

ecdata; Magma
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### References:

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