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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

Software:

ecdata; Magma
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References:

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