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Fundamental domains for Shimura curves

David R. Kohel, Helena A. Verrill (2003)

Journal de théorie des nombres de Bordeaux

We describe a process for defining and computing a fundamental domain in the upper half plane of a Shimura curve X 0 D ( N ) associated with an order in a quaternion algebra A / 𝐐 . A fundamental domain for X 0 D ( N ) realizes a finite presentation of the quaternion unit group, modulo units of its center. We give explicit examples of domains for the curves X 0 6 ( 1 ) , X 0 15 ( 1 ) , and X 0 35 ( 1 ) . The first example is a classical example of a triangle group and the second is a corrected version of that appearing in the book of Vignéras [13], due to Michon....

Fundamental groups of some special quadric arrangements.

Meirav Amram, Mina Teicher (2006)

Revista Matemática Complutense

Continuing our work on the fundamental groups of conic-line arrangements (Amram et al., 2003), we obtain presentations of fundamental groups of the complements of three families of quadric arrangements in P2. The first arrangement is a union of n conics, which are tangent to each other at two common points. The second arrangement is composed of n quadrics which are tangent to each other at one common point. The third arrangement is composed of n quadrics, n-1 of them are tangent to the n-th one...

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