Some Arithmetic Fuchsian Groups with Signature $(0;e_1,e_2,e_3,e_4)$

Traces in Arithmetic Fuchsian Groups

Genera of Arithmetic Fuchsian Groups

Introduction. The fundamental invariant of a Riemann surface is its genus. In this paper, using arithmetical means, we calculate the genus of certain Riemann surfaces defined by unit groups in quaternion algebras. First we recall a well-known general construction of Riemann surfaces. The group SL2(R) acts on the upper half-plane H by Möbius transformations. If G is a Fuchsian group, that is, a ...

On fundamental domains of arithmetic Fuchsian groups

Let K be a totally real algebraic number field and O an order in a quaternion algebra A over K. Assume that the group O1 of units in O with reduced norm equal to 1 is embedded into PSL2(R) as an arithmetic Fuchsian group. It is shown how Ford’s algorithm can be effectively applied in order to determine a fundamental domain of O1 as well as a complete system of generators of O1.

Lectures on Shimura Curves: Arithmetic Fuchsian Groups

The class of Fuchsian groups that we are (by far) most interested in are the arithmetic groups. The easiest way to describe arithmetic Fuchsian groups is that class of groups containing all groups commensurable with PSL2(Z) and also all groups which uniformize compact Shimura curves. This is however, not a very wellmotivated definition: structurally, what do classical modular curves and Shimura...

Derived Arithmetic Fuchsian Groups of Genus Two

We classify all cocompact torsion-free derived arithmetic Fuchsian groups of genus two by commensurability class. In particular, we show that there exist no such groups arising from quaternion algebras over number fields of degree greater than 5. We also prove some results on the existence and form of maximal orders for a class of quaternion algebras related to these groups. Using these results...