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 discrete subgroup of SL2(R), then it is possible to provide the quotient space G\H with the structure of a Riemann surface. A distinguished class of Fuchsian groups are the arithmetic ones. These are by definition groups commensurable with unit groups in quaternion orders. The best known example of an arithmetic Fuchsian group is the modular group SL2(Z). The genus of the (compactification of the) surfaces corresponding to certain subgroups of SL2(Z) is well investigated (see for example the first chapter in [16]). A more general investigation can be found in [4], where the authors, among other things, derive a list of all congruence subgroups of SL2(Z) which give Riemann surfaces with genus 0. Another related result is the determination of all arithmetic triangular groups in [17]. In this paper, we will consider the case of orders in quaternion division algebras. This case contains all arithmetic Fuchsian groups, except those commensurable with SL2(Z). There exists a general formula for the genus in this case. However, the implementation of this was only known explicitly in the simplest case of maximal orders in algebras over Q. The main purpose of the paper is to generalize this to arbitrary orders in rational quaternion algebras and also to maximal orders in algebras over quadratic fields. As an application of these explicit formulas, we give complete lists of all such orders for which the genus is less than or equal to 2. In Section 1, we give the necessary background and some notations. The following two sections contain general formulas for the area of fundamental