Congruence Subgroups of PSL(2, Z) of Genus Less than or Equal to 24

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 ...

0 M ar 2 00 3 Transfer operators for Γ 0 ( n ) and the Hecke operators for the period functions of PSL ( 2 , Z ) 1

In this article we report on a surprising relation between the transfer operators for the congruence subgroups Γ0(n) and the Hecke operators on the space of period functions for the modular group PSL(2,Z). For this we study special eigenfunctions of the transfer operators with eigenvalues ∓1, which are also solutions of the Lewis equations for the groups Γ0(n) and which are determined by eigenf...

The Vacuum Structure of N=2 Super–QCD with Classical Gauge Groups

We determine the vacuum structure of N=2 supersymmetric QCD with fundamental quarks for gauge groups SO(n) and Sp(2n), extending prior results for SU(n). The solutions are all given in terms of families of hyperelliptic Riemann surfaces of genus equal to the rank of the gauge group. In the scale invariant cases, the solutions all have exact Sdualities which act on the couplings by subgroups of ...

Congruence Subgroups of Groups Commensurable with PSL

Congruence Subgroups of the Modular Group

The congruence subgroups of the classical modular group which can be defined as the automorphs modulo q of some fixed matrix are studied, and their genera determined. Let T = SL{2, Z). A congruence subgroup of T is any subgroup containing a principal congruence subgroup T^), defined as the set of elements A of T such that A = I mod q, where q is a positive integer. Of these one of the most impo...