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...

The theory of nonabelian cohomology is used to show that the set of fixed points of a finite group acting on a Siegel modular space is a union of Shimura varieties

The aim of the paper is to give a preliminary overview of some of the results of the thesis prepared by the author. We propose a bijective classification of the subgroups of the modular group by pointed trivalent diagrams. Conjugacy classes of those subgroups are in one to one correspondence with unpointed trivalent diagrams. We also give in the form of generating series, the number of those tr...

Our main goal is to analyze the geometric and spectral properties of the Picard modular group with Gaussian integer entries acting on the two-dimensional complex hyperbolic space.

Hecke proved analytically that when h > 2 or when h = 2 CDS (m/q), q E Z, q > 3, then B(h) = (7: Im I > 0, / Re 7 1 < h/2, [ 7 j > 1) is a fundamental region for the group G(h) = (SA , T), where S,+ : 7 --f 7 + X and T: 7 ---t -l/~. He also showed that B(A) fails to be a fundamental region for all other A > 0 by proving that G(X) is not discontinuous. We give an elementary proof of these facts ...