Definition The modular group Γ (Gamma) = SL(2,Z) is the 2-dimensional special linear group over the integers. In other words, the modular group consists of all matrices
where a, b, c, and d are integers, and ad - bc = 1. The operation is the usual multiplication of matrices.
Table of contents |
2 Group-theoretic Properties 3 Applications to Number Theory 4 Congruence Subgroups |
The modular group is important because it forms a subgroup of the group of isometries of the hyperbolic plane. If we consider the upper half-plane model H of hyperbolic plane geometry, then the group of all orientation-preserving isometries of H consists of all Möbius transformations of the form
Relationship to Hyperbolic Geometry
where a, b, c, and d are real numbers and ad - bc = 1. Put differently, the group SL(2,R) acts on the upper half-plane H according to the following formula:
[...include here at least the expression of elements in terms of generators S and T...]
[...include here at least some mention of quadratic forms, the fundamental domain (modular curve) and modular forms...]
[...brief mention and definition of congruence subgroup, this really deserves its own article independent of Γ]Applications to Number Theory
Congruence Subgroups