The Modular Group

The modular group Γ is the set of all 2 × 2 matrices with integral elements and determinant 1. That is, Γ is the special linear group of 2 × 2 matrices over the integers, Γ = SL(2, Z). It forms a group under matrix multiplication. If M = a b c d ∈ Γ Then M defines a map f M (z) = az + b cz + d of the extended complex plane to itself. Here f M (−d/c) = ∞, and f M (∞) = a/c. Suppose that N = α β γ δ. Then by direct calculation we find that f N (f M (z)) = (αa + βc)z + (αb + βd) (γa + δc)z + (γb + δd) = f N M (z). While it might seem surprising that the composition of one such rational function with another would be connected with matrix multiplication, the mystery can be dispelled by considering how M and N transform the 2-dimensional vectors C 2. Suppose that M z 1 z 2 = w 1 w 2 , N w 1 w 2 = t 1 t 2. Then N M z 1 z 2 = t 1 t 2. Now suppose we consider these vectors in terms of projective geometry. Two vectors are then considered to be the same if their coordinates are proportional (i.e., the vectors are colinear). In other words, if c is a non-zero complex number, then z 1 z 2 and cz 1 cz 2 are considered to be the same. Thus a projective point z 1 : z 2 is determined by the ratio z = z 1 /z 2 if its coordinates, and the image w 1 : w 2 is determined by the ratio w = w 1 /w 2 of its coordinates. But then w = w 1 w 2 = az 1 + bz 2 cz 1 + dz 2 = az 1 /z 2 + b cz 1 /z 2 + d = az + b cz + d , so the map from z to w reflects a linear transformation in projective coordinates.