M Theory Lesson 1
Consider the following fractional transformations on the upper half plane. T is the map taking z to z + 1. A fundamental region for this map is a strip of width 1. We take the strip between -1/2 and 1/2 on the real line. The map S transforms the inside of the unit circle to the outside via z --> -1/z. Note that SS = 1 and (TS)^3 = 1. A fundamental region for the group so generated is the part of the selected strip above the unit circle. Maps compose via matrix multiplication for fractional transformations. The group generated by the map S fixes i and TS fixes the point w = exp(pi i/3), a root of unity.
By gluing this region into a cylinder with 2 singular points we obtain M(1,1), the orbifold moduli of the one punctured torus. The J invariant gives J(i) = 1 and J(w) = 0. We also take J(i oo) = oo. Then it is possible to describe the equivalence between elliptic curves by the relation J(tau) = J(tau'), where tau is the complex parameter which characterises the curve. So the moduli M(1,1) is parameterised by either tau in the upper half plane, or by z in the Riemann sphere CP^1 without the points 0,1,oo, which is the moduli M(0,4), obtained via a quotient of H using gamma(2).