Summary of introduction to the complex analytic theory of modular curves and forms:
- Equivalence of the categories of elliptic curves over C and lattices in C.
- From lattices+bases to the action of SL(2,Z) on the upper half plane.
- A fundamental domain for this action.
- The groups Gamma(N), Gamma1(N), Gamma0(N) (and congruence subgroups).
- The (moduli) spaces Y(N), Y1(N), Y0(N) and their interpretations and fundamental domains.
- The natural maps between the moduli spaces.
Literature: any book on modular forms.