Modular Curves

H is the upper half plane, a complex manifold. It will be helpful to interpret H in multiple ways. A lattice Λ ⊂ C is a free abelian group of rank 2, for which the map Λ ⊗Z R → C is an isomorphism. In other words, Λ is a subgroup of C of the form Zα⊕Zβ, where {α, β} is basis for C/R. Two lattices Λ and Λ′ are homothetic if Λ′ = θΛ for some θ ∈ C∗. This is an equivalence relation, and the equivalence classes are homothety classes. Let’s consider C as an oriented real vector space, meaning we have a privileged basis of ∧2 C modulo scaling by a positive real number. Then an oriented basis of a lattice Λ is a basis {a+ bi, c+ di} with ad− bc > 0. Any lattice Λ is homothetic to one of the form Z⊕Zτ , where τ ∈ H. The following is very easy: