Introduction to Modular Forms

Abstract

The word ‘modular’ refers to the moduli space of complex curves (= Riemann surfaces) of genus 1. Such a curve can be represented as ℂ/Λ where Λ ⊂ ℂ is a lattice, two lattices Λ₁ and Λ₂ giving rise to the same curve if Λ₂ = λΛ₁ for some non-zero complex number λ. (For properties of curves of genus 1, see the lectures of Cohen and Bost/Cartier in this volume.) A modular function assigns to each lattice Λ a complex number F (Λ) with F(Λ₁) = F(Λ₂) if Λ₂ = λΛ₁. Since any lattice Λ = ℤω₁+ℤω₂ is equivalent to a lattice of the form ℤτ+ℤ with τ (= ω₁/ω₂) a non-real complex number, the function F is completely specified by the values f(τ) = F(ℤτ + ℤ) with r in ℂ \ ℝ or even, since f(τ) = f(−τ), with τ in the complex upper half-plane ℌ = { τ ∈ ℂ ∣ J(τ) > 0}. The fact that the lattice Λ is not changed by replacing the basis {ω₁, ω₂} by the new basis aω₁ + bω₂, cω₁ + dω₂ (a, b, c, d ∈ ℤ, ad − bc = ±1) translates into the modular invariance property $f\left( {\frac{{a\tau + b}} {{c\tau + d}}} \right) = f\left( \tau \right)$.