here - Sites personnels de TELECOM ParisTech - Télécom ParisTech

5.2. Elliptic curves over the complex numbers 141Figure 5.1: A complex torus of dimension 1Definition 5.2.7 (Modular group). We denote by SL 2 (Z) the special linear group over Z, i.e.{( )}a bSL 2 (Z) = ∈ Matc d2 (Z) | det(Z) = 1 .We denote by Γ(1) the modular groupΓ(1) = SL 2 (Z)/ {±1} .An action of ( the modular ) group on points of the Poincaré upper halfplane is defined as follows.a bA matrix γ = ∈ Γ(1) acts on τ ∈ H asc dγτ = aτ + bcτ + d .Proposition 5.2.8 ([245, Lemma I.1.2]). The following map is a bijectionΓ(1)\H → L/C ∗ ,τ ↦→ Λ τ .In fact, a representative of each coset in Γ(1)\H can be chosen in the so-called fundamentaldomain.Proposition 5.2.9 ([245, Lemma I.1.5]). Let F denote the fundamental domainF = {τ ∈ H | |τ| ≥ 1 and |R(τ)| ≤ 1/2} .Then for every τ ∈ H, there exists γ ∈ Γ(1) such that γτ ∈ F.