here - Sites personnels de TELECOM ParisTech - Télécom ParisTech

142 Chapter 5. Complex multiplication and elliptic curvesUsing the Weierstraß ℘-function, it is easily seen that a complex torus is analytically isomorphicto an elliptic curve.Definition 5.2.10 (Weierstraß ℘-function [244, Section VI.3]). Let Λ be a lattice in C. TheWeierstraß ℘-function (relative to Λ) is defined by℘ (z; Λ) = 1 z 2 +∑ (1(z − ω) 2 − 1 )ω 2 .ω∈Λ, ω≠0Definition 5.2.11 (Eisenstein series [244, Section VI.3]). Let Λ be a lattice in C and k > 1 be apositive integer. The Eisenstein series of weight 2k (for Λ) is defined byG 2k (Λ) =∑ω∈Λ,ω≠01ω 2k .Proposition 5.2.12 ([244, Proposition VI.3.6]). Let Λ be a lattice in C and denote by g 2 and g 3the quantitiesThen g 3 2 − 27g 2 3 ≠ 0 and the curve defined byg 2 (Λ) = 60G 4 (Λ) ,g 3 (Λ) = 140G 6 (Λ) .is an elliptic curve. Moreover the mapE : y 2 = 4x 3 − g 2 x − g 3C/Λ → E(C) ,z ↦→ [℘(z) : ℘ ′ (z) : 1] ,is a complex analytic isomorphism of complex Lie groups.The discriminant and the j-invariant of the curve corresponding to τ ∈ H can then be definedas complex analytic functions:∆(τ) = g 3 2(τ) − 27g 3 (τ) 2 ,j(τ) = 1728 g 2(τ) 3∆(τ).The converse of this statement is harder to prove and is called the uniformization theorem forelliptic curves over C. It involves studying the j-invariant as a modular function.Theorem 5.2.13 (Uniformization [245, Corollary I.4.3]). Let a and b be complex numbers suchthat 4a 3 + 27b 2 ≠ 0. Then there exists a lattice Λ in C such that g 2 (Λ) = −4a and g 3 (Λ) = −4b.Hence, the mapC/Λ → E : y 2 = x 3 + ax + b ,z ↦→ [℘(z, Λ) : 1 2 ℘′ (z, Λ) : 1] ,is a complex analytic isomorphism.