here - Sites personnels de TELECOM ParisTech - Télécom ParisTech

140 Chapter 5. Complex multiplication and elliptic curves5.2.1 Complex toriOver the complex numbers, a general Weierstraß equation can be simplified to yield an equivalentequation of the form 2 E : y 2 = x 3 + ax + b .Two important invariants have then simple definitions.Definition 5.2.1 (Discriminant). Let E be a curve given by the Weierstraß equationE : y 2 = x 3 + ax + b .The discriminant of the Weierstraß equation is given by∆ = −16(4a 3 + 27b 2 ) .E is non-singular, and so is an elliptic curve, if and only if ∆ ≠ 0.Definition 5.2.2 (j-invariant). Let E be an elliptic curve given by the Weierstraß equationThe j-invariant of E is given byE : y 2 = x 3 + ax + b .j = −1728 (4a)3∆ .Two elliptic curves are isomorphic if and only if they have the same j-invariant.We are now going to give another description of elliptic curves over the complex numbers.Definition 5.2.3 (Lattice). A lattice Λ in C is a Z-module of rank 2 such that Λ ⊗ Z R = C.We denote by L be the set of lattices in C.Definition 5.2.4 (Multiplier ring). Let Λ be a lattice. The multiplier ring of Λ, denoted by R(Λ),is the set {α ∈ C | αΛ ⊆ Λ}.The quotient of C by a lattice is called a complex torus.Definition 5.2.5 (Complex torus). Let Λ be a lattice in C. We say that C/Λ is a complex torus.Such a complex torus is depicted in Figure 5.1. It is a basic fact that two complex tori areisomorphic if and only if the corresponding lattices are homothetic.We now relate lattices in C with the Poincaré upper halfplane through bases of a special form.Let Λ be a lattice in C and (ω 1 , ω 2 ) a basis of Λ such that τ = ω1ω 2satisfies I(τ) > 0. Then Λ isequivalent up to homothety toΛ τ = Zτ + Z .The value of τ depends on the initial choice of the basis, but it is easy to describe the equivalenceclasses of such numbers.Definition 5.2.6 (Poincaré upper halfplane). We denote by H the Poincaré upper halfplane, i.e.the set H = {x ∈ C | I(x) > 0}.2 This is true as soon as the characteristic is different from 2 and 3.