here - Sites personnels de TELECOM ParisTech - Télécom ParisTech

148 Chapter 5. Complex multiplication and elliptic curves3τ3τFigure 5.2: The lattice corresponding to τ = 1+i√ 23Definition 5.3.3. Let K be an imaginary quadratic field and O an order in K. Let E ∈ Ell(O)and Λ a lattice in C such that E ≃ E Λ . Let a be a proper fractional ideal of O. We define a ∗ E bya ∗ E = E a −1 Λ .The following proposition summarizes the above discussion.Proposition 5.3.4 ([245, Proposition II.1.2]). Let K be an imaginary quadratic field and Oan order in K. Then there is a simply transitive action of Pic(O) on Ell(O). In particular,#Ell(O) = h(O).5.3.2 Hilbert class polynomialLet σ ∈ Aut(C) be an automorphism of C and E an elliptic curve defined over C. We candefine E σ by letting σ act on the coefficients of a Weierstraß equation for E. In particular, thej-invariants of the two conjugated curves then verify j(E σ ) = j(E) σ .But σ also induces an isomorphism End(E σ ) ≃ End(E). So, if K is an imaginary quadraticfield (considered as a subfield of C), O an order in K and E an elliptic curve defined over C withEnd(E) ≃ O, then End(E σ ) ≃ O.Definition 5.3.5 (Hilbert class polynomial). Let K be an imaginary quadratic field and O anorder in K. The Hilbert class polynomial H O (X) of O is defined asH O (X) =∏(X − j(E)) .E∈Ell(O)There is only a finite number of elliptic curves defined over C with a given order as endomorphismring, so H O is well defined. The above discussion shows that H O (X) has rationalcoefficients. In fact, it can be shown that the j-invariant of a complex elliptic curve with complexmultiplication is an algebraic integer and thus that H O (X) has integral coefficients.