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5.3. Elliptic curves with complex multiplication 147Corollary 5.2.39. Let ∆ be a negative integer such that ∆ ≡ 0, 1 (mod 4). ThenH(∆) =∑d∈N, d 2 |∆, ∆/d 2 ≡0,1 (mod 4)h(∆/d 2 ) .Finally, there is a well defined map between the two different kinds of class groups.Theorem 5.2.40 ([61, Theorem 7.7], [32, Proposition 8.4.5]). Let K be an imaginary quadraticfield and O an order in K. Let ∆ be the discriminant of O. The following mapb(∆) → Prop(O) ,f = aX 2 + bXY + cY 2 ↦→ Za + Z(−b + √ ∆)/2 ,is well defined and induces an isomorphism of Pic(∆) onto Pic(O).Corollary 5.2.41. Let K be an imaginary quadratic field and O an order in K. Let ∆ be thediscriminant of O. Thenh(O) = h(∆) .5.3 Elliptic curves with complex multiplicationIn this section we describe the equivalence classes up to isomorphism of complex elliptic curveswith complex multiplication by a given order in an imaginary quadratic field. We then define theHilbert class polynomial and state the main theorem of complex multiplication.5.3.1 Complex multiplicationLet E be an elliptic curve defined over C and τ ∈ H an element of the Poincaré upper halfplanesuch that E(C) ≃ C/Λ τ . Then End(E) ≃ End(Λ τ ) and we deduce the following result on thestructure of End(E).Proposition 5.3.1 ([244, Theorem VI.5.5]). Let E be an elliptic curve defined over C and τ ∈ Hsuch that E(C) ≃ C/Λ τ . Then:1. if τ is quadratic, then End(E) is an order in Q(τ);2. otherwise End(E) = Z.A lattice arising from a quadratic number τ is depicted in Figure 5.2.We are interested in the first case, and more precisely in classifying curves (up to isomorphism)having complex multiplication by a given order O in an imaginary quadratic field K (with a fixedembedding into C).Definition 5.3.2. Let O be an order in K an imaginary quadratic field. We denote by Ell(O)the set of elliptic curves defined over C such that End(E) ≃ O.From the results of Subsection 5.2.1, this set can also be described as the set of lattices inC with multiplier ring O. For any such lattice there exists α ∈ C such that αΛ ⊂ K and so wecan choose Λ to be a fractional ideal of O. Every such elliptic curve then comes from a properfractional ideal of O, i.e. we have a well defined map Ell(O) → Pic(O) which is easily seen to beinjective and surjective. We can even explicitly give an action of Prop(O) on Ell(O) as follows.