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5.2. Elliptic curves over the complex numbers 143Finally, we have an equivalence of categories between complex tori and elliptic curves over thecomplex numbers.Theorem 5.2.14 ([244, Theorem VI.4.1]). The following maps are bijections:1.2.{α ∈ C | αΛ 1 ⊂ Λ 2 } → {φ : C/Λ 1 → C/Λ 2 | φ(0) = 0} ,defined by passing to the quotient;α ↦→ φ α ,{φ : E 1 → E 2 | φ(O E1 ) = O E2 } → {φ : C/Λ 1 → C/Λ 2 | φ(0) = 0} ,the natural inclusion associating an holomorphic map with an isogeny.These results are summarized in the following theorem.Theorem 5.2.15 ([244, Theorem VI.5.3]). The following categories are equivalent:1. elliptic curves over C up to isomorphism and isogenies;2. elliptic curves over C up to isomorphism and complex analytic maps sending 0 onto 0;3. lattices in C up to homothety and multiplication by a complex number.5.2.2 Orders in imaginary quadratic fieldsLet K be an imaginary quadratic field. Recall that an order is a subring of K which is also alattice, i.e. a Z-module of rank 2 in this case. Moreover, all orders are subrings of the ring ofintegers of K, the integral closure of Z in K.Proposition 5.2.16. Let K be an imaginary quadratic field and O K its ring of integers. ThenO K is the maximal order of K. This is the only order of K which is integrally closed. Furthermore,O K is a Dedekind ring.Proposition 5.2.17 (Conductor [160, Theorem 8.1.3], [61, Lemma 7.2]). Let K be an imaginaryquadratic field, O K its ring of integers and O an order. Then there exists an integer f ∈ N ∗ suchthat O = Z + fO K .Conversely, every integer f ∈ N ∗ gives a unique order.Proposition 5.2.18. Let K be an imaginary quadratic field, O K its ring of integers and O anorder. Then O is noetherian and of Krull dimension 1, i.e. all non-zero prime ideals are maximal.Definition 5.2.19 (Fractional ideal). Let K be an imaginary quadratic field and O an orderin K. A fractional ideal a is a non-zero O-submodule of K such that there exist α ∈ K ∗ withαa ⊂ O. The set of fractional ideals is denoted by Frac(O).In particular, the zero ideal is not a fractional ideal, but every non-zero ideal is. As O isnoetherian, its ideals are finitely generated and its fractional ideals are as well.If Λ is a lattice in K, then its multiplier ring R(Λ) is an order in K, and so Λ is a fractionalideal for some order. Conversely, a fractional ideal is a lattice in K. In fact, the fractional idealsof O are exactly the lattices in K whose multiplier rings are orders containing O. Addition,intersection and multiplication are defined on fractional ideals as they are on usual integral ideals,and they give well defined composition laws. Moreover, we can define the ideal quotient, alsocalled ideal colon.