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5.2. Elliptic curves over the complex numbers 139Definition 5.1.29 ([244, Section VII.5]). Let K be a local field. Let E an elliptic curve definedover K and Ẽ be the reduction of a minimal Weierstraß equation. We say that E has1. good reduction if Ẽ is smooth,2. bad reduction if Ẽ is singular.We say that E has potential good reduction if it has good reduction over an extension of K.Reduction can also be defined for points P ∈ E(K) in the same way. The next propositionshows that the corresponding map is injective for m-torsion points where m is co-prime to thecharacteristic of the residue field.Proposition 5.1.30 ([244, Proposition VII.3.1]). Let K be a local field with uniformizer π. Letm ≥ 1 be an integer co-prime to the characteristic of K. Let E be an elliptic curve defined overK with good reduction. Let Ẽ be a non-singular reduction of E. Then the reduction mapis injective.E(K)[m] → Ẽ(k)[m]Similarly, reduction can be defined for isogenies. Using the Weil pairing and the Isogenytheorem [244, Theorem III.7.7], which is valid over finite fields and number fields, it can be shownthat reduction from a number field preserves degrees of isogenies.Proposition 5.1.31 ([245, Proposition II.4.4]). Let L be number field and p a prime ideal of L.Let E 1 and E 2 be two elliptic curves with good reduction at p and Ẽ1 and Ẽ2 their reductions atp. Then the reduction mappreserves degrees and is injective.Hom(E 1 , E 2 ) → Hom(Ẽ1, Ẽ2)As far as endomorphisms are concerned, we have much more precise information.Theorem 5.1.32 ([68], [160, Theorem 13.4.12]). Let L be a number field and O an order in K animaginary quadratic number field. Let E be an elliptic curve defined over L with endomorphismring End(E) ≃ O. Let p be a prime of L over the rational prime p. Suppose that E has goodreduction Ẽ at p.Then Ẽ is supersingular if and only if p is inert or ramifies in K.Otherwise, write down the conductor f of O as f = p r f 0 where gcd(p, f 0 ) = 1. ThenEnd(Ẽ) ≃ Z + f 0O K , the order of conductor f 0 in K. In particular, if p splits and p ∤ f, thenthe reduction map End(E) → End(Ẽ) is an isomorphism.5.2 Elliptic curves over the complex numbersIn this section we present the basic theory of elliptic curves over the complex numbers, as wellas its links with the theory of complex tori, lattices and binary quadratic forms. All curves aresupposed to be defined over the complex numbers.