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5.3. Elliptic curves with complex multiplication 149Theorem 5.3.6 ([245, Theorem II.6.1], [160, Theorem 5.2.4]). Let E be an elliptic curve definedover the complex numbers with complex multiplication. Then j(E) is an algebraic integer.There exist three different proofs of this theorem [245, Section II.6]:1. a complex analytic one, considering j as a modular function;2. an l-adic one, showing that E must have good reduction at all primes;3. a p-adic one, showing that E can not have bad reduction at a prime.Moreover, we remark that j(E) σ only takes a finite number of different values, or moreprecisely that [Q(j(E)) : Q] ≤ h(O).5.3.3 The main theorem of complex multiplicationAccording to Proposition 5.2.30, Pic(O) is isomorphic toPic(O) ≃ Frac(O K , f)/ PF Z (O K , f) .But PF 1 (O K , f) ⊂ PF Z (O K , f) ⊂ Frac(O K , f) where PF 1 (O K , f) is the subgroup of Frac(O K , f)generated by principal ideals of the form αO K where α ∈ O K is such that α ≡ 1 (mod fO K ).This exactly means that PF Z (O K , f) is a congruence subgroup 4 and Pic(O) is a generalized idealclass group 5 . By the Existence theorem [216, Theorem VI.6.1], [145, Theorem 2.2], there existsan abelian extension of K, called the ring class field of O, such thatGal(H O /K) ≃ Pic(O) .Definition 5.3.7 (Ring class field). Let K be an imaginary quadratic field and O an order in K.The ring class field of O, denoted by H O , is defined to be the abelian extension of K such thatGal(H O /K) ≃ Pic(O) .The algebraic action of Gal(H O /K) on Ell(O) can be explicitly described in terms of theanalytic action of Pic(O). The key to this description is the Kronecker congruence relation.Proposition 5.3.8 (Kronecker congruence relation [145, Theorem 5.9], [245, Proposition II.4.2],[160, Theorem 10.1.1]). Let K be an imaginary quadratic field and O the order of conductor f inK. Let a 1 , a 2 , . . . , a h be representatives of the ideal classes in Pic(O). Let L be an extensionof K containing j(a 1 ), j(a 2 ), . . . , j(a h ). Let p be a rational prime co-prime to f, splitting aspO = pp ′ , which is not one of the finitely many primes of bad reduction of E 1 , E 2 , . . . , E h . Then,for any proper fractional ideal a ∈ Prop(O),where P is any prime ideal of L above pO K .j(a) p ≡ j(p −1 a) (mod P) ,Proof. There is a canonical analytic map coming from the inclusion a ⊂ p −1 a and Theorem 5.2.14states that there exists an isogeny λ such that the following diagram commutes:4 This is a notion from class field theory that we will not further describe here.5 The same remark as in Footnote 4 applies here.