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5.3. Elliptic curves with complex multiplication 151and such that the reduction of λ modulo any prime of L extending p is the p-th Frobeniushomomorphism composed with an automorphism of Ẽσ :˜λ = π p ◦ ɛ .If p is prime to the conductor of O, then that automorphism can be chosen to be the identity.Proof. Following the proof of Proposition 5.3.8, there exists an isogeny in characteristic zerowhose reduction is the composite of the p-th Frobenius homomorphism with an automorphism ofẼ σ . If p is prime to the conductor of O, then using Theorem 5.1.32, that automorphism (or ratherits inverse) can be lifted back to characteristic zero, changing only the analytic representation inthe commutative diagram.The Kronecker congruence relation can then be extended to every ideals.Theorem 5.3.10 ([145, Theorem 3.16], [245, Theorem II.4.3], [160, Theorem 10.3.5]). Let Kbe an imaginary quadratic field and O an order in K. Let σ ∈ Aut(C/K) and b a proper idealwhose Artin symbol on the ring class field is σ. Let a be a proper ideal. Thenj(a) σ = j(b −1 a) .Proof. We describe the approach of Kedlaya [145, Proposition 4.18]. Let L be an extension largeenough to contain H O and all the j(E i ). The Chebotarev density theorem ensures the existenceof infinitely many primes p ∈ O whose Artin symbol on L is σ and for which Proposition 5.3.8applies:j(a) σ ≡ j(a) p ≡ j(p −1 a) (mod P) ,where P is any prime ideal of L above pO K . All these primes have the same Artin symbol forH O . Therefore, by Artin reciprocity, they must lie in the same ideal class, so that j(p −1 a) isconstant. Then, the Kronecker congruence relation can be lifted back to L to obtain the desiredequality.Corollary 5.3.11 ([145, Corollary 3.17], [61, Theorem 11.1], [245, Theorem II.4.3], [160, Theorem10.3.5]). Let K be an imaginary quadratic field and O an order in K. Let E be anelliptic curve with complex multiplication by O. Then K(j(E)) is the ring class field of O and[K(j(E)) : K] = [Q(j(E)) : Q] = h(O).Proof. As [K(j(E)) : K] ≤ h(O), it is sufficient to prove that H O ⊂ K(j(E)).Following Kedlaya [145, Corollary 3.17] and Cox [61, Theorem 11.1], and according to Cox [61,Proposition 8.20], it is sufficient to show that the unramified rational primes under a prime ofdegree 1 of K(j(E)) are included in those that splits completely in H O , except for a finite numberof them.But, except for a finite number of them, if Q is a prime of degree 1 in K(j(E)), thenj(pa) p ≡ j(pa) (mod Q), and by the Kronecker congruence relation j(pa) p ≡ j(a) (mod Q). Ifwe moreover exclude the finite number of primes which divide the discriminant of the j-invariantof the curves with complex multiplication by O, then it implies that j(pa) = j(a) and, so, thatp is principal. Hence, it has a trivial Artin symbol and splits completely in H. Therefore,H ⊂ K(j(E)), whence the equality.The main theorem of complex multiplication then gives an explicit analytic description of thealgebraic action of the Galois group on torsion points 6 .6 The next theorem gives the idèlic formulation of the main theorem of complex multiplication. Multiplicationof fractional ideals by idèles will be defined in Subsection 6.2.3.