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150 Chapter 5. Complex multiplication and elliptic curvesCidCC/acan.C/p −1 aE(C)λp ∗ E(C)There exists an ideal b co-prime to p such that pb = (α) is principal. We can then extend ourdiagram toid α idC C C Ccan.C/a C/p −1 α can.a C/ba C/aλ µ νE(C) p ∗ E(C) b ∗ p ∗ E(C) = (α) ∗ E(C) E(C)for some isogeny µ : p ∗ E ↦→ b ∗ p ∗ E and an isomorphism ν : (α) ∗ E ↦→ E. It follows that thereduction of the composite of these maps is inseparable (this can be seen on differentials — whichwe will not present here — because ˜α = 0). But the degree of µ ◦ ν is N(b) which is co-prime to pand so the reduction of λ, which is of degree N(p) = p, must be purely inseparable. So ˜p ∗ E isisomorphic to Ẽp and j( ˜p ∗ E) = j(Ẽ)p .As a corollary, we remark that we can now lift the Frobenius homomorphism from characteristicp to characteristic zero provided that p splits in K and does not divide the conductor of O.Proposition 5.3.9 ([245, Proposition II.5.3], [160, Lemma 10.1.1]). Let K be an imaginaryquadratic field and O an order in K. Let a be a proper ideal of O and E an elliptic curve definedover L = K(j(E)) such that there exists an analytic representationθ : C/a → E(C) .For all but a finite number of primes p of degree 1 in K, if σ is the Frobenius homomorphism ofp in L, then we can find an analytic representationθ ′ : C/p −1 a → E σ (C)and an isogeny λ such that the following diagram commutes:C/aE(C)can.C/p −1 aθ θ ′λE σ (C)