Endomorphism rings of elliptic curves over finite fields by David Kohel

CHAPTER 3. COMPLEX MULTIPLICATION 34local-global correspondence of lattices, there is a well-defined lattice tΛ defined bytΛ = ⋂p∈M KK ∩ t p Λ p .Moreover, there are natural isomorphismsK/Λ ∼ = ⊕ pK p /Λ p and K/tΛ ∼ = ⊕ pK p /t p Λ p .This allows us to define an isomorphism t : K/Λ → K/tΛ by multiplication by t p onthe p-primary component.For the general case, we have an identification of adele rings A K = K ⊗ A Q , fromwhich we have a decomposition of rings A K = ∏ ′p K ⊗ Q p, restricted with respect tothe rings O K ⊗ Z p . For a prime p in Z, let Λ p = Λ ⊗ Z p . Then we can write an idelet ∈ J K as (t p ) p∈MQ , and t acts on Λ bytΛ = ⋂Again we have natural isomorphismsK/Λ ∼ = ⊕p∈M QK ∩ t p Λ p .p∈M QK p /Λ p , and K/tΛ ∼ = ⊕p∈M QK p /t p Λ p ,and we define an isomorphism t : K/Λ → K/tΛ by multiplication by t p on eachcomponent K p /Λ p .This definition coincides with that for the special case, and for any lattice Λ in K wenote that End(Λ) = O for some order O in K. If the conductor of O is m, then forall primes p of M K not dividing m, (O K ) p = O p and the lattice Λ p is well-defined.We have the liberty of decomposing an idele t ∈ J K as t = (t p ) p̸ |m ×(t q ) q|m ), and viewt as acting locally at p as in the previous case.Let σ ∈ Gal(C/Q). Corresponding to the automorphism of fields C ←− σC there is amorphism σ ∗ : Spec(C) −→ Spec(C). For any elliptic curve E/C we define E σ /C tobe the elliptic curve E base extended by σ ∗ .We can now state the main theorem of complex multiplication, in its idelic version.Theorem 20 Let K ⊆ C be a quadratic imaginary extension of Q, and let Λ be alattice in K. Let φ : C/Λ → E(C) be a complex analytic isomorphism to an ellipticcurve E. Let s be an idele of K and let σ be an automorphism of C such that[s, K] = σ| K ab. Then there exists a unique complex analytic isomorphismψ : C/s −1 Λ → E σ (C)