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6.3. Complex multiplication 179polarized complex CM abelian variety (A, i, ψ) of type (E, Φ, a). Then there are exactly [U 0 : U 1 ]isomorphism classes 15 of polarized complex CM abelian varieties of the same type.The above discussion also implies that, if α defines an automorphism of (A, i, ψ) for somecompatible polarization ψ, then αα = 1 and consequently that α is a root of unity [238, IV.14.2].Finally, Streng gave the following criterion for complex CM abelian varieties with complexmultiplication by the same CM field but different types.Proposition 6.3.21 ([252, Lemma I.5.6]). Let (A, i) and (B, j) be two complex abelian varietiesof types (K, Φ) and (K, Ψ). If Φ is primitive and Φ and Ψ are not equivalent, then A and B arenot isogenous.6.3.7 The main theorems of complex multiplicationAs was the case in dimension 1, the main theorems of complex multiplication describe the actionof automorphisms of C analytically. The proofs of the theorems, although they are more involvedand technical, are quite similar. We sketch the main steps of the proofs in this subsection.The classical proofs involve the theory of a-multiplications and a-transforms defined below.Definition 6.3.22 (a-multiplication [205, Definition II.7.17], [159, 3.2], [238, II.7]). Let (A, i) bean abelian variety with complex multiplication by an order R. Let a be an ideal in R. A surjectivehomomorphism λ : A → A a is an a-multiplication if every homomorphism α : A → A for α ∈ afactors through λ and it is universal for that property. An abelian variety B such that there existsan a-multiplication λ : A → B is called an a-transform 16 .For example, if (A, i) is a complex CM abelian variety of type (R, Φ, a) and b is an ideal inR, then the quotient map C g /Φ(a) → C g /Φ(b −1 a) is a b-multiplication [205, Example II.7.18.b].Conversely, if (A, i) and (B, j) are isogenous, then there exists an a-multiplication betweenthem [205, Proposition II.7.29], [159, Proposition 3.2.6].In any characteristic, it can be shown that an a-multiplication exists for every ideal a 17 [238,Proposition II.7], [205, Proposition II.7.20], [159, 3.2].The classical proofs of the main theorems of complex multiplication first prove them in theprincipal case and are then extended to a general abelian variety using commutative diagrams.Indeed, the theory of maximal orders is far better understood than that of non-maximal ones.The situation for non-maximal orders is more subtle and similar direct proofs seem more arduousto obtain.The Shimura–Taniyama formula, or congruence relation, then express a lift of the Frobeniusendomorphism in characteristic zero as an a-multiplication.Theorem 6.3.23 ([205, Corollary II.8.7], [159, Theorem 3.3.4], [238, Theorem III.13.1]). Let(A, i) be an abelian variety defined over a number field k with complex multiplication by themaximal order R of the CM algebra E. Let P be a prime of k where A has good reductionand such that p = P ∩ Z is unramified in E and let p = P ∩ O E r. Let σ ∈ Gal(l/E r ) be theFrobenius endomorphism of P in l, a Galois extension of Q containing k. Then there exists ana-multiplication λ : (A, i) → (A σ , i σ ) such that ˜λ is the q-th power Frobenius endomorphismwhere q = [O E r : p] and a = N Φ r(p).15 Here an isomorphism between (A, i, ψ) and (A, i, ψ ′ ) is understood as an isomorphism in the sense ofShimura [238, 4.1] or Lang [159, 3.4]: it is an isomorphism of abelian varieties such that the polarizationsψ and λ ∗ ψ ′ are proportional. In particular, degree is not preserved.16 Waterhouse defines the related notion of kernel ideals [280, 3.2].17 In fact, A a can even be directly defined as the quotient of A by the finite group scheme ker(a) = ⋂ α∈a ker(α).