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6.3. Complex multiplication 175• If A is isotypic, i.e. A ∼ B s where B is a simple abelian variety, then it has complexmultiplication 12 if and only if End 0 (A) contains a CM field F of degree 2g.• A has complex multiplication if and only if End 0 (A) contains a CM subalgebra.In fact, if A is a complex CM abelian variety and End 0 (A) contains a CM field F of degree2g, then A ∼ B s is isotypic. Moreover, the commutant of F in End 0 (A) is F [159, Theorem 3.1]and the center of End 0 (A) is End 0 (B) = K [159, Theorem 1.3.3]. Finally, the center of End Q (A)can also be described as (F r ) r [159, Theorem 1.5.3].We have a more precise result about the commutant of R in End(A) valid in any characteristic.Proposition 6.3.10 ([205, Corollary II.7.4]). Let A be an abelian variety with complex multiplicationby R. Then the commutant of R in End(A) is R.Working over the tangent space at zero of a complex CM abelian variety A, it can be shownthat the embedding i induces a rational representation and can in fact be given by a CM typeΦ, i.e. i(a) acts as Φ(a) [205, I.3.11], [159, 1.3]. We say that (A, i) is of type (E, Φ) or (R, Φ).Furthermore, if σ ∈ Aut(C), then (A σ , i σ ) is of type (R, σΦ).If A is an isotypic complex CM abelian variety, then there exists a CM field F of degree 2gand an embedding i : F → End 0 (A) which can be described by a CM type Φ and we also saythat (A, i) is of type (F, Φ), or (O, Φ) where O is an the order in F such that O = i(F ) ∩ End(A).If σ ∈ Aut(F ), then (A, i ◦ σ) is of type (O, Φσ).Proposition 6.3.11 ([205, Proposition I.3.13], [159, Theorem 1.3.5]). Let (A, i) be a complexCM abelian variety of type (E, Φ). Then A is simple if and only if E is a CM field and the typeΦ is primitive.Moreover, if (A, i) is a simple complex CM abelian variety of type (O, Φ) where O is an orderin the CM field K, then i is an isomorphism:andi(K) = End 0 (A) ,i(O) = End(A) .Finally, as was the case in dimension 1, complex CM abelian varieties can be described bylattices in their CM algebra.Theorem 6.3.12 ([205, I.3.11], [159, Theorem 1.4.1]). Let E be a CM algebra of degree 2gand R an order in E. Let (A, i) be a complex abelian variety with complex multiplication by R.Then there exists a lattice a in E which is a proper ideal of R, a CM type Φ and an analyticisomorphism θ such thatwhere V = C 2g .V/Φ(a) θ ≃ AWe say that the pair (A, i) is of type (E, Φ, a) or (R, Φ, a) with respect to θ.It can also be shown that the converse of Theorem 6.3.12 is true: every complex torus of theform V/Φ(a) admits a polarization and so is an abelian variety.Proposition 6.3.13 ([205, Example I.2.9], [159, Theorem 1.4.4]). Let E be a CM algebra, Φ aCM type, and a a lattice in E. Then the complex torus V/Φ(a) is polarizable.12 This is the more restrictive definition of Mumford [213, IV.22] and Lang [159, 1.2].