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6.3. Complex multiplication 177We have the obvious inclusiona −1 R ∗ ⊂ a ∗where R ∗ is the trace dual or codifferent of R. It is a proper ideal of R, but in general it is notinvertible. It is invertible if and only if every proper ideal is [125, Theorem 5.2]. If R is themaximal order O K of a number field K, then OK ∗ = d−1 K/Qis the usual absolute codifferent of K;moreover, the above inclusion becomes an equality [238, IV.14.3]:a ∗ = a −1 d −1K/Q .Every R-linear form k on V considered as a real vector space can be described as [238, I.3.3]k : w ↦→ 〈z, w〉 = 〈z, w〉 C + 〈z, w〉 C =g∑(z i w i + z i w i )for some z ∈ V . If (A, i) is a complex CM abelian variety of type (E, Φ, a) and k ∈ ̂Λ, then zmust be equal to Φ(ξ) for ξ ∈ E and hence k corresponds to Tr(ξ·) on E [238, II.6.3]. Thus,̂Λ corresponds to a ∗ and the dual abelian variety  is analytically described by V/Φ(a∗ ) usingthe above identification between V and VR ∗ . Moreover, for α ∈ E, the dual endomorphismî(α) : l ↦→ l ◦ i(α) is given by i(α) on V . Hence, if we define î(α) = i(α), then (Â, î) is of type(E, Φ, a ∗ ).Finally, if λ is a homomorphism from (A, i) of type (E, Φ, a) to (B, j) of type (E, Φ, b)described by α ∈ E such that αa ⊂ b, then the dual homomorphism ̂λ from ( ̂B, ĵ) to (Â, î) isdescribed by α.6.3.6 PolarizationsUntil the end of this subsection, all CM abelian varieties are defined over the complex numbers.Let (A, i) be a CM abelian variety of type (E, Φ). If ψ is a polarization on A, then we say that itis compatible with i if the Rosati involution associated with ψ leaves i(E) stable. In particular, ifA is simple, then any polarization is compatible with i.Furthermore, if ψ is a compatible polarization on (A, i) and σ ∈ Aut(C), then ψ σ is acompatible polarization of the same degree on (A σ , i σ ). In particular, if (A, i) is principallypolarized, then so is (A σ , i σ ). From now on, we write p.p.a.v. for “principally polarized CMabelian variety defined over the complex numbers”.Theorem 6.3.17 ([205, Example I.2.9], [159, Theorem I.4.5]). Let (A, i, ψ) be a polarized complexCM abelian variety of type (E, Φ, a) with respect to some analytic parametrization θ. Then thereexists an invertible element ξ ∈ E × verifying ξ = −ξ and I(φ(ξ)) > 0 for all φ ∈ Φ such that theRiemann form ω associated with ψ can be described on Φ(E) asand extended by R-linearity to V :ω(z, w) =i=1ω(Φ(x), Φ(y)) = Tr(ξxy)g∑(φ i ξ)(z i w i − z i w i ) = 〈Φ(ξ)z, w〉 .i=1With the above notation, the polarized complex CM abelian variety (A, i, ψ) is said to be oftype (E, Φ, a, ξ) with respect to θ.