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178 Chapter 6. Complex multiplication in higher generaConversely, every such element ξ ∈ E × defines a form with rational values on Λ which havebounded denominators, so a rational multiple of it is a Riemann form. We call such a form arational Riemann form.From the results of Subsection 6.3.5, we see that the polarization induced by ξ betweenC g /Φ(a) and C g /Φ(a ∗ ) is analytically described by multiplication by ξ. In particular, we haveξa ⊂ a ∗ , i.e. ξ ∈ (a ∗ : a). Its kernel is included in Φ(E) and given by K(ξ) = Λ(ξ)/a whereΛ(ξ) = {x ∈ V | Tr(ξxa) ⊂ Z} .Hence Λ(ξ) = ξ −1 a ∗ and K(ξ) = (ξ −1 a ∗ )/a. The degree of the polarization is therefore #K(ξ) =[ξ −1 a ∗ : a]. This is summarized in the following proposition.Proposition 6.3.18 ([30, 4.3], [273, Theorem 3]). Let (A, i, ψ) be a polarized complex CM abelianvariety of type (E, Φ, a, ξ). Thenξ ∈ (a ∗ : a) .The degree of the polarization is [a ∗ : ξa]. In particular, the polarization is principal if and only ifthe following equality holds:ξa = a ∗ .If A is simple and principal, then the last equality readsξaad K/Q = O K .A homomorphism from (A, i, ψ) of type (E, Φ, a, ξ) to (B, j, ψ ′ ) of type (E, Φ, b, ζ) is describedby an element α ∈ E such that αa ⊂ b. If ω 1 and ω 2 are the Riemann forms corresponding to thepolarizations, then ω 2 (z, w) = α ∗ ω 1 (z, w) = ω 1 (Φ(α)z, Φ(α)w), so that we have ξ = ααζ [159,3.5], [238, Proposition IV.14.3].In particular, two triples (A, i, ψ) of type (E, Φ, a, ξ) and (B, j, ψ ′ ) of type (E, Φ, b, ζ) areisomorphic if and only if there exists α ∈ E × such that b = αa and ξ = ααζ. Moreover, it isreadily seen that [a ∗ : ξa] = [b ∗ : ζb] so that such an isomorphism preserves the degree of thepolarization. To summarize, we get the following classification up to isomorphism.Proposition 6.3.19 (Classification of polarized CM abelian varieties over the complex numbersup to isomorphism [205, I.3.4], [237, 5.5.B]). The triples (A, i, ψ) of type (E, Φ) are classified upto isomorphism by the quadruples (E, Φ, a, ξ) up to a change by an element α ∈ E × as above.If the abelian varieties are simple, then every homomorphism necessarily commutes with theaction of i(E) and we can drop the dependency on i.This classification can be made more precise if we fix the couple (A, i) and the analyticparametrization θ, and so the type (E, Φ, a). First, if ξ and ζ define two polarizations on (A, i)of type (E, Φ, a), then ξζ −1 is left invariant by complex conjugation and so is a totally positiveinvertible element α 0 of E 0 , the subalgebra of E fixed by its involution [238, Proposition IV.14.2].Conversely, if α 0 is a totally positive invertible element of E 0 and ξ defines a compatiblepolarization on (A, i), so does some positive multiple of α0 −1 ξ. Second, if α ∈ E× defines anautomorphism of (A, i), then a = αa and α must be a unit in R, i.e. α ∈ R × . If moreover αdefines an isomorphism between polarized abelian varieties, then αα is a totally positive unit ofE 0 . This leads to the following proposition.Proposition 6.3.20 ([238, Proposition IV.14.5]). Let U 0 be the group of totally positive unitsof E 0 , and U 1 the subgroup of units of the form αα for α ∈ R × . Suppose that there exists a