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168 Chapter 6. Complex multiplication in higher generaDefinition 6.1.22 (Rosati involution [213, IV.20], [204, I.14], [65, Définition VI.10.2]). Let (A, ψ)be a polarized abelian variety. The Rosati involution † is defined asfor u ∈ End 0 (A).u † = ψ −1 ûψ ,The Rosati involution satisfies u †† = u, (u + v) † = u † + v † and (uv) † = v † u † . So this is indeedan anti-involution of the Q-algebra End 0 (A).Theorem 6.1.23 (Positivity [213, Theorem IV.21.1], [204, Theorem I.14.3]). Let (A, ψ) be apolarized abelian variety. Then the map (u, v) ↦→ Tr(u † v) is a rational positive definite quadraticform on End 0 (A).As was the case in dimension 1, the existence of the Rosati involution can be used to classifyall the possible structures of the endomorphism algebra End 0 (A) for A simple. Most of theclassification was conducted by Albert [5, 4, 7, 6] and can be found e.g. in [213, Application IV.21.I]or [270, 12.27].Over the complex numbers, the Rosati involution can be explicitly described in terms of theRiemann form associated with the polarization.Theorem 6.1.24 ([159, Theorem 3.4.3]). Let (A, ψ) be a polarized abelian variety defined overthe complex numbers and ω the Riemann form associated with ψ via the analytic parametrizationθ : V/Λ ˜→A. Then the transpose with respect to ω corresponds via θ to the Rosati involutionassociated with ψ.6.2 Class groups and unitsIn this section we generalize the discussion of Subsection 5.2.2 to class groups of orders in numberfields of any degree.6.2.1 DescriptionLet K be a number field of degree n over Q. We denote by O K the ring of integers of K. Recallthat O K is maximal, i.e. any order O of K is a subring of O K ; in fact, it is the integral closure ofany order O of K. We denote the integral closure, or normalization, using a tilde.Theorem 6.2.1 ([251, Theorem 3.20]). Let K be a number field, O K its ring of integers and Oan order. ThenÕ = O K .To avoid double subscripts, we indifferently use the notation O K and Õ. As in the quadraticcase, the maximal order O K is a Dedekind ring and general orders are noetherian and of Krulldimension 1.The definitions of fractional, proper and invertible ideals are the same as in Subsection 5.2.2.Note that it is not true anymore that a proper ideal is always invertible. In fact, counterexamplesarise as soon as n ≥ 3 [63, 1.4]. However, every fractional ideal becomes invertible for some orderwhen raised to the power n − 1 [63, Theorem C].Necessary conditions for a fractional ideal a of O to be invertible are still that it is proper, i.e.its multiplier ring R(a) = (a : a) is exactly O and that its inverse a −1 = (R(a) : a) is also proper.