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164 Chapter 6. Complex multiplication in higher generaPoincaré reducibility theorem [213, Theorem IV.19.1], [205, Theorem I.2.12], [65, ThéorèmeVI.8.1] shows that any abelian variety can be decomposed into a product of simple abelianvarieties.Proposition 6.1.13 ([213, Corollary IV.19.1], [65, VI.8.2]). Let A be an abelian variety. Thenthere exist abelian varieties A 1 , . . . , A n non isogenous to each other and positive integers r 1 , . . . , r nsuch thatA ∼ A r11 × Arn n .The abelian varieties A 1 , . . . , A n are unique up to isogeny and the integers r 1 , . . . , r n are unique.6.1.4 Picard variety and polarizationsLet A be an abelian variety. Let Div(A) denote the group of divisors of A, Div a (A) the subgroupof those algebraically equivalent to zero and Div l (A) the subgroup of those linearly equivalent tozero. The Picard group of A is Pic(A) = Div(A)/ Div l (A). The subgroup of equivalence classesof divisors algebraically equivalent to zero is denoted by Pic 0 (A) = Div a (A)/ Div l (A).The theorem of the square [213, Corollary II.6.4], [65, Théorème VI.3.3] implies that a grouphomomorphism between an abelian variety and its Picard group can be constructed from any linebundle.Theorem 6.1.14 ([213, Corollary II.6.4], [65, Théorème VI.4.2]). Let A be an abelian varietyand L a line bundle. Then the map ψ L defined asA → Pic 0 (A)x ↦→ [ τ ∗ xL ⊗ L −1]is a group homomorphism. It only depends on the algebraic equivalence class of L. If L is ample,then the kernel K(L) of ψ L is finite.The group Pic 0 (A) can then be given the structure of an abelian variety 7 .Theorem 6.1.15 (Picard variety [159, Section 3.4]). Let A be an abelian variety. There existsan abelian variety  called the Picard variety8 and a group isomorphism ≃ Pic 0 (A) .The Picard variety is also called the dual abelian variety of A. Indeed, there is a canonicalisomorphismA ≃ ̂ .Moreover, if u is a homomorphism between two abelian varieties A and B, then there exists a dualhomomorphism, or transpose, û : ̂B →  which verifies û = u, û + v = û+̂v for u, v ∈ Hom(A, B)and ̂vu = û̂v if u ∈ Hom(A, B) and v ∈ Hom(B, C). If u is an isogeny, then û is an isogeny ofthe same degree. On divisors, this is described by taking inverse image, i.e. by u ∗ . In particular,A and  have isomorphic endomorphism rings.7 This is done by quotienting A by the kernel of ψ L where L is an ample line bundle. The fact that the quotientis an abelian variety is definitely non-trivial.8 In fact, the Picard variety is defined together with a divisor on A × Â, but we will not use this fact here.