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6.1. Abelian varieties 165A homomorphism ψ L with L ample as above is then an isogeny between A and  and is calleda polarization of the abelian variety 9 . If ψ L is of degree one, it is called a principal polarization.A morphism λ of polarized abelian varieties between (A, ψ) and (B, ψ ′ ) is a morphism of abelianvarieties such that 10 λ ∗ (ψ ′ ) (= ̂λψ ′ λ) = ψ .Over the complex numbers, an analytic description of the dual complex torus ̂X can be givenfor any complex torus X.Proposition 6.1.16 ([213, II.9], [65, Proposition V.5.9], [16, Proposition 2.4.1]). Let X =V/Λ be a complex torus. Let V ∗ {}= Hom C(V, C) be the set of C-antilinear forms and ̂Λ =l ∈ V ∗ | I(l) ⊂ Z . Then the mapV ∗ /̂Λ → Pic 0 (X) ,l ↦→ L(0, e 2iπIl(·) ) ,is an isomorphism. Moreover, P ic 0 (X) is isomorphic to the group of characters Λ ∗ 1 = Hom(Λ, C 1 ).The real vector space V ∗ is canonically isomorphic to Hom R (V, R) using a similar correspondenceas for the forms ω and H: if l ∈ V ∗ , then we define k = Il; if k ∈ V ∗ R = Hom R(V, R), thenwe define l(z) = −k(iz) + ik(z) [65, Proposition 5.9], [16, 2.4]. Finally, V ∗ R is isomorphic to Λ∗ 1,both being defined by their values on a basis of Λ.If u : X → Y is a homomorphism of complex tori, then its dual is analytically represented byû : l ↦→ l ◦ u.Furthermore, every line bundle L is of the form L(H, α) and it can be shown that thepolarization ψ L associated with L only depends on H, or equivalently on the Riemann formω = I(H) associated with H. Therefore, on a complex abelian variety the polarization can bedefined as the choice of a Riemann form.The following proposition implies the theorem of the square over the complex numbers anddescribes the homomorphism from X into Pic 0 (X) associated with a Riemann form.Proposition 6.1.17 ([213, II.9], [65, Lemme 3.2]). Let X be a complex torus, L(H, α) a linebundle on X and x ∈ X. Thenτ ∗ xL(H, α) ≃ L(H, αe 2iπIH(·,˜x) ) .More precisely, the homomorphism φ L corresponding to ω is given analytically byX = V/Λ → ̂X = V ∗ /̂Λ ,x ↦→ H(·, ˜x) ,9 This is neither the more general definition of Milne [204, I.11] where it is only required that φ can be describedas above over an extension of the base field, nor that of Shimura [238, 4.1] or Lang [159, 3.4] which only considerthem up to the existence of positive integers m and m ′ such that mL and m ′ L ′ are algebraically equivalent, i.e. ifthe associated Riemann forms are proportional. In the latter case, there exists a so-called basic polar divisor Yin the polarization C such that any divisor X in the polarization is algebraically equivalent to a multiple of Y :X ≡ mY .10 This is once again more restrictive than the definitions of Shimura [238, 4.1] and Lang [159, 3.4] which only setλ ∗ (C ′ ) ⊂ C. Lang [159, 3.5] says that the polarizations correspond to each other for the above definition. However,for automorphisms of polarized abelian varieties, both definitions coincide.