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6.1. Abelian varieties 1676.1.5 Homomorphisms and the Rosati involutionIf A is a simple abelian variety, then every non zero endomorphism is surjective so that it is anisogeny and is invertible in End 0 (A) = End(A) ⊗ Z Q, the endomorphism algebra of A. Thus,End 0 (A) is a division algebra when A is simple.Moreover, if A and B are isogenous simple abelian varieties, then End 0 (A) ≃ End 0 (B).Finally, if A is not simple, using the decomposition of an abelian variety A into a product ofsimple abelian varieties A n11[65, Théorème VI.10.1]×· · ·×Anr rnot isogenous to each other, we get [213, Corollary IV.19.2],End 0 A ≃ Mat n1(End 0 (A 1 ) ) × · · · × Mat nr(End 0 (A r ) ) .Over the complex numbers, if A and B are two abelian varieties, they can be described ascomplex tori A ≃ V/Λ and B ≃ V ′ /Λ ′ . Moreover, as a consequence of the GAGA principle [234],or more precisely of a theorem of Chow [49], every complex analytic map between two complexabelian varieties is in fact given by rational maps and so is a morphism of abelian varieties. So itis enough to study homomorphisms between complex tori.Furthermore, a homomorphism φ between two complex tori X = V/Λ and X ′ = V ′ Λ ′ can belifted to a map ˜φ between the complex vector spaces V and V ′ which sends Λ into Λ ′ so that thefollowing diagram is commutative:˜φV V ′φV/Λ V ′ /Λ ′Therefore, we get different representations of Hom(X, X ′ ) [159, 1.1], [16, 1.2]:• a complex or analytic representation ρ a : Hom(X, X ′ ) → Hom C (V, V ′ ), φ ↦→ ˜φ,• and a rational representation ρ r : Hom(X, X ′ ) → Hom Z (Λ, Λ ′ ).Moreover, for the endomorphism algebra End 0 (X) = End(X) ⊗ Z Q of a complex torus X, therational representation ρ r ⊗ Z 1 : End 0 (X) ⊗ Z C → End C (Λ ⊗ Z C) is equivalent to the sum of thecomplex representation and its complex conjugate [16, Proposition 1.2.3]:ρ r ⊗ Z 1 ≃ ρ a ⊕ ρ a .In positive characteristic p > 0, the l-adic Tate module for any l co-prime to p, which is a freeZ l -module of rank 2g, plays the same role as Λ for the complex numbers [213, IV.19], [280].It can be shown that the Z-module of homomorphisms between two abelian varieties, and inparticular the endomorphism ring of an abelian variety, is always a free Z-module of finite rank.Theorem 6.1.21 ([213, Corollary IV.19.1], [204, Theorem I.10.15]). Let A and B be two abelianvarieties. Then Hom(A, B) is a free Z-module of rank ≤ 4 dim(A) dim(B).Over the complex numbers, if A ≃ V/Λ is simple, then Λ ⊗ Z Q is a vector space over End 0 (A)so that the rank of End(A) is in fact at most 2g [213, IV.19].