Products of CM elliptic curves - Universität Duisburg-Essen

ecause here f 2 |f 1 , so f = f 2 and f 1= [R(I) : R]. Next, apply (41) to L f1 = R andL 2 = (R : I). Since R(L 2 ) = R(I −1 ) = R(I) by (43), we see that in this case (41)gives(R : (R : I)) = [R(I) : R]((R : I)) −1 = [R(I) : R][R(I) : R] −1 (I −1 ) −1 = ISince I ∗ = (R : (R : I)) by [5], p. 476, we thus have that I = I ∗ , and so I is a divisorialideal in the sense of Remark 7(d).We now apply the preceding results to abelian varieties.Proposition 15 Let A/K be an abelian variety such that ˜R = End 0 (A) is a quadraticfield. Then every non-zero ideal of R = End(A) is a kernel ideal, and hence we havefor any two non-zero ideals I, J of R that Φ I,J := Φ H(I),H(J) defines an isomorphism(44)Φ I,J := Φ H(I),H(J) : Hom(A H(I) , A H(J) )∼→(I : J).Moreover, we have that A H(I) ≃ A H(J) ⇔ I ≃ J (as R-modules).Proof. Since R is an order of ˜R = F , we know by Corollary 14 that every non-zeroideal of R is divisorial and hence a kernel ideal by Remark 7(d). This proves thefirst assertion, and hence the other assertions follow from Proposition 10 and from thediscussion after (20).Corollary 16 In the above situation, let R ′ be an order of ˜R with R ⊂ R ′ . Then thereexists an abelian variety A ′ /K which is isogenous to A/K such that End(A ′ ) ≃ R ′ .Proof. Take I = [R ′ : R]R ′ ⊂ R. Then I is a non-zero R-ideal with R(I) = R ′ . ThenA ′ := A H(I) is isogenous to A and End(A ′ ) ≃ (I : I) = R(I) = R ′ by (44).Remark 17 (a) Note that if we drop the hypothesis R ⊂ R ′ in Corollary 16, thenthe corresponding statement is in general no longer true; cf. §3.3 below.(b) We can use the above Corollary 16 to construct an abelian variety A ′ with afinite subgroup scheme H ′ such that E(H ′ ) ⊄ End(A ′ ). In particular, H ′ is not anideal subgroup by Corollary 11.Indeed, suppose there exists an abelian variety A/K such that R := End(A) ⊂ Fbut R ≠ O F . Then by Corollary 16 there is an R-ideal I such that A ′ := A H(I) satisfiesEnd(A ′ ) ≃ O F . Consider H ′ = Ker(π ′ H(I) ). Since π′ H(I) : A′ → A is an isogeny, wehave (A ′ ) H ′ ≃ A, so E(H ′ ) ≃ End(A) = R. Thus O F = End(A ′ ) ⊄ E(H ′ ), and so H ′is not an ideal subgroup of A ′ .15