Products of CM elliptic curves - Universität Duisburg-Essen

Corollary 53 If A ≃ E 1 × . . . × E n , where E 1 , . . . , E n ∈ Isog(E/K), thenf A = lcm(f E1 , . . . , f En ).Proof. By Proposition 36 there is an E 0 ∈ Isog(E/K) such that f E0 = lcm(f E1 , . . . , f En ).Thus, if π i : E 0 → E i is an isogeny, then H i = Ker(π i ) is an ideal subgroup of E 0 by(46) (because f Ei |f E0 by construction). Thus, H i = H(I i ), for some R 0 -ideal I i , whereR 0 = End(E 0 ). Note that by (45) we have that End(E i ) ≃ E(H i ) = (I i : I i ) = R(I i ),and so f Ei = f R(Ii ).Put H = H 1 × . . . × H n . Then A H ≃ E H1 × . . . × E Hn ≃ E 1 × . . . × E n ≃ A.By Proposition 41 we have H = H(I), where I = (I 1 | . . . |I n ). Thus, by (95) wehave End(A) ≃ End(A H ) ≃ E(H) ≃ End R0 (M) op , where M = I 1 ⊕ . . . ⊕ I n ⊂ R0 n ,and so Z(End(A)) ≃ Z(End R0 (M)) op = (M : M) F = R F (M) by (94). Now clearly(M : M) F = R(I 1 ) ∩ . . . ∩ R(I n ). Since the latter has conductor lcm(f R(I1 ), . . . , f R(In ))by (37), the assertion follows.We next want to characterize the ideal subgroups of E n . The following characterizationshows that the necessary condition of Corollary 11 is also sufficient. Moreover,this result can also be viewed as a generalization of the criterion (46) of Theorem 18.Theorem 54 Let E/K is a CM elliptic curve, and let H be a finite subgroup schemeof E n . Then:(97)H is an ideal subgroup of E n ⇔ Z(End(E n )) ⊂ E(H) ⇔ f E(H) | f E .Before proving this, we note the following immediate application.Corollary 55 If A ∼ E n is an abelian variety which is isogenous to E n with f A |f E ,then there exist elliptic curves E 1 , . . . , E n ∈ Isog + (E/K) such that A ≃ E 1 × . . . × E n .Proof. Let π : E n → A be an isogeny, and let H = Ker(π). Then E n H ≃ A andEnd(A) ≃ E(H), so f A = f E(H) . It thus follows from the hypothesis and (97) that His an ideal subgroup. Thus, H = H(I), for some regular left ideal I of End(E n ), andhence by (95) we have A ≃ A H ≃ E 1 × . . . × E n , where E i := E H(Ii ), for suitable idealsI i of End(E). Since H(I i ) is an ideal subgroup of E, we have by (46) that f Ei |f E , andso E i ∈ Isog + (E/K), as claimed.Another application of Theorem 54 is the following result which can be viewedas a solution of the isomorphism problem which was posed in the introduction ofthis paper. Note that it applies to an arbitrary tuple E 1 , . . . , E n /K of isogenous CMcurves because by Proposition 36 we can always find an elliptic curve E/K such thatE 1 , . . . , E n ∈ Isog + (E/K).42