Products of CM elliptic curves - Universität Duisburg-Essen

Proof. To prove this, we shall use the criterion of Theorem 1. Now by Proposition49 we have that I E (E 1 ) ⊕ I E (E 2 ) ≃ I E (E 1) ′ ⊕ I E (E 2) ′ ⇔ R ⊕ I E (E 1 )I E (E 2 ) ≃ R ′ ⊕I E (E 1)I ′ E (E 2), ′ where R = R(I E (E 1 )) ∩ R(I E (E 2 )) and R ′ = R(I E (E 1)) ′ ∩ R(I E (E 2)).′Moreover, by Theorem (45) these modules are isomorphic if and only R = R ′ andI E (E 1 )I E (E 2 ) ≃ I E (E 1)I ′ E (E 2).′ Since R(I E (E i )) has conductor f i by (50), we seethat R has conductor f R = lcm(f 1 , f 2 ) by (37), and similarly f R ′ = lcm(f 1, ′ f 2). ′ ThusR = R ′ if and only if lcm(f 1 , f 2 ) = lcm(f 1, ′ f 2), ′ and so the assertion follows fromTheorem 1.If K = C, then above result can be restated in the following form which is essentiallyProposition 4.5 of [24].Corollary 64 Let L 1 , L 2 , L ′ 1, L ′ 2 ∈ Lat F be lattices in a quadratic field F and letf i = f R(Li ) and f ′ i = f R(L ′i ) be the conductors of their associated orders. ThenE L1 × E L2 ≃ E L ′1× E L ′2⇔ L 1 L 2 ≃ L ′ 1L ′ 2 and lcm(f 1 , f 2 ) = lcm(f ′ 1, f ′ 2).Proof. Put E i = E Li and E i ′ = E L ′i. Since End(E Li ) ≃ R(L i ) by (64), we havef Ei = f i and similarly f E ′i= f i. ′ Let R = R(L 1 ) ∩ R(L 2 ) ∩ R(L ′ 1) ∩ R(L ′ 2), andchoose n ∈ N such that L := nR ⊂ L i ∩ L ′ i, for i = 1, 2. Put E = E L . SinceEnd(E) ≃ R ⊂ R(L i ) ≃ End(E Li ), we see that E i ∈ Isog + (E/C) and similarlyE i ′ ∈ Isog + (E/C). By Corollary 27 we have I E (E i ) ≃ L −1i L and I E (E i) ′ ≃ (L ′ i) −1 L.Thus I E (E 1 )I E (E 2 ) ≃ I E (E 1)I ′ E (E 2) ′ ⇔ L −11 L −12 L ≃ (L ′ 1) −1 (L ′ 2) −1 L ⇔ (L 1 L 2 ) −1 ≃(L ′ 1L ′ 2) −1 , the latter because R(L) ⊂ R(L −11 ) ∩ R((L ′ 1) −1 ). Thus I E (E 1 )I E (E 2 ) ≃I E (E 1)I ′ E (E 2) ′ ⇔ L 1 L 2 ≃ L ′ 1L ′ 2, and hence it is clear that the corollary follows fromProposition 63.We next prove Theorem 4, which is clearly a special case of the following moreprecise result.Theorem 65 Let E/K be a CM elliptic curve of discriminant ∆ = ∆ E . Then thereexist bijections between the following sets:(i) the set Isog + (E/K) of isomorphism classes of elliptic curves E ′ /K with E ′ ∼ Eand f E ′|f E ;(ii) the set of non-zero ideal classes of End(E);(iii) the set of proper equivalence classes of positive definite binary quadratic formsq with discriminant ∆(q) = ∆;(iv) the set of End(R)-submodules M of End(E) 2 of rank 2 with R F (M) = R;(v) the set of isomorphism classes of abelian surfaces A/K with A ∼ E 2 and centralconductor f A = f E ;(vi) the set of isomorphism classes of abelian surfaces A/K with A ∼ E 2 anddiscriminant ∆(A/K) = −∆.49