Products of CM elliptic curves - Universität Duisburg-Essen

Proof. First note that for any two finite subgroups H 1 , H 2 of E we have(70)I(H 1 )I(H 2 ) ⊂ I(H 1 + H 2 ) ⊂ I(H 1 ) ∩ I(H 2 ).Indeed, since H i ≤ H 1 + H 2 , we have I(H 1 + H 2 ) ⊂ I(H i ) by (6) and so the secondinclusion of (70) follows. To prove the first inclusion, let f i ∈ I(H i ), so H i ≤ Ker(f i ).Since f 1 f 2 = f 2 f 1 , we have H i ≤ Ker(f i ) ≤ Ker(f 1 f 2 ) and so H 1 + H 2 ≤ Ker(f 1 f 2 ).Thus f 1 f 2 ∈ I(H 1 + H 2 ), and so the first inclusion of (70) holds.Put n i = |H i |, e i = [R : E(H i )] and N i = [R : I(H i )], so N i = e i n i by (69) (togetherwith the hypothesis that E(H i ) ⊂ R). Since H i ≤ Ker([n i ]), we have n i ∈ I(H i ), andso Rn i ⊂ I(H i ), which implies that N i = [R : I(H i )]|[R : Rn i ] = n 2 i . Thus, since(n 1 , n 2 ) = 1, we see that also (N 1 , N 2 ) = 1 and hence that (e 1 , e 2 ) = 1.In addition we have that I(H 1 ) + I(H 2 ) = R because n i ∈ I(H i ) and (n 1 , n 2 ) = 1.Thus, by elementary ideal theory we have that I(H 1 )I(H 2 ) = I(H 1 ) ∩ I(H 2 ), and soequality holds throughout in (70). Using (68) and (39), we obtain for H = H 1 + H 2thatE(H)R = R(I(H)) = R(I(H 1 )I(H 2 )) = R(I(H 1 ))R(I(H 2 )) = R · R = R,because R(I(H i )) = RE(H i ) = R by (68) and the hypothesis. Thus E(H 1 + H 2 ) ⊂ R.Next we note that H 1 ∩H 2 = 0 (because |H 1 ∩H 2 | | (n 1 , n 2 ) = 1), and so |H 1 +H 2 | =n 1 n 2 . Moreover, since I(H 1 + H 2 ) = I(H 1 ) ∩ I(H 2 ), we see that [R : I(H 1 + H 2 )] =[R : I(H 1 ) + I(H 2 )] = N 1 N 2 , the latter because I(H 1 ) + I(H 2 ) = R. Thus, by (69)(applied to H = H 1 + H 2 ) we obtain n 1 e 1 n 2 e 2 = N 1 N 2 = [RE(H) : E(H)]n 1 n 2 , and so[R : E(H)] = e 1 e 2 . Thus, if f = f R , then E(H) has conductor fe 1 e 2 = lcm(fe 1 , fe 2 )because (e 1 , e 2 ) = 1. Since f E(Hi ) = fe i , it thus follows from (37) that E(H) =E(H 1 ) ∩ E(H 2 ), as claimed.Coming back to the case that E = E L , we now generalize the above formula (66)for I(H) by giving a similar interpretation of the lattice H(H 1 , H 2 ).For this, we first observe that the identification map Ψ L of (64) can be generalizedas follows. If L 1 , L 2 ∈ C are any two lattices, then by [18], §1.4, we have a bijection(71)Ψ L1 ,L 2: (L 2 : L 1 ) C ∼ → Hom(E L1 , E L2 )which is given by the rule λ ↦→ π λ = π L 1,L 2λ∈ Hom(E L1 , E L2 ), where π λ is defined byπ λ (ρ L1 (z + L 1 )) = ρ L2 (λz + L 2 ), for λ ∈ (L 2 : L 1 ) C = {λ ∈ C : λL 1 ⊂ L 2 }.Note that these maps Ψ are multiplicative in the sense that if L 3 is a third lattice,then we have(72)π L 1,L 3λ 2 λ 1= π L 2,L 3λ 2◦ π L 1,L 2λ 1, if λ 1 ∈ (L 2 : L 1 ) C and λ 2 ∈ (L 3 : L 2 ) C .We now prove the following generalization of (66).26