Products of CM elliptic curves - Universität Duisburg-Essen

Proposition 30 Let L ∈ Lat F , where F is an imaginary quadratic field, and letH i = ρ L (L Hi /L) be two finite subgroups of E L . Then we have(73)H(H 1 , H 2 ) = Ψ 0 L((L H2 : L H1 )),where Ψ 0 L : F ∼ → End 0 (E L ) is the canonical extension of Ψ L to F .Proof. First note that if H = ρ L (L H /L), then we have(74)π H = π L,L H1 and π ′ H = π L H,Ln H, where n H = |H|.Indeed, since π L,L H1 : E L → E LH is surjective and has kernel H = ρ L (L H /L), we seethat (E LH , π L,L H1 ) ≃ ((E L ) H , π H ) is a quotient of E L by H. Thus, the first equalityof (74) holds. Moreover, since n H = [L H : L] by (65), we have n H L H ⊂ L and son H ∈ (L : L H ). Thus, π L H,Ln Hexists, and by (72) we have π L H,Ln H◦ π H = πn L,LH= [n H ] EL ,and so π H ′ = πL H,Ln H; cf. (4).We next observe that(75)Φ H1 ,H 2(Ψ LH1 ,L H2(λ)) = Ψ 0 L(λ), for all λ ∈ (L H2 : L H1 ) C .Indeed, from the definitions and (74) and (72) we obtainΦ H1 ,H 2(Ψ LH1 ,L H2(λ)) = 1n Hπ ′ Hπ L H 1 ,L H2λπ H = 1n Hπ L,Ln H λ = 1n HΨ L (n H λ) = Ψ 0 L(λ),which proves (75). From this, equation (73) follows immediately because Φ H1 ,H 2◦Ψ LH1 ,L H2is an isomorphism from (L H2 : L H2 ) to H(H 1 , H 2 ); cf. (71) and Remark 9(a).Remark 31 (a) It follows from (73), (41) and (65) that(ψ 0 L) −1 (H(0, H)) = (L H : L) = [RE(H) : E(H)]L H L −1 ,where R = End(E L ) ≃ R(L). Thus, we can partially recover the lattice L H (whichdefines the analytic curve C/L H ≃ (E L ) H ) from the “algebraic lattice” H(0, H).(b) If E/K is any CM curve over K ⊂ C, then it follows from (73) that for anytwo finite subgroup schemes H 1 , H 2 of E we have that(76)(I(H 1 ) : I(H 2 )) = [RR 1R 2 : RR 2 ]RH(H 1 , H 2 ),[R 1 R 2 : R 2 ]where R = End(E) and R i = E(H i ). Indeed, as in the proof of Corollary we canassume that K = C, E = E L and H i = ρ L (L i /L). Then in view of (73), (67) and(65), the formula (76) follows from the following formula (which is easily deduced from(41)):(77)((L : L 2 ) : (L : L 1 )) = [RR 1 R 2 : RR 2 ]RL 1 L −12 = [RR 1R 2 : RR 2 ](L 1 : L 2 )R,[R 1 R 2 : R 2 ]in which R = R(L) and R i = R(L i ).27