Products of CM elliptic curves - Universität Duisburg-Essen

integral quadratic formq L,α,β (x, y) = q L (xα + yβ), for x, y ∈ Z,and hence q L defines an equivalence class of positive binary quadratic forms. Moreover,we have by [3], §II.7 that(78)∆(q L ) = ∆(R(L)) and cont(q L ) = 1.We now prove:Proposition 37 Let E/K be a CM elliptic curve and E 1 , E 2 ∈ Isog + (E). If f i = f Ei ,then(79)q E1 ,E 2∼ cq L , where L = I E (E 1 )I E (E 2 ) −1 and c = lcm(f 1, f 2 )gcd(f 1 , f 2 ) .In particular, c = cont(q E1 ,E 2) and(80)∆(q E1 ,E 2) = −lcm(|∆ E1 |, |∆ E2 |) = lcm(f 1 , f 2 ) 2 ∆ F , where F = End 0 (E).Proof. Let π i : E → E i be an isogeny and put H i = Ker(π i ) and n i = deg(π i ).Moreover, put Φ = Φ H1 ,H 2and H = H(H 1 , H 2 ). We first show that(81)q E1 ,E 2(h) = cq H (Φ(h)), for all h ∈ Hom(E 1 , E 2 );here we used the identification E i = E Hi .Since the H i ’s are ideal subgroups by (46), we have by (57) that N(L) = c n 1n 2.Moreover, since we have by (54) that deg(f) = N(f), for all f ∈ R = End(E),we see from the definition of Φ that N(Φ(h)) = N( 1 n 2π 2 hπ 1 ) = 1 deg(πn 2 2 hπ 1 ) =21deg(πn 2 2 ) deg(h) deg(π 1 ) = n 12n 2deg(h), and soq E1 ,E 2(h) = deg(h) = n 2n 1N(Φ(h)) = c N(Φ(h))N(H) , for h ∈ Hom(E 1, E 2 ).This proves (81) and hence also (79) because by (47) and (41) we have that H =(I(H 1 ) : I(H 2 )) = f 1I(H f 1)I(H 2 ) −1 = f 1L, and so q f H ∼ q L .Since q L is primitive by (78), we have from (79) that cont(q E1 ,E 2) = c · cont(q L ) =c. Moreover, by (78) we have that ∆(q L ) = ∆(R(L)). Now since R(I(H i )) =E(H i ) ≃ End(E i ) has conductor f i , it follows from (39) and (37) that R(L) =R(I(H 1 ))R(I(H 2 )) has conductor f = (f 1 , f 2 ), and so ∆(q L ) = f 2 ∆ F . Thus wesee that ∆(q E1 ,E 2) = c 2 ∆(q L ) = (cf) 2 ∆(f) = lcm(f 1 , f 2 ) 2 ∆ F , and so (80) follows.31