Products of CM elliptic curves - Universität Duisburg-Essen

we have q A ∼ xy ⊥ (−q A ). Thus, q A ∼ xy ⊥ (−q) ⇔ xy ⊥ (−q A ) ∼ xy ⊥ (−q) ⇔ q Aand q are in the same genus, the latter by Remark 27 of [17]. Since the number ofsuch forms q A equals the number of forms in the principal genus (and is given by theformula of (105)), the assertion follows.We also note that the above Proposition 67 allows us to reformulate Theorem 3 inthe case of surfaces as follows.Theorem 69 Let A/K be an abelian suface which is isogenous to E × E, whereE/K is a CM elliptic curve. Then there exist CM elliptic curves E 1 , E 2 /K such thatA ≃ E 1 × E 2 if and only if ∆(A/K)|∆ E0 , for some elliptic curve E 0 /K with E 0 ∼ E.Proof. If A ≃ E 1 × E 2 , then by Theorem 3 we know that f A |f E0 , for some E 0 ∼ E.Thus, by (104) we have that ∆(A/K) = −f 2 A ∆ F |f 2 E 0∆ F = ∆ E0 , as claimed.Conversely, suppose that ∆(A/K)|∆ E0 , for some E 0 ∼ E. Then by Proposition67 we have that ∆(A/K) = −f 2 A ∆ F , and so it follows that f A |f E0 . By Theorem 3 wethus obtain that A ≃ E 1 × E 2 , for some elliptic curves E 1 , E 2 /K.From the above Theorem 4 (or from Theorem 65) we can deduce the followingresult which is essentially the same as Theorem 3.1 of [24]:Theorem 70 Let K be an algebraically closed field of characteristic 0. Then there isa bijection between the following sets:(i) the set Q/SL 2 (Z) of proper equivalence classes of positive definite binary quadraticforms;(ii) the set of isomorphism classes of abelian surfaces A/K with Picard numberρ(A) := rank(NS(A)) = 4.Remark 71 In the paper [24], the abelian surfaces A/C with ρ(A) = 4 are calledsingular abelian surfaces; cf. [24], p. 459. This terminology unfortunately conflictswith classical terminology of singular abelian varieties used in the 19th century: theseare abelian varieties with the property that End(A) ≠ Z; cf. Hurwitz[15], p. 167, 187(and the references therein) and Humbert [14].Proof of Theorem 70. It follows from the classification theory of endomorphisms ofsimple abelian surfaces in characteristic 0 (cf. [20], p. 202) that if ρ(A) = 4, then Acannot be simple and so one sees easily that A ∼ E 2 , where E/K is a CM ellipticcurve. Thus the set (ii) is the same as the set(ii ′ ) the set of isomorphism classes of abelian surfaces A/K with A ∼ E 2 , for someCM elliptic curve E/K.To describe the bijection, fix for each discriminant ∆ < 0 an elliptic curve E ∆ with∆ E∆ = ∆ (which exists by Proposition 33(a)). LetA(∆) := {A/K : A ∼ E 2 ∆ and ∆(A/K) = −∆}/≃52