Products of CM elliptic curves - Universität Duisburg-Essen

Theorem 58 Let E/K be a CM elliptic curve, where K is an algebraically closedfield. If A ∼ E n is an abelian variety which is isogenous to E n , then there existelliptic curves E 1 , . . . , E n ∼ E such that A ≃ E 1 × . . . × E n .Remark 59 In the case that K = C, Theorem 58 was first proven by Lange[19], usingthe results of Shioda and Mitani[24] (who proved the case n = 2). A different proof ofthis was given by Schoen; cf. [22], Satz 2.4. His proof is based on the above Theorem45(b) of Borevich and Faddeev[1] and hence is closely related to the one given below(in the case K = C).In his paper, Schoen[22] also proves the following interesting “converse” to Theorem58 in the case K = C. Suppose that A/C is an abelian variety with the property thatif B ∼ A, then B ≃ A 1 × . . . × A n , for some simple abelian varieties A i . Then eitherA is simple or A ∼ E n , where E/C is a CM elliptic curve.Before proving Theorem 58, let us see how Theorem 54 follows from it.Proof of Theorem 54 (using Theorem 58). First note that the second equivalenceof (97) is trivial because we have End(E) = Z(End(E n )) ⊂ E(H) ⇔ End(E) ⊂Z(E(H)) ⇔ f E(H) |f E .Next we note that the one direction (⇒) of the first equivalence of (97) is true byCorollary 11. It thus remains to prove the other direction.Thus, suppose that Z(End(E n )) ⊂ E(H). Since E is a CM elliptic curve, we havethat End(E n ) = End(E n ⊗ K), and hence also that End(A) = End(A ⊗ K). (Tosee this, note that End 0 (E n ) = End 0 (A) (because A ∼ E n ), and that End 0 (E n ) =End 0 (E n ⊗ K), and so the assertion follows because End(A ⊗ K)/End(A) is alwaystorsionfree.) Thus, we have that E(H ⊗ K) ≃ End(A ⊗ K) = End(A) ≃ E(H), and soZ(End(E n ⊗ K)) ⊂ Z(E(H ⊗ K)). Thus f AH⊗K = f E(H⊗K) |f E⊗K = f E .By Theorem 58 we have that A H⊗K ≃ E 1 ×. . .×E n , for some elliptic curves E i /K.By Corollary 53 we have f Ei |f AH⊗K |f E , and so E i ∈ Isog + (E ⊗K/K). Thus, condition(101) holds for H ⊗ K, and so H ⊗ K is an ideal subgroup of (E ⊗ K) n = E n ⊗ K byProposition 57. It thus follows from Remark 7(d) that H is an ideal subgroup of E n .Proof of Theorem 58. We shall divide the proof into several cases.Case 1. K = C.Here E ≃ E L , where L ∈ Lat F is a lattice in F = End 0 (E); cf. §3.2. Thus E n ≃ C n /L n ,where L n = L ⊕ . . . ⊕ L ⊂ F n . Let π : E n → A be an isogeny. Then Ker(π) = L ′ /L n ,for some subgroup L ′ ⊂ C n . Since [L ′ : L n ] = deg(π) < ∞, it follows that L ′ is a latticein F n (in particular, L ′ ⊂ F n ) and hence by Corollary 47 we know that L ′ has the form(91). Put L ′′ = L ′ 1 ⊕ . . . ⊕ L ′ n ⊂ F n , where L ′ i = R i for 1 ≤ i ≤ n − 1 and L ′ n = I andthe R i ’s and I are given by (91). Thus L ′ = L ′′ g, for some g ∈ Aut F (F n ) ⊂ Aut C (C n ),and so A ≃ C n /L ′ ≃ C n /L ′′ ≃ C/L ′ 1 ×. . .×C/L ′ n, and so the assertion follows becauseL ′ i ∈ Lat F (and hence C/L ′ i ∼ E L ).44