Products of CM elliptic curves - Universität Duisburg-Essen

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Theorem 43 Let H 1 , . . . , H n and H ′ 1, . . . , H ′ n be ideal subgroups of A. ThenA H1 × . . . × A Hn ≃ A H ′1× . . . × A H ′ n⇔ I(H 1 ) ⊕ . . . ⊕ I(H n ) ≃ I(H ′ 1) ⊕ . . . ⊕ I(H ′ n).Proof. Since H := H 1 × . . . × H n and H ′ := H ′ 1 × . . . × H ′ n are ideal subgroups of A nby Proposition 42, we have by (21) that A n H ≃ An H ′ ⇔ I(H) ≃ I(H′ ). Now by (87) wehave I(H) = (I 1 | . . . |I n ) and I(H ′ ) = (I ′ 1| . . . |I ′ n), where I i = I(H i ) and I ′ i = I(H ′ i).Thus, since T := T A,n is an isomorphism, and since T (I(H)) = T ((I 1 | . . . |I n )) =I n (I 1 ⊕ . . . ⊕ I n ) and T (I(H ′ )) = I n (I ′ 1 ⊕ . . . ⊕ I ′ n), we see that I(H) ≃ I(H ′ ) (asEnd(A n )-modules) ⇔ I n (I 1 ⊕ . . . ⊕ I n ) ≃ I n (I ′ 1 ⊕ . . . ⊕ I ′ n) (as M n (End(A))-modules)⇔ I 1 ⊕ . . . ⊕ I n ≃ I ′ 1 ⊕ . . . ⊕ I ′ n (as End(A)-modules), the latter by Proposition 40.This proves the assertion.Note that Theorem 1 follows easily from this and the results of section 3, as weshall now see.Proof of Theorem 1. Let π i : E → E i and π ′ i : E → E ′ i be isogenies. Since f Ei |f E andf E ′i|f E , we know by (46) that H i = Ker(π i ) and H ′ i = Ker(π i ) are ideal subgroups,and so the assertion follows from Theorem 43 because E i ≃ E Hi , E ′ i ≃ E H ′iandI E (E i ) ≃ I(H i ) and I E (E ′ i) ≃ I(H ′ i).4.2 The theorems of Steinitz and of Borevich and FaddeevIn order to derive further properties about abelian varieties which are isogenous to aproduct A n , we shall use the results due to Steinitz[25] and to Borevich and Faddev[1]about the R-module structure of the submodules of R n .Theorem 44 (Steinitz) If R be a Dedekind domain, then every submodule of R nis isomorphic to a direct sum of R-ideals. Moreover, if I 1 , . . . , I n and J 1 , . . . J m areR-ideals, thenI 1 ⊕ . . . ⊕ I n ≃ J 1 ⊕ . . . ⊕ J m ⇔ m = n and I 1 · · · I n ≃ J 1 · · · J m .Proof. See [8], p. 85.This theorem does not generalize to arbitrary orders in a number field F , for alreadythe first assertion of theorem may be false. As Borevich and Faddeev[2] observed, oneneeds the extra condition that the order R be cyclic in the sense that O F /R is acyclic R-module. In their papers, they prove the following generalization of Steinitz’stheorem.Theorem 45 (Borevich/Faddeev) Let R be an order in a Dedekind domain O.Then:36
(a) The order R is cyclic if and only if for all n ≥ 1 we have that every R-submoduleof R n is isomorphic to a direct sum of R-ideals.(b) Let R be a cyclic order, and let M be an R-submodule of R n of rank n. Thenthere exist R-ideals I 1 , . . . , I n such that(89)M ≃ I 1 ⊕ . . . ⊕ I n and R(I 1 ) ⊂ R(I 2 ) ⊂ . . . R(I n ),and the orders R(I 1 ) ⊂ . . . ⊂ R(I n ) and the ideal class of the product I 1 · · · I n areuniquely determined by the isomorphism class of M. More precisely, if J 1 , . . . , J m areR-ideals with R(J 1 ) ⊂ . . . ⊂ R(J m ), then we have(90)I 1 ⊕ . . . ⊕ I n ≃ J 1 ⊕ . . . ⊕ J m ⇔ n = m and R(I k ) = R(J k ), for 1 ≤ k ≤ n,and I 1 · · · I n ≃ J 1 · · · J m .Proof. (a) This is the main theorem of Borevich/Faddeev[2].(b) The existence of the I 1 , . . . , I n satisfying (89) is proven in [1], Theorem 3.The uniqueness of the orders R(I 1 ), . . . , R(I n ) and of the product I 1 · · · I n is proven inTheorems 5 and 6 of [1], and the assertion (90) is the content of [1], Theorem 7.Remark 46 (a) Since a Dedekind domain is trivially a cyclic order (in itself), it clearthat Theorem 45 generalizes Theorem 44.(b) Every order R ∆ = Z + Zω ∆ in a quadratic field F = Q( √ ∆) is cyclic becauseO F = Z + Zω F = R ∆ + R ∆ ω F . Thus, Theorem 45(b) applies to all orders in quadraticfields.(c) It follows from the above Theorem 45 (cf. [1], Theorem 8) that the rule(R 1 , . . . , R n ; I) ↦→ R 1 ⊕ . . . ⊕ R n−1 ⊕ Iinduces a bijection between the following sets:(i) the set of lists (R 1 , . . . , R n ; I) where R ⊂ R 1 ⊂ . . . ⊂ R n ⊂ O Fcontaining R and I ∈ Pic(R n ) is a class of invertible R n -ideals;are orders(ii) the set of isomorphism classes of finitely generated torsion-free R-modules ofrank n.Corollary 47 Let V be an n-dimensional F -vector space, where F ⊃ Q is a quadraticfield, and let L be a lattice in V , i.e. L ⊂ V is a finitely generated subgroup whichcontains a basis of V . Then R F (L) := (L : L) F = {x ∈ F : xL ⊂ L} is an orderof F , and L is an R F (L)-module. Furthermore, there exists a sequence of ordersR 1 = R F (L) ⊂ R 2 ⊂ . . . ⊂ R n , an invertible R n -ideal I and a basis x 1 , . . . , x n of Vsuch that(91)L = R 1 x 1 + R 2 x 2 + . . . + R n−1 x n−1 + Ix n .37
Page 1: Products of CM elliptic curves1 Int
Page 7 and 8: Indeed, if If is a kernel ideal, th
Page 9 and 10: Indeed, by (18) we have I 1 f = I 2
Page 11 and 12: Proof. To prove (29), let h ∈ Hom
Page 13 and 14: 2.4 The quadratic caseWe now specia
Page 15 and 16: ecause here f 2 |f 1 , so f = f 2 a
Page 17 and 18: For this, put f = [R ′ : R], I =
Page 19 and 20: From (54) we can conclude that(55)d
Page 21 and 22: If H is any finite subgroup scheme
Page 23 and 24: E. As we shall see, it is very illu
Page 25 and 26: From this we see on the one hand th
Page 27 and 28: Proposition 30 Let L ∈ Lat F , wh
Page 29 and 30: (b) Deuring[10], p. 263. Note that
Page 31 and 32: integral quadratic formq L,α,β (x
Page 33 and 34: Corollary 39 Let E 1 /K and E 2 /K
Page 35: Proof. Write π i = π Hi : A → A
Page 39 and 40: for k = 1, . . . , n − 2. Moreove
Page 41 and 42: 4.3 Products of CM elliptic curvesW
Page 43 and 44: Corollary 56 Let E/K be a CM ellipt
Page 45 and 46: Case 2. char(K) = 0.As was mentione
Page 47 and 48: Let h A/K (X) ∈ Z[X] (respectivel
Page 49 and 50: Proof. To prove this, we shall use
Page 51 and 52: where xy denotes the quadratic form
Page 53 and 54: denote the set of isomorphism class

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