Products of CM elliptic curves

Ernst Kani

1 Introduction

Let K be an arbitrary field, and let E_{1}, …, E_{n} be isogenous CM elliptic curves over K. The basic problem considered in this paper is to find suitable criteria for determining whether or not a given abelian variety A′ is isomorphic to the product variety A = E_{1} × … × E_{n}. A special case of this problem is to determine this in the case that A′ = E′_{1} × … × E′_{n} is also a product variety.

In the case that K = ℂ and n = 2, Shioda and Mitani[24] presented a solution of this subproblem in terms of the period lattices L_{i} and L′_{i} of the elliptic curves E_{i} = ℂ/L_{i} and E′_{i} = ℂ/L′_{i}, and a similar criterion is implicit in the work of Schoen[22] for arbitrary n.

In this paper we present a criterion that is partially similar to the complex-analytic approach but has the advantage that it works over an arbitrary ground field. Here the role of the (isomorphism class) of the lattice L_{i} is replaced by the ideal class

I_{E}(E_{i}) := Hom(E_{i}, E)π_{i},

where E is a fixed suitable elliptic curve which is isogenous to E_{i} and π_{i} : E → E_{i} is any isogeny. It is immediate that I_{E}(E_{i}) is an End(E)-ideal whose ideal class does not depend on the choice of the isogeny π_{i}. Here, “suitable” means that the conductor f_{E} = f_{End(E)} of E is a multiple of those of E_{i} and E′_{i} for all i. We then have:

Theorem 1 Let E/K be a CM-elliptic curve and let E_{1}, …, E_{n}, E′_{1}, …, E′_{n} be elliptic curves isogenous to E with f_{E_{i}}|f_{E} and f_{E′_{i}}|f_{E}, for 1 ≤ i ≤ n. Then

E_{1} × … × E_{n} ≃ E′_{1} × … × E′_{n} ⇔ I_{E}(E_{1}) ⊕ … ⊕ I_{E}(E_{n}) ≃ I_{E}(E′_{1}) ⊕ … ⊕ I_{E}(E′_{n})

as End(E)-modules.

Note that this result is actually a special case of a very general result about isomorphisms of product abelian varieties; cf. Theorem 43.

At first sight the criterion of Theorem 1 does not seem to specialize to that of [24], Proposition 4.5. However, by using results due to Steinitz[25] and Borevich/Fadeev[1] on the struture of R-modules when R is a quadratic order, one can easily deduce their result from that of Theorem 1; cf. subsection 4.4.

It turns out that general isomorphism problem can be reduced to the previously considered subproblem in many cases, for we have the following result.

Theorem 2 Let E/K be a CM elliptic curve, where K is either algebraically closed or finite. If A/K is an abelian variety which is isogenous to E^{n}, for some n ≥ 1, then there exist CM elliptic curves E_{1}, …, E_{n}/K such that A ≃ E_{1} × … × E_{n}.

In the case that K = ℂ, this theorem was proved by Shioda and Mitani[24] when n = 2 (see also Ruppert[21]), and their work was extended to the general case by Lange[19]. Moreover, Schoen[22] gave a beautiful analysis of this theorem; cf. Remark 59 below.

Over a general ground field K, the analogue of Theorem 2 seems to be more subtle. The following generalization of Theorem 2 uses the concept of the central conductor f_{A} = f_{End(A)} which is the conductor of the centre Z(End(A)) as an order in F = Z(End^{0}(A)); cf. §4.3 below.

Theorem 3 Let E/K be a CM elliptic curve, where K is an arbitrary field, and let A/K be an abelian variety which is isogenous to E^{n}, for some n ≥ 1. Then there exist CM elliptic curves E_{1}, …, E_{n}/K such that A ≃ E_{1} × … × E_{n} if and only if f_{A}|f_{E_{0}}, for some elliptic curve E_{0}/K with E_{0} ∼ E.

Note that if we combine Theorem 3 with Theorem 1, then we get an indirect solution of the isomorphism problem mentioned at the beginning. A more intrinsic solution (similar to Theorem 1) is given below; cf. Corollary 56.

In the case that n = 2 and K = ℂ, Shioda and Mitani[24] also showed that one can classify the abelian surfaces A with A ∼ E^{2} by equivalence classes of binary quadratic forms. While their theorem does not directly carry over to an arbitrary ground field, the following refinement is true in general. To state it, it is useful to employ the following terminology. If E/K is a CM elliptic curve, then its discriminant is the discriminant Δ_{E} = Δ(End(E)) of the order End(E); thus Δ_{E} = f_{E}^{2}Δ_{F}, where Δ_{F} is the discriminant of the imaginary quadratic field F = End^{0}(E). Moreover, if A/K is an abelian surface, then its discriminant Δ(A/K) is the discriminant of its Néron-Severi group NS(A) (with respect to the intersection pairing). We then have the following result:

Theorem 4 Let E/K be a CM elliptic curve of discriminant Δ = Δ_{E}. Then there is a bijection between:

(i) the set of proper equivalence classes of positive definite binary quadratic forms q with discriminant Δ(q) = Δ;

(ii) the set of isomorphism classes of abelian surfaces A/K with A ∼ E^{2} and discriminant Δ(A/K) = −Δ.

Note that in the above bijection, the binary quadratic forms q need not be primitive, i.e. their content cont(q) need not be equal to 1.

The above Theorem 4 will be deduced in §4.4 from the following generalization to abelian varieties of arbitrary dimension n.

Theorem 5 Let E/K be a CM elliptic curve with discriminant Δ_{E} = f_{E}^{2}Δ_{F}. If n ≥ 2, then there are natural bijections between the following sets:

(i) The set of sequences (E′; f_{1}, …, f_{n−2}) where E′ ∼ E is an isomorphism class of elliptic curves with f_{E′}|f_{E} and the f_{i}’s are positive integers with f_{E′}|f_{1}| … |f_{n−2}|f_{E}.

(ii) the set of sequences (I; f_{1}, …, f_{n−2}) where I is an isomorphism class of nonzero End(E)-ideals whose associated order R(I) has conductor f_{R(I)}|f_{1}| … |f_{n−2}|f_{E}.

(iii) the set of sequences (q; c_{1}, …, c_{n−2}) where q is a proper equivalence class of positive binary quadratic forms of discriminant Δ and c_{1}| … |c_{n−2}|cont(q).

(iv) the set of isomorphism classes of End(E)-submodules M of End(E)^{n} of rank n with (M : M)_{F} := {f ∈ F : fM ⊂ M} = End(E);

(v) the set of isomorphism classes of abelian varieties A ∼ E^{n} with central conductor f_{A} = f_{E}.

Note that the above theorem can easily be extended to classify the isomorphism classes of abelian varieties A ∼ E^{n} with f_{A}|f_{E}; cf. Remark 62(b) below.

The basic technique for proving these theorems is the method of Deuring[10], Shimura and Taniyama[23], and Waterhouse[26] of constructing isogenies: for a given left ideal I of End(A), this method defines a finite subgroup scheme H(I) of A and hence an isogeny π_{I} : A → A/H(I). This theory, together with some extensions, is presented in some detail in §2.

In order to be able to apply this theory, it is useful to know which finite subgroup schemes H of A are of the form H = H(I) for some ideal I of End(A); such subgroup schemes are called ideal subgroups in this paper. A key result here is the following theorem (see also Theorem 54 below) which classifies the ideal subgroups of A = E^{n}.

Theorem 6 Let E/K is a CM elliptic curve, and let H be a finite subgroup scheme of E^{n}. Then H = H(I) for some left ideal I of End(E^{n}) if and only if the central conductor f_{E^{n}/H} of the quotient E^{n}/H divides the conductor f_{E} of E.

This paper is organized as follows. In §2 we review and extend the theory of Deuring, Shimura/Taniyama and Waterhouse. This is then worked out in detail in §3 for the case of a CM elliptic curve. Here Theorem 18, Corollary 19 and Proposition 36 are basic tools for the rest of paper. In §4 we study products of abelian varieties: the general case is analyzed in §4.1 and then applied to products of CM elliptic curves in §4.3. To this end we also review and extend the results of Steinitz and of Borevich/Faddeev[1] in §4.2. Finally, in §4.4 we consider the case of abelian surfaces and show how the present results are related to those of Shioda and Mitani[24].

This research was partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC), and also by the Graduiertenkolleg of the Institute of Experimental Mathematics (IEM) of the University of Duisburg/Essen. I would like to express my appreciation to Gerd Frey and to the IEM for their hospitality, and to thank him for helpful comments on this paper.

2 Isogenies, subgroups and ideals

2.1 Kernel ideals and ideal subgroups

In this section we review and augment the method of constructing isogenies of via ideals; cf. Waterhouse[26], §3.2. This method is due to Deuring[10] in the case of elliptic curves, and was generalized to abelian varieties by Shimura and Taniyama[23], §7.

Throughout this paper, K is an arbitrary field, and A/K is an abelian variety. All morphisms are tacitly K-morphisms, and all subvarieties are K-subvarieties.

We first review some basic facts about finite subgroup schemes. If H is finite subgroup scheme of A, then the quotient variety A_{H} = A/H with quotient morphism

π_{H} : A → A_{H} := A/H

exists by [20], p. 111, and is again an abelian variety. Thus, (A_{H}, π_{H}) is characterized by the universal property that for any H-invariant morphism f : A → X there is a unique morphism f_{H} : A_{H} → X such that f = f_{H} ∘ π_{H}. Note that π_{H} is an isogeny of degree deg(π_{H}) = |H|, where |H| denotes the rank of the finite subgroup scheme H.

Conversely, if π : A → A′ is an isogeny of abelian varieties, then Ker(π) is a finite subgroup scheme of rank |Ker(π)| = deg(π). Since π is faithfully flat (use [11], Ex.III.9.3(a) or [4], 7.3/1), it follows that π : A → A′ is a quotient of A, i.e. there is an isomorphism φ : A′ →^{∼} A_{Ker(π)} such that π_{Ker(π)} = φ ∘ π. Thus, we have a bijection between finite subgroup schemes H of A and isogenies π : A → A′, modulo isomorphisms.

If H_{1} and H_{2} are any two (not necessarily finite) subgroup schemes of A, then we write H_{1} ≤ H_{2} if the canonical inclusion morphism j_{H_{1}} : H_{1} ↪ A of H_{1} factors over that of H_{2}, i.e. if j_{H_{1}} = j_{H_{2}} ∘ h, for some h : H_{1} → H_{2} (necessarily an immersion). Since a homomorphism h : A → A′ of abelian varieties is H_{1}-invariant if and only if Ker(h_{1}) ≤ Ker(h), it follows from the universal property of quotients that if H_{1} and H_{2} are finite, then the condition H_{1} ≤ H_{2} is equivalent to the existence of a morphism (necessarily an isogeny) π_{H_{1},H_{2}} := (π_{H_{2}})_{H_{1}} : A_{H_{1}} → A_{H_{2}} such that π_{H_{2}} = π_{H_{1},H_{2}} ∘ π_{H_{1}}. In particular, we see that |H_{1}| = deg(π_{H_{1}}) | deg(π_{H_{2}}) = |H_{2}| in this case. From this it follows that the relation ≤ is a partial order on the set of finite subgroup schemes of A; in particular, H_{1} ≤ H_{2} and H_{2} ≤ H_{1} ⇒ H_{1} = H_{2} (because π_{H_{1},H_{2}} is an isomorphism if |H_{1}| = |H_{2}|).

For any finite collection H_{1}, …, H_{n} of subgroup schemes of A, their intersection H_{1} ∩ … ∩ H_{n} := H_{1} ×_{A} … ×_{A} H_{n} is a subgroup scheme of A, which is the greatest lower bound of H_{1}, …, H_{n} with respect to the partial order ≤. Note that ∩H_{i} is finite if at least one of the H_{i} is finite. We can thus extend the definition of to an infinite collection {H_{i}}_{i∈I} of subgroup schemes provided that at least one of the H_{i} is finite, because then the definition reduces to a finite subcollection (because the finite subgroup schemes satisfy d.c.c.).

We now present the Deuring/Shimura/Taniyama/Waterhouse method of constructing finite subgroup schemes via left ideals. However, instead of fixing an identification of End(A) with an abstract ring R (as in [23] and [26]), we shall work directly with

R = R_{A} := End(A) and R̃ = R̃_{A} := End^{0}(A) = End(A) ⊗ ℚ

because A is usually fixed. Note that R is canonically embedded in R̃. Later in §2.3 we shall study what happens when we replace A by an isogenous variety A′.

Let I be a regular left ideal I of R = End(A), i.e. I is a left R-ideal which contains an isogeny. Then as in [26], §3.2, we put

H(I) = ⋂_{f∈I} Ker(f)

which is finite subgroup scheme of A. Note that I is finitely generated and that

(1) H(I) = Ker(h_{1}) ∩ … ∩ Ker(h_{r}), if I = ∑_{i=1}^{r} Rh_{i},

because for all f, g ∈ R we have

(2) Ker(g) ≤ g^{−1}(Ker(f)) = Ker(fg) and Ker(f) ∩ Ker(g) ≤ Ker(f + g).

From (1) it follows immediately that for any two regular left R-ideals I_{1}, I_{2} we have

H(I_{1} + I_{2}) = H(I_{1}) ∩ H(I_{2}) (3)

because I_{1} + I_{2} is generated by I_{1} ∪ I_{2}.

Here we complement this construction by the following “dual construction”. Given a finite subgroup scheme H of A, put

I(H) = Hom(A_{H}, A)π_{H} = {f ∈ R : H ≤ Ker(f)},

where the second equality follows from the universal property of (A_{H}, π_{H}). It is immediate that I(H) is a left ideal of R. Moreover, I(H) is a regular ideal because H ≤ Ker([n_{H}]_{A}), where n_{H} = |H| (and [n]_{A} = n · 1_{A} denotes the multiplication-by-n map), and so [n_{H}]_{A} ∈ I(H). For later reference we observe that it thus follows from the above equality that there exists a unique π′_{H} ∈ Hom(A_{H}, A) such that

(4) π′_{H} ∘ π_{H} = [n_{H}]_{A} and that hence also π_{H} ∘ π′_{H} = [n_{H}]_{A_{H}}.

If I_{1}, I_{2} and I are regular left R-ideals and H_{1}, H_{2} and H are finite subgroup schemes of A, then we have

(5)  I_{1} ⊂ I_{2} ⇒ H(I_{2}) ≤ H(I_{1}),
(6)  H_{1} ≤ H_{2} ⇒ I(H_{2}) ⊂ I(H_{1}),
(7)  I ⊂ I(H(I)),
(8)  H ≤ H(I(H)).

Indeed, (5) and (6) are clear from the definitions. Moreover, if f ∈ I, then Ker(f) ≥ H(I) = ∩_{g∈I}Ker(g), so f ∈ I(H(I)) and hence I ⊂ I(H(I)), which proves (7). Similarly, since H ≤ Ker(f), for all f ∈ I(H), we have H ≤ ∩_{f∈I(H)}Ker(f) = H(I(H)), which yields (8).

From the above properties we see immediately that

H(I) = H(I(H(I))) and I(H) = I(H(I(H))). (9)

Indeed, by (8) (applied to H = H(I)) we have H(I) ≤ H(I(H(I)). On the other hand, since I ⊂ I(H(I)) by (7), it follows from (5) that H(I) ≥ H(I(H(I)), and so the first equation of (9) follows. The second is proved similarly.

Definition. A regular left R-ideal I is called a kernel ideal if we have that I = I(H(I)). A finite subgroup scheme H of A is called an ideal subgroup (scheme) if we have H = H(I(H)).

Remark 7 (a) Waterhouse[26], p. 533, calls an ideal I a kernel ideal if we have I = {f ∈ R : fH(I) = 0}. Since fH(I) = 0 ⇔ H(I) ≤ Ker(f), it is clear that {f ∈ R : fH(I) = 0} = I(H(I)), and so his definition agrees with the one above. (He does not define “ideal subgroups”.)

(b) It follows from the definition and (9) that I is a kernel ideal if and only if I = I(H), for some finite subgroup scheme H of A. Similarly, we see that H is an ideal subgroup if and only if H = H(I) for some regular left R-ideal I.

(c) If f ∈ R is an isogeny, then it clear that

(10) H(Rf) = Ker(f) and I(Ker(f)) = Rf

(the latter by the universal property), and so Rf is a kernel ideal and Ker(f) is an ideal subgroup. More generally, we have for any regular left R-ideal I, any finite subgroup scheme H of A and isogeny f ∈ R that

H(If) = f^{−1}(H(I)) and I(f^{−1}(H)) = I(H)f. (11)

Indeed, the first equality follows from the fact that intersections commute with inverse images, and the second follows because the faithful flatness of f implies that

(12) H ≤ Ker(g) ⇔ f^{−1}(H) ≤ f^{−1}(Ker(g)) = Ker(gf), ∀g ∈ R.

From (11) it thus follows that

I(H(If)) = I(H(I))f and H(I(f^{−1}(H))) = f^{−1}(H(I(H))), (13)

and so we see that if I is a kernel ideal, then so is If. Similarly, if H is an ideal subgroup, then so is f^{−1}(H). In fact, the converses of these assertions are also true:

(14)  I is a kernel ideal ⇔ If is a kernel ideal;
(15)  H is an ideal subgroup ⇔ f^{−1}(H) is an ideal subgroup.

Indeed, if If is a kernel ideal, then by the first part of (13) we have I(H(I))f = I(H(If)) = If, and so I(H(I)) = I because f is a unit in R̃ ⊃ R. Thus I is a kernel ideal, which proves (14). Similarly, if f^{−1}(H) is an ideal subgroup, then by the second part of (13) we have f^{−1}(H(I(H))) = H(I(f^{−1}(H))) = f^{−1}(H), and so H(I(H)) = H by the faithful flatness of f. Thus H is an ideal subgroup, which proves (15).

(d) For any regular left R-ideal we have

I(H(I)) ⊂ I^{∗} := ⋂_{Rf̃⊃I} Rf̃, (16)

where the intersection runs over all f̃ ∈ R̃ such that Rf̃ ⊃ I. Indeed, if f̃ = f/n with f ∈ R and n ∈ ℕ, then I ⊂ Rf̃ implies that In ⊂ Rf and so by (11), (5), (6) and (10) we have that I(H(I))n = I(H(In)) ⊂ I(H(Rf)) = Rf, and hence I(H(I)) ⊂ Rf̃. This verifies (16).

In particular, if I = I^{∗}, i.e. if I is a divisorial ideal (cf. [5], p. 476), then it follows from (8) and (16) that I = I(H(I)). Thus every divisorial ideal is a kernel ideal. In particular, if R is commutative, then every invertible R-ideal I is a kernel ideal by [5], p. 118, 476. Thus, if R is a Dedekind domain, then all non-zero ideals are kernel ideals.

(e) If K′/K is any field extension, then we have an injective ring homomorphism

β_{K′/K} : End(A) → End(A ⊗ K′)

given by base-change f ↦ f ⊗ K′. If this is surjective (hence an isomorphism), then it is clear from the definitions that for a finite subgroup scheme H of A and R-ideal I we have

H(I) ⊗ K′ = H(β_{K′/K}(I)) and I(H ⊗ K′) = β_{K′/K}(I(H))

because Ker(f ⊗ K′) = Ker(f) ⊗ K′, for all f ∈ End(A). Note, however, that in general A′ := A ⊗ K′ has more finite subgroup schemes than A, i.e. A′ may have finite subgroup schemes which are not of the form H ⊗ K′.

2.2 The invariant I_{A}(B)

Given an abelian variety B which is isogenous to A (notation: B ∼ A), we shall define an “invariant” I_{A}(B) which is an isomorphism class of left R-ideals, where, as before, R = R_{A} = End(A). This invariant is defined by the rule

I_{A}(B) = Hom(B, A)π = I(Ker(π)) where π : A → B is any isogeny.

As we shall see presently, the R-module isomorphism class of the right hand side does not depend on the choice of the isogeny π : A → B. For this, we first observe the following.

As before, let H_{1} and H_{2} be two finite subgroup schemes of A and let I_{1} and I_{2} two regular left R-ideals. It is then easy to see (cf. [26], p. 532) that we have

(17) A_{H_{1}} ≃ A_{H_{2}} ⇔ f^{−1}(H_{1}) = [n]^{−1}(H_{2}), for some f ∈ R ∩ R̃^{×} and n ∈ ℤ,
(18)  I_{1} ≃ I_{2} ⇔ I_{1} = I_{2}f̃, for some f̃ ∈ R̃^{×}.

From this we see that I_{1} ≃ I_{2} ⇒ A_{H(I_{1})} ≃ A_{H(I_{2})} (19)

because the hypothesis implies by (18) that I_{1} = I_{2}f̃, where f̃ = f/n with f ∈ R ∩ R̃ an isogeny, and so I_{1}n = I_{2}f. Thus, by (12) we have [n]^{−1}(H(I_{1})) = H(I_{1}n) = H(I_{2}f) = f^{−1}(H_{2}), and hence A_{H(I_{1})} ≃ A_{H(I_{2})} by (17). This proves (19).

Similarly, we have A_{H_{1}} ≃ A_{H_{2}} ⇒ I(H_{1}) ≃ I(H_{2}). (20)

Indeed, by (17) the hypothesis implies that there exist f, n such that f^{−1}(H_{1}) = [n]^{−1}(H_{2}), and so by (11) we have I(H_{1})f = I(f^{−1}(H_{1})) = I([n]^{−1}(H_{2})) = I(H_{2})n, so I(H_{2}) = I(H_{1})f̃ with f̃ = \frac{f}{n}, and hence I(H_{1}) ≃ I(H_{2}), which proves (20).

While in general the converses of (19) and (20) are false, we note that if I_{1} and I_{2} are kernel ideals, then it follows from (19) and (20) that the converse of (19) holds. Similarly:

(21) A_{H_{1}} ≃ A_{H_{2}} ⇔ I(H_{1}) ≃ I(H_{2}), if H_{1} and H_{2} are ideal subgroups.

We now apply this to the invariant I_{A}(B). It follows from (20) that the isomorphism class of I_{A}(B) does not depend on the choice of the isogeny π : A → B, for if π_{1} : A → B is another, then A_{Ker(π)} ≃ B ≃ A_{Ker(π_{1})}, and so by (20) we have I(Ker(π)) ≃ I(Ker(π_{1})), as asserted.

As was mentioned above, it can happen that I_{A}(B_{1}) ≃ I_{A}(B_{1}) yet B_{1} ≄ B_{2}. However, if B_{1} and B_{2} have the “ideal property” that there exist isogenies π_{i} : A → B_{i} such that Ker(π_{i}) is an ideal subgroup, then we have by the above discussion that

I_{A}(B_{1}) ≃ I_{A}(B_{2}) ⇔ B_{1} ≃ B_{2}, provided that B_{1}, B_{2} have the ideal property.

Remark 8 It is useful to observe that the “ideal property” of B can be decided by considering a single isogeny π : A → B because if H_{1} and H_{2} are two finite subgroup schemes, then we have:

(22) If A_{H_{1}} ≃ A_{H_{2}}, then H_{1} is an ideal subgroup ⇔ H_{2} is an ideal subgroup.

Indeed, by (17) the hypothesis means that f^{−1}(H_{1}) = [n]^{−1}(H_{2}), for some isogeny f ∈ R and n ∈ ℕ and so the conclusion follows from (15).

Similarly, the property of being a kernel ideal is a property of the isomorphism class: if I, J are two regular left R-ideals, then we have:

(23) If I ≃ J, then I is a kernel ideal ⇔ J is a kernel ideal.

Indeed, by (18) we have I_{1}f = I_{2}n, for some isogeny f ∈ R and n ∈ ℕ, and so the conclusion follows from (14).

2.3 Homomorphisms

The discussion of the previous subsection drew attention to the importance of ideal subgroups, and so it is of interest to classify them. Now while it is frequently the case that all regular left R-ideals are kernel ideals (for example, if R is a Dedekind domain; cf. Remark 7(d)), it rarely happens that all finite subgroups schemes of A are ideal subgroups. The reason for this is that if H = H(I) is an ideal subgroup, then End(A_{H}) has additional properties which are not necessarily satisfied by End(A_{H}) for an arbitrary finite subgroup scheme H.

To explain this in more detail, let H_{1} and H_{2} be two finite subgroup schemes of A, and consider the subset H(H_{1}, H_{2}) := Im(Φ_{H_{1},H_{2}}) ⊂ R̃ which is the image of the map

Φ_{H_{1},H_{2}} : Hom(A_{H_{1}}, A_{H_{2}}) → R̃ = End^{0}(A)

defined by the rule

Φ_{H_{1},H_{2}}(h) = \frac{1}{n_{H_{2}}} π′_{H_{2}} ∘ h ∘ π_{H_{1}}, for h ∈ Hom(A_{H_{1}}, A_{H_{2}}). (24)

Here, as before n_{H_{2}} = deg(π_{H_{2}}) = |H_{2}| and π′_{H_{2}} is defined by (4).

Remark 9 (a) Note that Φ = Φ_{H_{1},H_{2}} is injective because it follows from (4) that

(25) π_{H_{2}}Φ(h)π′_{H_{1}} = n_{H_{1}}h, for all h ∈ Hom(A_{H_{1}}, A_{H_{2}}).

Thus, Φ defines an isomorphism (of additive groups)

Φ : Hom(A_{H_{1}}, A_{H_{2}}) →^{∼} H(H_{1}, H_{2})

which extends to an isomorphism

Φ^{0} : Hom^{0}(A_{H_{1}}, A_{H_{2}}) := Hom(A_{H_{1}}, A_{H_{2}}) ⊗ ℚ →^{∼} R̃.

(To see that Φ^{0} is surjective, note that if f̃ = \frac{f}{n} ∈ R̃ with f ∈ R, n ∈ ℕ, then h̃ = \frac{1}{nn_{H_{1}}} π_{H_{2}}fπ′_{H_{1}} ∈ Hom^{0}(A_{H_{1}}, A_{H_{2}}) and Φ^{0}(h̃) = f̃.) Thus, H(H_{1}, H_{2}) is a lattice of

R̃, i.e. it is an additive subgroup of R̃ which contains a ℚ-basis of R̃.

Note that the lattices H(H_{1}, H_{2}) can be viewed as a generalization of I(H)- construction. Indeed, if we take H_{2} = 0, then we have

H(H, 0) = Hom(A_{H}, A)π_{H} = I(H)

because Φ_{H,0}(h) = hπ_{H} as here n_{H_{2}} = 1, A_{H_{2}} = A and π_{H_{2}} = π′_{H_{2}} = 1_{A}.

(b) The map Φ_{H_{1},H_{2}} is multiplicative in the sense that if H_{3} is another finite subgroup scheme of A, then for h_{i} ∈ Hom(A_{H_{i}}, A_{H_{i+1}}), i = 1, 2, we have that

Φ_{H_{2},H_{3}}(h_{2})Φ_{H_{1},H_{2}}(h_{1}) = Φ_{H_{1},H_{3}}(h_{2} ∘ h_{1}). (26)

Indeed, writing π_{i} = π_{H_{i}}, π′_{i} = π′_{H_{i}}, and n_{i} = n_{H_{i}}, for i = 1, 2, 3, we have by using (4) that Φ_{H_{2},H_{3}}(h_{2})Φ_{H_{1},H_{2}}(h_{1}) = \frac{1}{d_{3}} π′_{3}h_{2}π_{2} \frac{1}{d_{2}} π′_{2}h_{1}π_{1} = \frac{1}{d_{3}} π′_{3}h_{2}h_{1}π_{1} = Φ_{H_{1},H_{3}}(h_{2} ∘ h_{1}).

In particular, if H_{1} = H_{2} = H, then Φ_{H} = Φ_{H,H} defines a ring isomorphism

Φ_{H} : End(A_{H}) →^{∼} E(H) := H(H, H),

and so E(H) is a subring of R̃. (Note that Φ_{H}(1_{A_{H}}) = \frac{1}{n_{H}} π′_{H}π_{H} = 1_{A}.) We observe that since Φ_{H}(π_{H}hπ′_{H}) = n_{H}h, ∀h ∈ R, we have the inclusions

n_{H}R ⊂ E(H) ⊂ \frac{1}{n_{H}} R. (27)

(c) If H ≤ H′, then Φ_{H,H′}(π_{H,H′}) = 1_{A} because Φ_{H,H′}(π_{H,H′}) = \frac{1}{n_{H′}} π′_{H′}π_{H,H′}π_{H} = \frac{1}{n_{H′}} π′_{H′}π_{H′} = 1_{A}. Thus, if H_{1} ≥ H′_{1} and H_{2} ≤ H′_{2}, then it follows from (26) that

Φ_{H_{1},H_{2}}(h) = Φ_{H_{2},H′_{2}}(π_{H_{2},H′_{2}})Φ_{H_{1},H_{2}}(h)Φ_{H′_{1},H_{1}}(π_{H′_{1},H_{1}}) = Φ_{H′_{1},H′_{2}}(π_{H_{2},H′_{2}} ∘ h ∘ π_{H′_{1},H_{1}}),

for all h ∈ Hom(A_{H_{1}}, A_{H_{2}}), and so it follows that

H_{1} ≥ H′_{1}, H_{2} ≤ H′_{2} ⇒ H(H_{1}, H_{2}) ⊂ H(H′_{1}, H′_{2}). (28)

We now want to describe H(H_{1}, H_{2}) in terms of the ideals I(H_{i}). For this, it is useful to introduce the following notation. If S, T are subsets of R̃, put

(S : T) := {f ∈ R̃ : T f ⊂ S} = {f ∈ R̃ : tf ∈ S, ∀t ∈ T }.

We then have the following description of H(H_{1}, H_{2}).

Proposition 10 If H_{1}, H_{2} are finite subgroup schemes of A and if I_{1}, I_{2} are regular left R-ideals, then

(29)  H(H_{1}, H_{2}) ⊂ (I(H_{1}) : I(H_{2})),
(30)  (I_{1} : I_{2}) ⊂ H(H(I_{1}), H(I_{2})).

Moreover, if H_{2} is an ideal subgroup, then equality holds in (29), and if I_{1} is a kernel ideal, then equality holds in (30).

Proof. To prove (29), let h ∈ Hom(A_{H_{1}}, A_{H_{2}}) and f = f′π_{H_{2}} ∈ I(H_{2}), where f′ ∈ Hom(A_{H_{1}}, A). Then by (4) we have fΦ(h) = f′π_{H_{2}} \frac{1}{n_{H_{2}}} π′_{H_{2}}hπ_{H_{1}} = f′hπ_{H_{1}} ∈ I(H_{2}), and so the inclusion (29) follows.

In order to prove (30), we first observe that

H(H_{1}, H_{2}) = \{\frac{f}{n} : f ∈ R, n ∈ ℕ and [n]^{−1}(H_{1}) ≤ f^{−1}(H_{2})\} . (31)

Indeed, if f̃ ∈ H(H_{1}, H_{2}), then f̃ = \frac{f}{n}, where n = n_{H_{2}} and f = π′_{H_{2}}hπ_{H_{1}}, for some h ∈ Hom(A_{H_{1}}, A_{H_{2}}). Then π_{H_{2}}f = nhπ_{H_{1}} = h(nπ_{H_{2}}), so f^{−1}(H_{2}) = Ker(π_{H_{2}}f) = Ker(hnπ_{H_{1}}) ≥ Ker(nπ_{H_{1}}) = [n]^{−1}(H_{1}). Thus, the left hand side of (31) is contained in the right hand side.

Conversely, suppose that f ∈ R and n ∈ ℕ satisfy [n]^{−1}(H_{2}) ≤ f^{−1}(H_{2}), i.e. Ker(nπ_{H_{1}}) ≤ Ker(π_{H_{2}}f). Then by the universal property of quotients ∃h ∈ Hom(A_{H_{1}}, A_{H_{2}}) such that π_{H_{2}}f = hnπ_{H_{1}} = nhπ_{H_{1}}. Then n_{H_{2}}f = π′_{H_{2}}π_{H_{2}}f = π′_{H_{2}}nhπ_{H_{1}}, so f = \frac{n}{n_{H_{2}}} π′_{H_{2}}hπ_{H_{1}} = nΦ_{H_{1}.H_{2}}(h). Thus f̃:= \frac{f}{n} = Φ_{H_{1},H_{2}}(h) ∈ H(H_{1}, H_{2}), and so we have verified that equality holds in (31).

Now we prove (30). For this, let f̃ = \frac{f}{n} ∈ (I_{1} : I_{2}) with f ∈ R, n ∈ ℕ, and write H_{i} = H(I_{i}) and π_{i} = π_{H_{i}} for i = 1, 2. We first claim that

Ker(nπ_{1}) ≤ Ker(αf), ∀α ∈ I_{2}.

Indeed, since \frac{α}{n}f = αf̃ =: β ∈ I_{1}, we have αf = nβ. Since H_{1} = H(I_{1}) ≤ Ker(β), we thus have Ker(nπ_{1}) ≤ Ker(nβ) = Ker(αf), which proves the above claim.

Now since H_{2} = H(I_{2}) = ∩_{i}Ker(α_{i}), if I_{2} = ∑_{i} Rα_{i}, we see that Ker(π_{2}f) = f^{−1}(H_{2}) = ∩_{i}f^{−1}(Ker(α_{i})) = ∩_{i}Ker(α_{i}f). It thus follows from the above claim that Ker(nπ_{1}) ≤ Ker(π_{2}f), and hence f̃ ∈ H(H_{1}, H_{2}) by (31). This proves (30).

By combining (29) and (30) we obtain

H(H_{1}, H_{2}) ⊂ (I(H_{1}) : I(H_{2})) ⊂ H(H(I(H_{1})), H(I(H_{2}))) ⊂ H(H_{1}, H(I(H_{2}))), (32)

where the last inclusion follows from (28) (together with (8)). Thus, if H_{2} is an ideal subgroup, i.e. if H_{2} = H(I(H_{2})), then equality holds throughout, and hence equality holds in (29). Similarly, combining (29) and (30) yields

(I_{1} : I_{2}) ⊂ H(H(I_{1}), H(I_{2})) ⊂ (I(H(I_{1})) : I(H(I_{2}))) ⊂ (I(H(I_{1})) : I_{2}) (33)

and so equality holds throughout if I_{1} is a kernel subgroup.

As an application, we obtain that ideal subgroups H satisfy the extra condition that Z(R) ⊂ E(H), where Z(R) = {z ∈ R : xz = zx, ∀x ∈ R} denotes the centre of R. This condition is usually not true for arbitrary subgroup schemes of A, as we shall see in Remark 17(b) below.

Corollary 11 If H is an ideal subgroup of A, then Z(R) ⊂ E(H).

Proof. By hypothesis, H = H(I), for some regular left R-ideal I, and so (I : I) ⊂ E(H) by (30). Now Z(R) ⊂ (I : I) because I is a left R-ideal (for z ∈ Z(R) ⇒ Iz = zI ⊂ I), and so the assertion follows.

The following result, which is a variant of a result of [26], p. 534, shows that the subgroup scheme associated to a product of ideals has a natural interpretation in terms of composition of maps.

Proposition 12 Let I be a regular left R-ideal and let J be a regular left R′-ideal, where R′ = E(H(I)). Then IJ is a regular left R-ideal and

H(IJ) = Ker(π_{H(Φ_{I}^{−1}(J))} ∘ π_{H(I)}), (34)

where Φ_{I} = Φ_{H(I)} : End(A_{H(I)}) →^{∼} R′ and π_{H(Φ_{I}^{−1}(J))} : A_{H(I)} → (A_{H(I)})_{H(Φ_{I}^{−1}(J))} is the

canonical quotient map.

Proof. Put H = H(I). Since J ⊂ E(H) ⊂ (I(H) : I(H)) by (29), we have IJ ⊂ I(H)J ⊂ I(H) ⊂ R, and hence IJ is an R-ideal. Moreover, IJ is regular because by hypothesis ∃α ∈ I ∩ R̃^{×} and β ∈ J ∩ R̃^{×}, and so αβ ∈ IJ ∩ R̃^{×}, which means that IJ is regular.

To prove (34), note first that it follows from (25) that nΦ_{I}^{−1}(J) = πJπ′, where n = n_{H}, π = π_{H}, and π′ = π′_{H}, and so Φ_{I}^{−1}(J)π = πJ. From this it follows that

Ker(πf) = ∩_{g∈I}Ker(gf), ∀f ∈ J, (35)

where we view πf ∈ Hom(A, A_{H}) since πf = f_{1}π for some f_{1} ∈ Φ_{I}^{−1}(J) ⊂ End(A_{H}). To verify (35), write f = f_{2}/n with f_{2} ∈ R and n ∈ ℕ. Then

[n]^{−1}(Ker(πf)) = Ker(πf_{2}) = f_{2}^{−1}(Ker(π)) = f_{2}^{−1}(∩_{g∈I}Ker(g))
= ∩_{g∈I}f_{2}^{−1}(Ker(g)) = ∩_{g∈I}Ker(gf_{2}) = [n]^{−1}(∩_{g∈I}Ker(gf)),

the latter because gf_{2} = gfn and gf ∈ R, for all g ∈ I. From this, equation (35) follows because [n] is faithfully flat.

Using (35), we therefore obtain that

Ker(π_{H(Φ_{I}^{−1}(J))}π) = π^{−1}(Ker(π_{H(Φ_{I}^{−1}(J))})) = π^{−1}(∩_{f∈Φ_{I}^{−1}(J)}Ker(f))
= ∩_{f∈Φ_{I}^{−1}(J)}Ker(fπ)) = ∩_{f′∈Φ_{I}^{−1}(J)π}Ker(f′)) = ∩_{f′∈πJ}Ker(f′))
= ∩_{f∈J}Ker(πf)) = ∩_{f∈J}f^{−1}(Ker(π)) = ∩_{f∈J}f^{−1}(∩_{g∈I}Ker(g))
= ∩_{f∈J} ∩_{g∈I} Ker(gf) = H(IJ),

which proves (34).

2.4 The quadratic case

We now specialize the discussion to the case that R̃ is a quadratic field F ⊃ ℚ. Since R is finitely generated (as a ℤ-module), it follows that R is an order of F = R̃, i.e. R is a subring of F which is lattice. We first recall some basic facts about such orders and lattices (cf. [3], §II.7 or [6], ch. 7).

Every order R of F is contained in the maximal order 𝔒_{F}, the ring of integers of F, and is uniquely determined by its conductor f_{R} := [𝔒_{F} : R]. Indeed, if Δ(R) := f_{R}^{2}Δ_{F}, where Δ_{F} = Δ(𝔒_{F}) is the discriminant of F (or, more correctly, of 𝔒_{F}), then we have

(36) R = R_{Δ} := ℤ + ℤω_{Δ}, where ω_{Δ} = \frac{1}{2}(Δ + \sqrt{Δ}).

Thus, R is also uniquely determined by its discriminant Δ(R). Conversely, for each integer f there is unique order of conductor f. Moreover, if R_{1}, R_{2} are orders in F then we have

(37) R_{1} ⊂ R_{2} ⇔ f_{R_{2}}|f_{R_{1}} and hence f_{R_{1}R_{2}} = (f_{R_{1}}, f_{R_{2}}), f_{R_{1}∩R_{2}} = lcm(f_{R_{1}}, f_{R_{2}}).

Let Lat_{F} denote the set of lattices of F, i.e. the set of finitely generated subgroups L of F such that LF = F. If L ∈ Lat_{F}, then R(L) := (L : L) is an order of F, and for a given order R, the set

Lat(R) = {L ∈ Lat_{F} : R(L) = R}

is the set of invertible R-submodules of F, and hence forms an abelian group under the multiplication of lattices. Here the identity is R and the inverse of L is

L^{−1} = (R(L) : L) = σ(L)N(L)^{−1}, (38)

where σ ∈ Gal(F/ℚ) is the unique nontrivial automorphism of F and N(L) ∈ ℚ^{×} is the norm of L. Moreover, the group

Pic(R) = Lat(R)/{fR : f ∈ F^{×}}

is a finite abelian group whose order is denoted by h(R) = h(Δ(R)).

For later reference we recall the following useful formulae (cf. [6], p. 151):

R(L_{1}L_{2}) = R(L_{1})R(L_{2}) and N(L_{1}L_{2}) = N(L_{1})N(L_{2}). (39)

In addition, we have following formulae (40) and (41) will be used several times below. Since there does not seem to be a suitable reference, we provide a proof of these identities. Note that (41) is stated without proof on p. 71 of [12].

Lemma 13 If L_{1}, L_{2} ∈ Lat_{F} are any two lattices of F, then

(L_{1} : L_{2})L_{2} = (R(L_{1}) : R(L_{2}))L_{1}. (40)

Thus, if f_{i} = f_{R(L_{i})} = [O_{F} : R(L_{i})], for i = 1, 2, and if f = (f_{1}, f_{2}), then we have

(L_{1} : L_{2}) = [R(L_{1})R(L_{1}) : R(L_{1})]L_{1}L_{2}^{−1} = \frac{f_{1}}{f} L_{1}L_{2}^{−1}. (41)

Proof. Put R_{i} = R(L_{i}) and R_{0} = R_{1}R_{2} = R(L_{1}L_{2}). Note that (L_{1} : L_{2}) is an R_{0}-module, for if c ∈ (L_{1} : L_{2}) and r_{i} ∈ R(L_{i}), then cr_{1}r_{2} ∈ (L_{1} : L_{2}) because r_{1}cr_{2}L_{2} ⊂ r_{1}cL_{2} ⊂ r_{1}L_{1} ⊂ L_{1}. In particular, (R_{1} : R_{2}) is also an R_{0}-module (because R(R_{i}) = R_{i}).

Now let c ∈ (L_{1} : L_{2}). Then (cL_{1}^{−1}L_{2})R_{2} = cL_{1}^{−1}L_{2} = cL_{2}L_{1}^{−1} ⊂ L_{1}L_{1}^{−1} = R_{1}, so cL_{2}L_{1}^{−1} ⊂ (R_{1} : R_{2}), and hence (L_{1} : L_{2})L_{2}L_{1}^{−1} ⊂ (R_{1} : R_{2}). Thus (L_{1} : L_{2})L_{2}R_{1} = (L_{1} : L_{2})L_{2}^{−1}L_{1} ⊂ (R_{1} : R_{2})L_{1}. But since (L_{1} : L_{2}) is an R_{1}-module (as R_{1} ⊂ R_{0}), we have that (L_{1} : L_{2})L_{2}R_{1} = (L_{1} : L_{2})L_{2}, and so the left hand side of (40) is contained in the right hand side.

To prove the other inclusion, let r ∈ (R_{1} : R_{2}). Then (rL_{1}L_{2}^{−1})L_{2} = rL_{1}R_{2} = rR_{2}L_{2} ⊂ R_{1}L_{1} = L_{1}, so rl_{1}L_{2}^{−1} ⊂ (L_{1} : L_{2}), and hence (R_{1} : R_{2})L_{1}L_{2}^{−1} ⊂ (L_{1} : L_{2}). Thus (R_{1} : R_{2})L_{1}R_{2} = (R_{1} : R_{2})L_{1}L_{2}^{−1}L_{2} ⊂ (L_{1} : L_{2})L_{2}, and so the other inclusion of (40) holds because (R_{1} : R_{2})L_{1}R_{2} = (R_{1} : R_{2})L_{1} (since (R_{1} : R_{2}) is an R_{2}-module). This proves (40).

The formula (41) follows immediately from (40) once we have shown that

(R(L_{1}) : R(L_{1})) = \frac{f_{1}}{f} R(L_{1})R(L_{2}) and \frac{f_{1}}{f} = [R(L_{1})R(L_{2}) : R(L_{1})]. (42)

Indeed, multiplying (40) by L_{2}^{−1}, we obtain with (42) that (L_{1} : L_{2}) = \frac{f_{1}}{f} R_{1}R_{2}L_{1}L_{2}^{−1} = \frac{f_{1}}{f} L_{1}L_{2}^{−1}, where the last equality follows from the obvious fact that L_{i}^{±1} is an R_{i}- module, for i = 1, 2.

It thus remains to verify (42). For this, we first note that since f = f_{R_{0}} by (39), we have that [R_{0} : R_{1}] = \frac{f_{1}}{f}, which is the second equality of (42). Thus, \frac{f_{1}}{f} R_{0} is largest R_{0}-module which is contained in R_{1}, and hence (R_{1} : R_{2}) ⊂ \frac{f_{1}}{f} R_{0} because (R_{1} : R_{2}) is an R_{0}-module which is contained in R_{1}. On the other hand, since \frac{f_{1}}{f} R_{2} ⊂ \frac{f_{1}}{f} R_{0} ⊂ R_{1}, we have the opposite inclusion \frac{f_{1}}{f} R_{0} ⊂ (R_{1} : R_{2}), which proves (42).

Corollary 14 Every non-zero ideal of an order in F is a divisorial ideal.

Proof. Let R be an order of F and let I be a nonzero R-ideal. Then I ∈ Lat_{F} and R ⊂ R(I). Applying (41) to L_{1} = R and L_{2} = I yields

(R : I) = [R(I) : R]I^{−1} (43) because here f_{2}|f_{1}, so f = f_{2} and \frac{f_{1}}{f} = [R(I) : R]. Next, apply (41) to L_{1} = R and L_{2} = (R : I). Since R(L_{2}) = R(I^{−1}) = R(I) by (43), we see that in this case (41) gives

(R : (R : I)) = [R(I) : R]((R : I))^{−1} = [R(I) : R][R(I) : R]^{−1}(I^{−1})^{−1} = I

Since I^{∗} = (R : (R : I)) by [5], p. 476, we thus have that I = I^{∗}, and so I is a divisorial ideal in the sense of Remark 7(d).

We now apply the preceding results to abelian varieties.

Proposition 15 Let A/K be an abelian variety such that R̃ = End^{0}(A) is a quadratic field. Then every non-zero ideal of R = End(A) is a kernel ideal, and hence we have for any two non-zero ideals I, J of R that Φ_{I,J} := Φ_{H(I),H(J)} defines an isomorphism

Φ_{I,J} := Φ_{H(I),H(J)} : Hom(A_{H(I)}, A_{H(J)}) →^{∼} (I : J). (44)

Moreover, we have that A_{H(I)} ≃ A_{H(J)} ⇔ I ≃ J (as R-modules).

Proof. Since R is an order of R̃ = F, we know by Corollary 14 that every non-zero ideal of R is divisorial and hence a kernel ideal by Remark 7(d). This proves the first assertion, and hence the other assertions follow from Proposition 10 and from the discussion after (20).

Corollary 16 In the above situation, let R′ be an order of R̃ with R ⊂ R′. Then there exists an abelian variety A′/K which is isogenous to A/K such that End(A′) ≃ R′.

Proof. Take I = [R′ : R]R′ ⊂ R. Then I is a non-zero R-ideal with R(I) = R′. Then A′ := A_{H(I)} is isogenous to A and End(A′) ≃ (I : I) = R(I) = R′ by (44).

Remark 17 (a) Note that if we drop the hypothesis R ⊂ R′ in Corollary 16, then the corresponding statement is in general no longer true; cf. §3.3 below.

(b) We can use the above Corollary 16 to construct an abelian variety A′ with a finite subgroup scheme H′ such that E(H′) ⊄ End(A′). In particular, H′ is not an ideal subgroup by Corollary 11.

Indeed, suppose there exists an abelian variety A/K such that R := End(A) ⊂ F but R ≠ 𝔒_{F}. Then by Corollary 16 there is an R-ideal I such that A′ := A_{H(I)} satisfies End(A′) ≃ 𝔒_{F}. Consider H′ = Ker(π′_{H(I)}). Since π′_{H(I)} : A′ → A is an isogeny, we have (A′)_{H′} ≃ A, so E(H′) ≃ End(A) = R. Thus O_{F} = End(A′) ⊄ E(H′), and so H′ is not an ideal subgroup of A′.

3 CM elliptic curves

3.1 Kernel ideals and ideal subgroups

We now apply the theory of the previous section to the case that A = E is elliptic curve over K with complex multiplication. By this we mean that E/K is an elliptic curve such that

End^{0}(E) = End^{0}(E ⊗ \overline{K}) = F

is an imaginary quadratic field F, where \overline{K} denotes the algebraic closure of K. Note that this definition is slightly more restrictive than that of (say) [18], since we assume here that all \overline{K}-endomorphisms are already defined over K.

If E/K is a CM elliptic curve, then R = End(E) is an order in F. Thus, R is uniquely characterized by its conductor f_{E} := [𝔒_{F} : R] or by its discriminant Δ_{E} := Δ(R), which we call the conductor and discriminant of E, respectively.

The main results about kernel ideals and ideal subgroups are summarized in the following theorem which can be viewed as a refinement of results of Deuring[10] and Waterhouse[26].

Theorem 18 Let E/K be a CM elliptic curve and R = End(E). Then:

(a) Every non-zero R-ideal I is a kernel ideal. Thus

Φ_{I_{1},I_{2}} := Φ_{H(I_{1}),H(I_{2})} : Hom(E_{H(I_{1})}, E_{H(I_{2})}) →^{∼} (I_{1} : I_{2}) (45)

is an isomorphism for all non-zero ideals I_{1}, I_{2} of R.

(b) If H be a finite subgroup scheme of E, then

(46) H is an ideal subgroup ⇔ f_{E_{H}}|f_{E}.

Thus, if H_{1}, H_{2} are finite subgroup schemes of E such that f_{E_{H_{i}}}|f_{E}, for i = 1, 2, then

Φ_{H_{1},H_{2}} : Hom(E_{H_{1}}, E_{H_{2}}) →^{∼} H(H_{1}, H_{2}) = (I(H_{1}) : I(H_{2})), (47)

and we have E_{H_{1}} ≃ E_{H_{2}} ⇔ I(H_{1}) ≃ I(H_{2}). (48)

Proof. (a) This is a special case of Proposition 15.

(b) We first observe that it is enough to verify (46), for then (47) follows immediately from Proposition 10 and (48) follows from (21).

To prove (46), we first note that since E(H) ≃ End(E_{H}), it follows from (37) that f_{E_{H}}|f_{E} ⇔ R ⊂ E(H). Thus, the one direction (⇒) follows directly from Corollary 11.

Conversely, suppose that f_{E_{H}}|f_{E} or, equivalently, that R ⊂ R′ := E(H). By (15) it is enough to show that H_{n} := [n]^{−1}(H) is an ideal subgroup, for some n ∈ ℕ.

For this, put f = [R′ : R], I = fR′, and E′ = E_{H(I)}. Then by (45) we have End(R′) ≃ (I : I) = R′. Moreover, if π := π_{H(I)} : E → E′ and n := deg(π), then by (4) we have π_{H}[n] = π_{H}π′π, where π′ = π′_{H(I)}, and so [n]^{−1}(H) = Ker(π_{H}[n]) = π^{−1}(H′), where H′ = Ker(π_{H}π′) is a finite subgroup scheme of E′.

Since π_{H}π′ : E′ → E_{H} is an isogeny, it follows that (E′)_{H′} ≃ E_{H}, and hence End((E′)_{H′}) ≃ End(E_{H}) ≃ E(H) = R′ ≃ End(E′). Thus, by the abovementioned result of Deuring/Waterhouse (which is proved via ℓ-adic representations; cf. [10] or [26], p. 541), there is an ideal I′ of End(E′) such that H′ = H(I′). If Ĩ′ = Φ_{I}(I′) is the corresponding ideal of E(H(I)) = R′, then by Proposition 12 we have that H(IĨ′) = Ker(π_{H(I′)} ∘ π_{H(I)}) = π^{−1}(H′) = [n]^{−1}(H). Thus, [n]^{−1}(H) is an ideal subgroup of E, and hence so is H (by what was said above).

We can apply the above results to obtain information about the following subset Isog^{+}(E/K) of the set Isog(E/K) of elliptic curves isogenous to E/K.

Notation. Let Isog(E/K) = {E′/K : E′ ∼ E}/ ≃ be the set of isomorphism classes of elliptic curves E′/K which are isogenous to E, and let

Isog^{+}(E/K) = {E′ ∈ Isog(E/K) : f_{E′}|f_{E}}.

Corollary 19 If E/K is a CM elliptic curve, then the map E′ ↦ I_{E}(E′) induces a bijection

I_{E}^{+} : Isog^{+}(E/K) →^{∼} Id(R_{E}))/≃,

where Id(R_{E})/ ≃ denotes the set of isomorphism classes of non-zero ideals of R_{E} = End(E).

Proof. By the discussion of subsection 2.2 we know that the given rule defines a map I_{E} : Isom(E/K) → Id(R)/≃. We denote its restriction to Isom^{+}(E/K) by I_{E}^{+}.

To show that I_{E}^{+} is surjective, let I ∈ Id(R), and put E′ = E_{H(I)}. Then by (45) we have that End(E′) ≃ (I : I) = R(I) ⊃ R, This means that f_{E′}|f_{E}, and so E′ ∈ Isog^{+}(E/K). Moreover, I_{E}(E′) ≃ I(H(I)) = I, and hence I_{E}^{+} is surjective.

To show that I_{E}^{+} is injective, let E_{1}, E_{2} ∈ Isog^{+}(E/K) such that I_{E}(E_{i}) ≃ I_{E}(E_{2}). Thus, by definition, I(H_{1}) ≃ I(H_{2}), where H_{i} = Ker(π_{i}) and π_{i} : E → E_{i} are any two isogenies, and we have f_{E_{i}}|f_{E}. Since E_{i} = E_{H_{i}}, we have by (48) that I(H_{1}) ≃ I(H_{2}), and so I_{E}^{+} is injective.

Remark 20 It follows from (45) that for any R-ideal I we have that

(49) End(E_{H(I)}) ≃ (I : I) = R(I) and so f_{E_{H(I)}} = f_{E}[R(I) : R_{E}]^{−1}.

Moreover, by (47) we have that

(50) End(E′) ≃ R(I_{E}(E′)) and f_{E′} = f_{R(I_{E}(E′))}, for all E′ ∈ Isog^{+}(E/K).

From this we see that if we restrict the map I_{E}^{+} to the subset

Isog^{∗}(E/K) = {E′ ∈ Isog(E/K) : f_{E′} = f_{E}},

then we obtain the (well-known) bijection

Isog^{∗}(E/K) →^{∼} {I ∈ Id(R_{E}) : R(I) = R_{E}}/≃ →^{∼} Pic(R). (51)

Thus, Isog^{∗}(E/K) is a finite set of cardinality h(Δ_{E}) := h(R_{Δ_{E}}). Moreover, since

Isog^{+}(E/K) = ˙ ⋃_{f|f_{E}} {E′ ∈ Isog(E/K) : f′_{E} = f} = ˙ ⋃_{f|f_{E}}Isog(E_{f}/K),

where E_{f} ∼ E is an elliptic curve of conductor f|f_{E} (which exists by Corollary 16), we see that Isog^{+}(E/K) is also finite set of cardinality

#(Isog^{+}(E/K)) = ∑_{f|f_{E}} h(Δ_{E}/f^{2}). (52)

On the other hand, the set Isog(E/K) is often infinite; for example, this is the case when K is algebraically closed.

We can also give a “numerical criterion” to detect ideal subgroups; cf. Corollary 23 below. For this we first prove the following result which is also of independent interest.

Proposition 21 If I is a non-zero ideal of R = End(E), then

|H(I)| = deg(π_{H(I)}) = [R : I]. (53)

Remark 22 If I is not invertible (i.e. if R(I) ≠ R), then the above formula (53) contradicts the formula on p. 211 of Deuring[10], who asserts that |H(I)| = [R(I) : I] in place of |H(I)| = [R : I].

In fact, Deuring’s proof of his statement contains an error. While his proof in the case that R(I) = R is correct, his proof of the general case is not. To be precise, on p. 218 he uses a result of W. Weber incorrectly because that result only applies to invertible ideals (and is, in fact, false otherwise).

Proof of Proposition 21. We first note that if f ∈ R = End(E), then

deg(f) = |Ker(f)| = N(f), (54)

where N = N_{F/ℚ} denotes the usual field norm. Indeed, by e.g. [20], p. 180, we know that P_{f}(f) = 0 where P_{f}(X) = X^{2} + tX + deg(f), for some t ∈ ℤ, and so (54) follows.

From (54) we can conclude that

deg(π_{H(I)}) | gcd({N(f) : f ∈ I}) = N(I) = [R(I) : I] (55)

because if f ∈ I, then H(I) ≤ Ker(f) and so |H(I)| = deg(π_{H(I)})| deg(f) = N(f) by (54). Thus, the indicated divisibility holds. The last equality of (55) is just the definition of N(I), and the first equality is well-known. (It follows from one of the steps needed to verify the bijection between equivalence classes of binary quadratic forms and equivalence classes of invertible R′-modules (where R′ = R(I)).)

Suppose first that I is invertible (i.e. R(I) = R). Then σ(I) is also an invertible Rideal (where σ ∈ Gal(F/ℚ), σ ≠ 1) and N(σ(I)) = N(I). Moreover, Iσ(I) = N(I)R by (38). Since E(E_{H(I)}) = (I : I) = R(I) = R, we can apply Proposition 12 to J = σ(I) to obtain H(Iσ(I)) = Ker(π_{H}(I′)) ∘ π_{H(I)}), where I′ = Φ_{I}^{−1}(σ(I)). Thus, |H(Iσ(I))| = |H(I′)| · |H(I)|. Since H(Iσ(I)) = Ker([N(I)]), we thus obtain from (54) and (55) that N(I)^{2} = |H(I′)| · |H(I)| | N(σ(I))N(I) = N(I)^{2}, and so it follows that we must have equality throughout, so |H(I)| = N(I) = [R : I]. This proves (53) when I is invertible.

In order to prove (53) in the general case, we first verify it in the special case that I = fR′, where R′ ⊃ R is an order and f = [R′ : R]. More precisely, we show that

|H(fR′)| = f = [R : fR′], if R ⊂ R′ and f = [R′ : R]. (56)

First note that the second equality is clear because [R′ : fR′] = f^{2}. To prove the first equality, we shall induct on the number ν = ν(f) of primes (counted with multiplicities) which divide f. If ν = 1, then f = p is a prime. Since p ∈ pR′, we have H(pR′) ≤ Ker([p]), and so by the universal property there exists g ∈ Hom(E′, E) such that [p] = gπ, where E′ = E_{H(pR′)} and π = π_{H(pR′)}. Thus deg(π) | deg([p]) = p^{2}. Now deg(π) ≠ 1 because otherwise E′ ≃ E, which is impossible since End(E′) ≃ E(H(pR′)) = R′ ≄ R = End(E). Similarly, deg(π) ≠ p^{2} because otherwise g is an isomorphism and then again E′ ≃ E, contradiction. Thus deg(π) = p, and so (56) holds in this case.

Now suppose ν > 1. Then f = f′p, where p is a prime and ν(f′) ≥ 1. Let R_{1} be the unique order such that R ⊂ R_{1} ⊂ R′ and [R_{1} : R] = p (and hence [R′ : R_{1}] = f′). Put π_{1} = π_{H(pR_{1})} : E → E_{1} := E_{H(pR_{1})}. By what was just shown, deg(π_{1}) = p. Now consider I′ = Φ_{pR_{1}}^{−1}(f′R′) ⊂ End(E_{1}). By the induction hypothesis (applied to E_{1} in place of E) we have deg(π_{I′}) = f′. Thus, since pR_{1}f′R′ = fR′, we have by Proposition 12 that H(fR′) = Ker(π_{H(I′)} ∘ π_{1}), so deg(π_{fR′}) = deg(π_{I′}) deg(π_{1}) = f′p = f, which proves (56).

We can now verify (53) for an arbitrary R-ideal I ≠ 0. Put R′ = R(I) and f = [R′ : R]. Since I is an R′-ideal contained in R, we have I ⊂ fR′, and so I′ := \frac{1}{f}I ⊂ R′. Thus I′ is an invertible R′-ideal. Since E(H(fR′)) = R′, we see that I′′ = Φ_{fR′}^{−1}(I′) is an invertible End(E_{2})-ideal, where E_{2} := E_{H(fR′)}, and so, by what was proved above, deg(π_{H(I′′)}) = [R′ : I′] = [fR′ : I] because fI′ = I. Now since (fR′)I′ = R′I = I, we have by Proposition 12 that H(I) = Ker(π_{I′′} ∘ π_{fR′}), and so deg(π_{H(I)}) = deg(π_{I′′}) deg(π_{fR′}) = [fR′ : I][R : fR′] by (56). Thus deg(π_{H(I)}) = [R : I], which proves (53) in general.

Corollary 23 If H is a finite subgroup scheme, then |H| | [R : I(H)], and equality holds if and only if H is an ideal subgroup.

Proof. Since H ≤ H(I(H)), we have |H| | |H(I(H))| = [R : I(H)] by (53). Moreover, H is an ideal subgroup ⇔ H = H(I(H)) ⇔ |H| = |H(I(H))| ⇔ |H| = [R : I(H)], and so the assertion follows.

Corollary 24 Let H_{1} and H_{2} be two ideal subgroups of E and let f_{i} = f_{E_{H_{i}}}. Then the norms of the ideals I(H_{i}) and of the lattice H(H_{1}, H_{2}) are given by

(57) N(I(H_{i})) = \frac{f_{E}}{f_{i}} |H_{i}| and N(H(H_{1}, H_{2})) = \frac{lcm(f_{1}, f_{1})}{gcd(f_{1}, f_{2})} \frac{|H_{1}|}{|H_{2}|}.

Proof. Put n_{i} = |H_{i}| and L_{i} = I(H_{i}). Since R(L_{i}) = E(H_{i}) ⊃ R by (46) and (47), we have [R(L_{i}) : R] = f_{E}/f_{i}, and so N(L_{i}) = [R(L_{i}) : L_{i}] = \frac{f_{E}}{f_{i}} [R : I(H_{i})] = \frac{f_{E}}{f_{i}} n_{i} by Corollary 23. This proves the first equality of (57).

Now by (47) and (41) we have H(H_{1}, H_{2})) = (L_{1} : L_{2}) = \frac{f_{1}}{f} L_{1}L_{2}^{−1}, where f = gcd(f_{1}, f_{2}). Thus, using the first equality of (57), we obtain

N(H(H_{1}, H_{2})) = \frac{f_{1}^{2}}{f^{2}} \frac{N(L_{1})}{N(L_{2})} = \frac{f_{1}^{2}}{f^{2}} \frac{(f_{E}/f_{1})n_{1}}{(f_{E}/f_{2})n_{2}} = \frac{f_{1}f_{2}}{f^{2}} \frac{n_{1}}{n_{2}} ,

which proves the second equation of (57) because lcm(f_{1}, f_{2}) = \frac{f_{1}f_{2}}{f}.

For later purposes we briefly consider how the constructions I(·) and H(·) behave under specializations.

Proposition 25 Let E/K be a CM elliptic curve defined over a number field and 𝔭 be a prime ideal of K with residue field k = 𝔒_{K}/𝔭. Assume that E has good reduction E_{k}/k at 𝔭. Then:

(a) E_{k}/k is a CM elliptic curve if and only if p = char(k) splits in F := End^{0}(E). If this is the case, then the conductor of E_{k} is f_{E_{k}} = f_{E}/p^{r}, where p^{r}||f_{E}.

(b) Assume that the condition of (a) holds, and that p ∤ f_{E}, so that the reduction map defines isomorphisms

r_{k} : End(E) →^{∼} End(E_{k}) and r_{k}^{0} : End^{0}(E) →^{∼} End^{0}(E_{k}).

If H is any finite subgroup scheme of E, then its reduction H_{k} is a subgroup scheme of the same rank, i.e. |H_{k}| = |H|, and we have

r_{k}(I(H)) ⊂ I(H_{k}) and r_{k}^{0}(E(H)) ⊂ E(H_{k}). (58)

Furthermore, both inclusions are equalities if H is an ideal subgroup or if (p, |H|) = 1. In addition, we have

(59) H(I)_{k} = H(r_{k}(I)), for all non-zero ideals I of End(E).

Proof. (a) Lang[18], Theorem 13.12 (p. 182).

(b) We first recall how the reduction map r_{k} is defined. Let 𝔒_{𝔭} = (𝔒_{K})_{𝔭} denote the local ring (valuation ring) at 𝔭, and let Ẽ/𝔒_{𝔭} denote the minimal (Néron) model of E/K (with respect to 𝔭). Then r_{k} is defined as the composition of the base-change maps

r_{k} = r_{k}^{E} : End(E) ←^{∼} End(Ẽ) → End(E_{k}).

Now let H be a finite subgroup scheme of E. To define H_{k}, we shall use the hypothesis that E/K has good reduction at 𝔭, i.e. that Ẽ/S := Spec(𝔒_{𝔭}) is an abelian scheme (of relative dimension 1). Then E_{H} ∼ E also has good reduction at 𝔭 (by the criterion of Néron-Ogg-Shafarevich or otherwise), and so π_{H} extends by the Néron property to a homomorphism π̃_{H} : Ẽ → Ẽ_{H}. By [4], 7.3/6, π̃_{H} is an isogeny. Moreover, since π_{H} is proper (because Ẽ_{H}/S is proper), it follows that π̃_{H} is finite and flat by (the argument of) [4], 7.3/2, and hence H̃ := Ker(π̃_{H}) is a subgroup scheme of Ẽ which is finite and flat over S and whose generic fibre is H. Thus, its special fibre H_{k} = H̃ ×_{S} Spec(k) is a finite subgroup scheme of E_{k} of the same rank as H, i.e. |H| = |H_{k}|. Note also that the above shows that (E_{k})_{H_{k}} is the reduction of E_{H}.

As a special case of this construction we see that

(60) Ker(g)_{k} = Ker(g̃) ×_{S} Spec(k) = Ker(r_{k}(g)), for all g ∈ End(E).

From this, equation (59) follows immediately because intersections (i.e. fibre products) commute with base-change.

To prove the first part of (58), let h ∈ I(H), so h = gπ_{H} with g ∈ Hom(E_{H}, E) and hence h̃ = g̃π̃_{H} and r_{k}(h) = h̃_{k} = g̃_{k}(π̃_{H})_{k} = g_{k}π_{H_{k}} ∈ I(H_{k}) because g_{k} ∈ Hom((E_{H})_{k}, E_{k}) (and (E_{H})_{k} = (E_{k})_{H_{k}}). This proves the first inclusion of (58).

The second inclusion follows immediately from the fact that

(61) r_{k}^{0}(Φ_{H}(h)) = Φ_{H_{k}}(r_{k}^{E_{H}}(h)), for all h ∈ End(E_{H}),

which in turn follows from the functoriality of the construction of r_{k}, for we have that n_{H}r_{k}^{0}(Φ_{H})(h) = r_{k}(π′_{H}hπ_{H}) = (π′_{H})_{k}h_{k}(π_{H})_{k} = π′_{H_{k}}r^{E_{H}}(h)π_{H_{k}} = n_{H_{k}}Φ_{H_{k}}(r_{k}^{E_{H}}(h)).

Note that it follows from (61) that

r_{k}^{0}(E(H)) = E(H_{k}) ⇔ p ∤ f_{E(H)} = f_{E_{H}} (62) because by part (a) we know that r_{k}^{E_{H}} is surjective if and only if p ∤ f_{E_{H}}.

Now suppose that H is an ideal subgroup scheme. Since f_{E_{H}}|f_{E} by (46), it follows from (62) that we have equality in the second part of (58). Moreover, since H = H(I) for some ideal I, we see from (59) that H_{k} = H(r_{k}(I)) is also an ideal subgroup. Put R = End(E). Since r_{k}(R) = End(E_{k}), we obtain from Corollary 23 that [R : I(H)] = |H| = |H_{k}| = [r_{k}(R) : I(H_{k})], and so we must have equality in the first part of (58) as well.

Finally, to prove that equality holds in (58) when (|H|, p) = 1, we first observe that although the map H ↦ H_{k} is not injective in general, we do have

H_{1} ≤ H_{2} ⇔ (H_{1})_{k} ≤ (H_{2})_{k}, if (|H_{1}|, p) = 1. (63)

To see this, recall first that we also have a reduction map r_{k}^{E(K)} : E(K) → E_{k}(k) on the group of rational points and that by base-change this extends to a map \overline{r}_{k} : E(\overline{K}) → E(\overline{k}) whose kernel has no non-trivial points of order prime to p = char(k). We thus have an isomorphism

\overline{r}_{k}^{(p)} : E(\overline{K})_{tor}^{(p)} →^{∼} E_{k}(\overline{k})_{tor}^{(p)},

where E(\overline{K})_{tor}^{(p)} denotes the group of torsion points in E(\overline{K}) of order prime to p, and E_{k}(\overline{k})_{tor}^{(p)} is defined similarly. Now if H is a finite group scheme of E with (|H|, p) = 1, then H̃ is an etale group scheme, and so we can identify H_{k} ⊗ \overline{k} and H ⊗ \overline{K} with subgroups of E_{k}(\overline{k}) and of E(\overline{K}) of the same order and we have \overline{r}_{k}^{(p)}(H ⊗\overline{K}) = H_{k} ⊗\overline{k}.

From this, the assertion (63) follows immediately, provided that also (p, |H_{2}|) = 1. To prove it without this hypothesis, note that we can reduce the assertion to this case as follows. Put n_{1} = |H_{1}| = |(H_{1})_{k}|. Since H_{1} ≤ Ker([n_{1}]), we see that H_{1} ≤ H_{2} ⇔ H_{1} ≤ H_{2}[n_{1}] := H_{2} ∩ Ker([n_{1}])), and similarly, (H_{1})_{k} ≤ (H_{2})_{k} ⇔ (H_{1})_{k} ≤ (H_{2})_{k}[n_{1}] = (H_{2}[n_{1}])_{k}. Thus, since p ∤ |H_{2}[n_{1}]|, it follows that (63) is true.

We are now ready to prove that equality holds in (58) when p ∤ |H|. Indeed, since f_{E(H)}|n^{2}f_{E} by (27), it follows from (62) that equality holds in the second part of (58). To prove this also for the first part, let g ∈ I(H_{k}), so H_{k} ≤ Ker(g). Since r_{k} : End(E) → End(E_{k}) is an isomorphism, there exists a unique g_{1} ∈ End(E) such that r_{k}(g_{1}) = g. By (60) we know that Ker(g_{1})_{k} = Ker(g), and so it follows from (63) that H ≤ Ker(g_{1}). Thus g_{1} ∈ I(H) and so g = r_{k}(g_{1}) ∈ r_{k}(I(H)). This shows that equality holds in the first part of (58) as well.

3.2 The case K = ℂ

In the case that K = ℂ, every elliptic curve E/ℂ has an analytic description, i.e. there exists a lattice L ⊂ ℂ and an isomorphism of compact Riemann surfaces E_{ℂ} ≃ ℂ/L, where E_{ℂ} denotes the compact Riemann surface associated to the (algebraic) curve E. As we shall see, it is very illuminating to relate the previous constructions to the lattices appearing in the complex theory.

To make this analytic description more precise, recall first that if L ⊂ ℂ is any lattice, then the existence of the Weierstrass ℘-function ℘_{L} shows that the Riemann surface ℂ/L can be identified with a unique elliptic curve E_{L} ⊂ ℙ^{2}(ℂ) (given by the Weierstrass equation y^{2} = 4x^{3} − g_{2}(L)x − g_{2}(L)) and that we have an isomorphism

ρ_{L} : ℂ/L →^{∼} E_{L}(ℂ) given by z + L ↦ (℘_{L}(z) : ℘′_{L}(z) : 1) ∈ ℙ^{2}(ℂ).

Conversely, given any E/ℂ, then E is isomorphic to a Weierstrass curve E′ ⊂ ℙ^{2}(ℂ), and for each such E′ : y^{2} = 4x^{3} −ax−b there is a unique lattice L such that g_{2}(L) = a and g_{2}(L) = b; cf. Cox[7], p. 224. In particular, E_{L} = E′ ≃ E.

Via this isomorphism we have a natural identification

Ψ_{L} : (L : L)_{ℂ} = {λ ∈ ℂ : λL ⊂ L} →^{∼} End(E_{L}) (64)

given by λ ↦ π_{λ}, where π_{λ} = π_{λ}^{L} ∈ End(E_{L}) is defined by π_{λ}(ρ_{L}(z +L)) = ρ_{L}(λz +L); cf. [18], §1.4.

From this one sees easily that E_{L} is a CM elliptic curve if and only if (L : L)_{ℂ} ≠ ℤ. If this is the case, then (L : L)_{ℂ} is an order in some (unique) imaginary quadratic field F ⊂ ℂ and L = λL_{0}, for some lattice L_{0} ⊂ F and λ ∈ ℂ^{×}. Since E_{L} ≃ E_{L_{0}}, we can and will henceforth assume that L = L_{0} ∈ Lat_{F}.

If E_{L} is CM elliptic curve with L ∈ Lat_{F}, then its finite subgroup schemes H can be identified with lattices L_{H} ⊃ L, as we shall now see. Via this identification we can determine I(H) and H(I) as follows.

Proposition 26 Let L ∈ Lat_{F}, where F is an imaginary quadratic field. For every lattice L′ ∈ Lat_{F} with L ⊂ L′, the group H_{L′} = ρ_{L}(L′/L) is a finite subgroup scheme of E_{L}, and conversely every finite subgroups scheme H of E_{L} is of the form H = H_{L_{H}}, for a unique lattice L_{H} ∈ Lat_{F} with L ⊂ L_{H}. Moreover,

(E_{L})_{H} ≃ E_{L_{H}}, E(H) ≃ R(L_{H}) and |H| = [L_{H} : L]. (65)

In addition, if I is a non-zero ideal of End(E_{L}), then we have

(66)  I(H) = Ψ_{L}((L : L_{H})),
(67)  H(I) = ρ_{L}((L : Ψ_{L}(I))/L).

Proof. Since every finite subgroup scheme of E_{L} is reduced (and etale), we can identify them with the (abstract) finite subgroups of E_{L}(ℂ) ≃ ℂ/L. Thus, if L′ ∈ Lat_{F} with L ≤ L′, then [L′ : L] < ∞ and so L′/L is a finite subgroup of ℂ/L. Clearly, every finite subgroup H of ℂ/L has the form H = L_{H}/L, for a unique subgroup L_{H} with L ≤ L_{H} ≤ ℂ. Since L_{H} ⊂ \frac{1}{n}L, where n = |H|, it follows that L_{H} ∈ Lat_{F}.

Since the inclusion L ⊂ L_{H} defines a surjective analytic homomorphism π : ℂ/L → ℂ/L_{H} with kernel L_{H}/L, it follows that (E_{L})_{H} ≃ E_{L_{H}} and |H| = |L_{H}/L| = [L_{H} : L]. Thus E(H) = Φ_{H}(End((E_{L})_{H})) ≃ End((E_{L})_{H}) ≃ End(E_{L_{H}}) = Ψ_{L_{H}}(R(L_{H})) ≃ R(L_{H}), which proves (65).

Let 0 ≠ λ ∈ R(L). Since Ker(π_{λ}) = ρ_{L}((\frac{1}{λ}L)/L), we see that λ ∈ Ψ_{L}^{−1}(I(H)) ⇔ ρ_{L}(L_{H}/L) ≤ Ker(π_{λ}) ⇔ L_{H} ≤ \frac{1}{λ}L ⇔ L_{H}λ ≤ L ⇔ λ ∈ (L : L_{H}), which proves (66).

To prove (67), put I′ = Ψ_{L}^{−1}(I). Then by definition

H(I) = ⋂_{λ∈I′} Ker(π_{λ}) = ⋂_{0≠λ∈I′} ρ_{L}((λ^{−1}L)/L) = ρ_{L}(H̃(I′)/L), where H̃(I′) = ⋂_{0≠λ∈I′} λ^{−1}L.

Now H̃(I′) = (L : I′) because x ∈ (L : I′) ⇔ xλ ∈ L, ∀λ ∈ I′, λ ≠ 0 ⇔ x ∈ ∩_{0≠λ∈I′}λ^{−1}L, and so (67) follows.

Corollary 27 If L, L′ ∈ Lat_{F} are two lattices, then I_{E_{L}}(E_{L′}) ≃ L(L′)^{−1}.

Proof. Since E_{L} ≃ E_{nL} and L(L′)^{−1} ≃ nL(L′)^{−1}, for any n ∈ ℕ, we may assume without loss of generality that L ⊂ L′. Put H = ρ_{L}(L′/L). Since (E_{L})_{H} ≃ E_{L′}, we have I_{E_{L}}(E_{L′}) ≃ I(H) = ψ_{L}((L : L′)) ≃ (L : L′) by (66). This proves the assertion because (L : L′) ≃ L(L′)^{−1} by (41).

We can use the above proposition to give not only a quick proof of Proposition 21 and of its Corollary 23 in the case that K = ℂ but also an important refinement of these results.

Proposition 28 Let E/K be a CM elliptic curve, where K is an arbitrary field. If R = End(E), then for any finite subgroup scheme H of E with char(K) ∤ |H|

(68)  R(I(H)) = E(H)R,
(69)  [R : I(H)] = [E(H)R : E(H)] · |H|.

Proof. By base-change (cf. Remark 7(e)), it is enough to consider the case that K = \overline{K} is algebraically closed. Moreover, since E ≃ E_{0} ⊗ K for some E_{0}/\overline{P}, where P ⊂ K is the prime subfield (and since all finite subgroups schemes are defined of \overline{P}), we can assume that either K = \overline{ℚ} or K = \overline{𝔽}_{p}. Case 1: K = \overline{ℚ}. By base-change again, we can assume here that K = ℂ. Then E ≃ E_{L}, for some lattice L ∈ Lat_{F} and some imaginary quadratic field F, and H = ρ_{L}(L_{H}/L), for some L_{H} ∈ Lat_{F} with L ⊂ L_{H}.

Put L_{1} = L and L_{2} = L_{H}, and R_{i} = R(L_{i}) and R′ = R_{1}R_{2}. Then R_{1} = Ψ_{L}^{−1}(R) ≃ R and R_{2} ≃ E(H) by (65). Thus [E(H)R : E(H)] = [R′ : R_{2}] = \frac{f_{2}}{f}, where f_{i} = f_{R_{i}} and f = f_{R′} = (f_{1}, f_{2}). Now by (66) and (41) we have Ψ_{L}^{−1}(I(H)) = (L_{1} : L_{2}) = \frac{f_{1}}{f} L_{1}L_{2}^{−1}.

From this we see on the one hand that R((L_{1} : L_{2})) = R(L_{1})R(L_{2}^{−1}) = R′ by (39), which proves (68). On the other hand we obtain

\frac{f_{1}}{f} [R : I(H)] = [R′ : (L_{1} : L_{2})] = N(\frac{f_{1}}{f} L_{1}L_{2}^{−1}) = \frac{f_{1}^{2}}{f^{2}} N(L_{1})N(L_{2})^{−1}.

But since L_{1} ⊂ L_{2} we have N(L_{1})N(L_{2})^{−1} = \frac{f_{2}}{f_{1}} [L_{2} : L_{1}] because if we choose n ∈ ℕ such that nL_{2} ⊂ 𝔒_{F}, then \frac{f_{1}}{f_{2}} N(L_{1})N(L_{2})^{−1} = \frac{f_{1}}{f_{2}} N(nL_{1})N(nL_{2})^{−1} = [𝔒_{F} : nL_{1}][𝔒_{F} : nL_{2}]^{−1} = [nL_{2} : nL_{1}] = [L_{2} : L_{1}]. Thus \frac{f_{1}}{f} [R : I(H)] = \frac{f_{1}f_{2}}{f^{2}} [L_{2} : L_{1}], and so (69) follows because |H| = [L_{2} : L_{1}] by (65). Case 2: K = \overline{𝔽}_{p} By the Deuring Lifting Theorem ([18], p. 184) there is a number field K′/ℚ and a CM elliptic curve E′/K′ and a prime 𝔭|p of K′ such that E′ has good reduction E′_{k} at 𝔭 and E′_{k} ⊗ \overline{𝔽}_{p} ≃ E and such that the reduction map induces an isomorphism r_{k} : End(E′) →^{∼} End(E′_{k}) ≃ End(E), where k = 𝔒_{K′}/𝔭 is the residue field. Note that we have that p ∤ f_{E′}; cf. Proposition 25(a).

By enlarging the ground field K′ if necessary, we may assume without of loss of generality that the n-torsion points of E are K′-rational, where n = |H|. Then the n-torsion points of E′_{k} are k-rational, and so we can identify H with a subgroup of E′_{k}(k) of order n, which therefore lifts uniquely to a subgroup H′ of E′(K′) which we view as a subgroup scheme of E′ of rank n. Thus (H′)_{k} = H in the notation of Proposition 25(b), and so by Proposition 25 we have

I(H) = r_{k}(I(H′)) and r_{k}^{0}(E(H′)) = E(H).

Since (68) is true for H′ by Case 1, it follows from the above that

R(I(H)) = r_{k}^{0}(R(I(H′))) = r_{k}^{0}(E(H′)R′) = E(H)R,

where R′ := End(E′) and R := r_{k}(R′) = End(E′_{k}). Thus (68) holds for H.

Similarly, from the above we obtain that [R′ : I(H′)] = [R : I(H)] and also that [E(H′)R′ : E(H′)] = [E(H)R : E(H)]. Thus, since (69) is true for H′ by Case 1, it follows that (69) also holds for H because

[R : I(H)] = [R′ : I(H′)] = [E(H′)R′ : E(H′)]|H′| = [E(H)R : E(H)]|H|.

The above formulae (68) and (69) allow us to prove the following useful fact.

Proposition 29 In the situation of Proposition 28, let H_{1} and H_{2} be two finite subgroups of E with (|H_{1}|, |H_{2}|) = 1 and E(H_{i}) ⊂ R, for i = 1, 2. If char(K)) ∤ |H_{i}|, for i = 1, 2, then

E(H_{1} + H_{2}) = E(H_{1}) ∩ E(H_{2}).

Proof. First note that for any two finite subgroups H_{1}, H_{2} of E we have

I(H_{1})I(H_{2}) ⊂ I(H_{1} + H_{2}) ⊂ I(H_{1}) ∩ I(H_{2}). (70)

Indeed, since H_{i} ≤ H_{1} + H_{2}, we have I(H_{1} + H_{2}) ⊂ I(H_{i}) by (6) and so the second inclusion of (70) follows. To prove the first inclusion, let f_{i} ∈ I(H_{i}), so H_{i} ≤ Ker(f_{i}). Since f_{1}f_{2} = f_{2}f_{1}, we have H_{i} ≤ Ker(f_{i}) ≤ Ker(f_{1}f_{2}) and so H_{1} + H_{2} ≤ Ker(f_{1}f_{2}). Thus f_{1}f_{2} ∈ I(H_{1} + H_{2}), and so the first inclusion of (70) holds.

Put n_{i} = |H_{i}|, e_{i} = [R : E(H_{i})] and N_{i} = [R : I(H_{i})], so N_{i} = e_{i}n_{i} by (69) (together with the hypothesis that E(H_{i}) ⊂ R). Since H_{i} ≤ Ker([n_{i}]), we have n_{i} ∈ I(H_{i}), and so Rn_{i} ⊂ I(H_{i}), which implies that N_{i} = [R : I(H_{i})]|[R : Rn_{i}] = n_{i}^{2}. Thus, since (n_{1}, n_{2}) = 1, we see that also (N_{1}, N_{2}) = 1 and hence that (e_{1}, e_{2}) = 1.

In addition we have that I(H_{1}) + I(H_{2}) = R because n_{i} ∈ I(H_{i}) and (n_{1}, n_{2}) = 1. Thus, by elementary ideal theory we have that I(H_{1})I(H_{2}) = I(H_{1}) ∩ I(H_{2}), and so equality holds throughout in (70). Using (68) and (39), we obtain for H = H_{1} + H_{2} that

E(H)R = R(I(H)) = R(I(H_{1})I(H_{2})) = R(I(H_{1}))R(I(H_{2})) = R · R = R,

because R(I(H_{i})) = RE(H_{i}) = R by (68) and the hypothesis. Thus E(H_{1} + H_{2}) ⊂ R.

Next we note that H_{1}∩H_{2} = 0 (because |H_{1}∩H_{2}| | (n_{1}, n_{2}) = 1), and so |H_{1}+H_{2}| = n_{1}n_{2}. Moreover, since I(H_{1} + H_{2}) = I(H_{1}) ∩ I(H_{2}), we see that [R : I(H_{1} + H_{2})] = [R : I(H_{1}) + I(H_{2})] = N_{1}N_{2}, the latter because I(H_{1}) + I(H_{2}) = R. Thus, by (69) (applied to H = H_{1} + H_{2}) we obtain n_{1}e_{1}n_{2}e_{2} = N_{1}N_{2} = [RE(H) : E(H)]n_{1}n_{2}, and so [R : E(H)] = e_{1}e_{2}. Thus, if f = f_{R}, then E(H) has conductor fe_{1}e_{2} = lcm(fe_{1}, fe_{2}) because (e_{1}, e_{2}) = 1. Since f_{E(H_{i})} = fe_{i}, it thus follows from (37) that E(H) = E(H_{1}) ∩ E(H_{2}), as claimed.

Coming back to the case that E = E_{L}, we now generalize the above formula (66) for I(H) by giving a similar interpretation of the lattice H(H_{1}, H_{2}).

For this, we first observe that the identification map Ψ_{L} of (64) can be generalized as follows. If L_{1}, L_{2} ∈ ℂ are any two lattices, then by [18], §1.4, we have a bijection

Ψ_{L_{1},L_{2}} : (L_{2} : L_{1})_{ℂ} →^{∼} Hom(E_{L_{1}}, E_{L_{2}}) (71)

which is given by the rule λ ↦ π_{λ} = π_{λ}^{L_{1},L_{2}} ∈ Hom(E_{L_{1}}, E_{L_{2}}), where π_{λ} is defined by

π_{λ}(ρ_{L_{1}}(z + L_{1})) = ρ_{L_{2}}(λz + L_{2}), for λ ∈ (L_{2} : L_{1})_{ℂ} = {λ ∈ ℂ : λL_{1} ⊂ L_{2}}.

Note that these maps Ψ are multiplicative in the sense that if L_{3} is a third lattice, then we have

π_{λ_{2}λ_{1}}^{L_{1},L_{3}} = π_{λ_{2}}^{L_{2},L_{3}} ∘ π_{λ_{1}}^{L_{1},L_{2}}, if λ_{1} ∈ (L_{2} : L_{1})_{ℂ} and λ_{2} ∈ (L_{3} : L_{2})_{ℂ}. (72)

We now prove the following generalization of (66).

Proposition 30 Let L ∈ Lat_{F}, where F is an imaginary quadratic field, and let H_{i} = ρ_{L}(L_{H_{i}}/L) be two finite subgroups of E_{L}. Then we have

H(H_{1}, H_{2}) = Ψ_{L}^{0}((L_{H_{2}} : L_{H_{1}})), (73)

where Ψ_{L}^{0} : 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_{1}^{L,L_{H}} and π′_{H} = π_{n_{H}}^{L_{H},L}, where n_{H} = |H|.

Indeed, since π_{1}^{L,L_{H}} : E_{L} → E_{L_{H}} is surjective and has kernel H = ρ_{L}(L_{H}/L), we see that (E_{L_{H}}, π_{1}^{L,L_{H}}) ≃ ((E_{L})_{H}, π_{H}) is a quotient of E_{L} by H. Thus, the first equality of (74) holds. Moreover, since n_{H} = [L_{H} : L] by (65), we have n_{H}L_{H} ⊂ L and so n_{H} ∈ (L : L_{H}). Thus, π_{n_{H}}^{L_{H},L} exists, and by (72) we have π_{n_{H}}^{L_{H},L} ∘ π_{H} = π_{n_{H}}^{L,L} = [n_{H}]_{E_{L}}, and so π′_{H} = π_{n_{H}}^{L_{H},L}; cf. (4).

We next observe that

(75) Φ_{H_{1},H_{2}}(Ψ_{L_{H_{1}},L_{H_{2}}}(λ)) = Ψ_{L}^{0}(λ), for all λ ∈ (L_{H_{2}} : L_{H_{1}})_{ℂ}.

Indeed, from the definitions and (74) and (72) we obtain

Φ_{H_{1},H_{2}}(Ψ_{L_{H_{1}},L_{H_{2}}}(λ)) = \frac{1}{n_{H}} π′_{H}π_{λ}^{L_{H_{1}},L_{H_{2}}}π_{H} = \frac{1}{n_{H}} π_{n_{H}λ}^{L,L} = \frac{1}{n_{H}} Ψ_{L}(n_{H}λ) = Ψ_{L}^{0}(λ),

which proves (75). From this, equation (73) follows immediately because Φ_{H_{1},H_{2}} ∘ Ψ_{L_{H_{1}},L_{H_{2}}} is an isomorphism from (L_{H_{2}} : L_{H_{2}}) to H(H_{1}, H_{2}); cf. (71) and Remark 9(a).

Remark 31 (a) It follows from (73), (41) and (65) that

(ψ_{L}^{0})^{−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} (which defines 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 any two finite subgroup schemes H_{1}, H_{2} of E we have that

(I(H_{1}) : I(H_{2})) = \frac{[RR_{1}R_{2} : RR_{2}]}{[R_{1}R_{2} : R_{2}]}RH(H_{1}, H_{2}), (76)

where R = End(E) and R_{i} = E(H_{i}). Indeed, as in the proof of Corollary we can assume 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)):

((L : L_{2}) : (L : L_{1})) = [RR_{1}R_{2} : RR_{2}]RL_{1}L_{2}^{−1} = \frac{[RR_{1}R_{2} : RR_{2}]}{[R_{1}R_{2} : R_{2}]}(L_{1} : L_{2})R, (77)

in which R = R(L) and R_{i} = R(L_{i}).

3.3 Endomorphism rings

In the sequel it is sometimes useful to know which orders of an imaginary quadratic field F can be endomorphism rings of CM elliptic curves E/K or, more precisely, to describe the set of conductors f_{E′} for E′ ∈ Isog(E/K). Note that the answer for the corresponding question for the subset Isog^{+}(E/K) follows from Corollary 19: an order R of F = End^{0}(E) is the endomorphism ring of some E′ ∈ Isog^{+}(E/K) if and only if we have f_{R}|f_{E}. However, the characterization of the set of conductors in Isog(E/K) is more delicate and depends on the nature of the ground field K.

We begin with the case that K is algebraically closed. Here the set of CM curves in Isog(E/K) can be characterized as follows.

Proposition 32 Let K be an algebraically closed field, and let E_{1}/K and E_{2}/K be two CM curves. Then E_{1} ∼ E_{2} if and only if End^{0}(E_{1}) ≃ End^{0}(E_{2}).

Proof. Clearly, if E_{1} ∼ E_{2}, then End^{0}(E_{1}) ≃ End^{0}(E_{2}). To prove the converse, suppose first that K = ℂ. If End^{0}(E_{1}) ≃ End^{0}(E_{2}) =: F, then by the discussion of the previous section we know that E_{i} ≃ E_{L_{i}}, for some L_{i} ∈ Lat_{F}, and then Hom(E_{1}, E_{2}) ≃ (L_{2} : L_{1}) ≠ 0. Thus E_{1} ∼ E_{2}.

From this, the assertion follows for an arbitrary field K of characteristic 0. Indeed, any two CM curves over K are defined over \overline{ℚ} (cf. [18], p. 40), i.e. E_{i} = E_{i}^{0} ⊗ K for some E_{i}^{0}/\overline{ℚ}. Since F = End(E_{i}) = End(E_{i}^{0}) = End(E_{i}^{0} ⊗ ℂ), and Hom(E_{1}^{0}, E_{2}^{0}) = Hom(E_{1}^{0} ⊗ ℂ, E_{2}^{0} ⊗ ℂ), we conclude from what was just proved that E_{1}^{0} ∼ E_{2}^{0} and hence also E_{1} ∼ E_{2}.

Now suppose that char(K) = p ≠ 0. Then E_{i} = E_{i}^{0} ⊗ K, for some CM curves E_{i}^{0}/𝔽_{p}^{r} (and some r ≥ 1); cf. [18], p. 184. By the Deuring Lifting Theorem ([18], p. 184), there exists a number field L and CM elliptic curves Ẽ_{i}/L whose reduction is E_{i}^{0} and such that End^{0}(Ẽ_{i}) = End^{0}(E_{i}^{0}). By the characteristic 0 result, there is a finite extension L′/L such that Ẽ_{1} ⊗ L′ ∼ Ẽ_{2} ⊗ L′, and hence E_{1}^{0} ⊗ \overline{𝔽}_{p} ∼ E_{2}^{0} ⊗ \overline{𝔽}_{p}, and hence also E_{1} ∼ E_{2}.

In view of the previous result, the following result classifies the possible conductors of elliptic curves in Isom(E/K) (when K is algebraically closed).

Proposition 33 Let K be an algebraically closed field and let Δ < 0 be a discriminant.

(a) If char(K) = 0, then there exists a CM elliptic curve E/K with Δ_{E} = Δ.

(b) If char(K) = p ≠ 0, then there exists a CM elliptic curve E/K with Δ_{E} = Δ if and only if (\frac{Δ}{p}) = 1.

Proof. (a) This follows easily from the complex theory (together with the reduction steps as in the proof of Proposition 32); cf. Deuring[10], p. 263.

(b) Deuring[10], p. 263. Note that the condition (\frac{Δ}{p}) = 1 is equivalent to the following two conditions: (i) p splits in F = ℚ(\sqrt{Δ}) and (ii) p ∤ f_{R_{Δ}}.

If K is a finite field, then the results of Waterhouse[26] furnish a solution to the above question.

Proposition 34 Let K = 𝔽_{q} be a finite field and let E/𝔽_{q} be a CM elliptic curve. Let

h_{E/K}(X) = X^{2} − a_{E/F} X + q

be the characteristic polynomial of the Frobenius endomorphism of E/K, and put Δ(h_{E/K}) = a_{E/K}^{2} − 4q. If Δ ∈ ℤ, then there is an elliptic curve E′ ≃ E with Δ_{E′} = Δ if and only if Δ(h_{E/K})/Δ = n^{2}, for some n ∈ ℕ.

Proof. Let ϕ_{E/K} ∈ R := End(E) denote the Frobenius endomorphism of E/K, and let R′ ⊂ R̃ = End^{0}(E) be an order. Then by [26], Theorem 4.2, there is an elliptic curve E′/K with E′ ∼ E and End(E′) ≃ R′ if and only if ϕ_{E/K} ∈ R′. Since ℤ[ϕ_{E/K}] is a order of discriminant Δ(h_{E/K}), we see that the latter condition is equivalent to Δ(h_{E/K}) = Δn^{2} (with n = [R′ : ℤ[ϕ_{E/K}]).

For fields of characteristic 0, we have the following partial analogue of Proposition 34. Here we shall use the modular function j which is defined on the upper half plane ℌ and which can be viewed as a function on the set of lattices of ℂ by the rule j(L) = j(ω_{1}/ω_{2}), if L = ℤω_{1} + ℤω_{2} ⊂ ℂ with Im(ω_{1}/ω_{2}) > 0.

Proposition 35 Let K ⊂ ℂ, and let R be a quadratic lattice of discriminant Δ = Δ(R) < 0. Then there is a CM elliptic curve E/K with End(E) ≃ R if and only if ℚ(\sqrt{Δ}, j(R)) ⊂ K.

Proof. Suppose first that E/K is a CM elliptic curve with End(E) ≃ R. Put F = End^{0}(E) = ℚ(\sqrt{Δ}). Then by §3.2 there is a lattice L ∈ Lat_{F} such that E ⊗ ℂ ≃ E_{L}, and R(L) ≃ R. Then j(L) = j_{E} ∈ K. Moreover, by [18], DIFF 3 (p. 119), we know that F ⊂ K (because End(E) = End(E_{L})), and so F (j(L)) ⊂ K. But F (j(L)) = F (j(R)) by the First Main Theorem of complex multiplication (cf. [7], Theorem 11.1), and so ℚ(\sqrt{Δ}, j(R)) = F (j(L)) ⊂ K.

Conversely, suppose that ℚ(\sqrt{Δ}, j(R)) ⊂ K. Since R is a lattice in F := ℚ(\sqrt{Δ}) and since j(R) ∈ K, there is an elliptic curve E/K such that E ⊗ ℂ = E_{R}. Since End(E_{R}) ≃ R ⊂ F and F ⊂ K, it follows from [18], DIFF 3, again that End(E) = End(E_{R}) ≃ R. Thus, E/K is a CM curve with End(E) ≃ R.

The following result shows that the set of conductors appearing in Isog(E/K) forms a sublattice of ℕ with respect to the partial order f_{E}|f_{E′}.

Proposition 36 Let E/K be a CM elliptic curve. If E_{1}, …, E_{n} ∈ Isog(E/K), then there is an elliptic curve E′ ∼ E such that f_{E′} = lcm(f_{E_{1}}, …, f_{E_{n}}).

Proof. By induction, it is clearly enough to verify the case n = 2.

Since E_{i} ∼ E, we can view R_{i} := End(E_{i}) as a subring of F = End^{0}(E) which is uniquely determined by the condition that [𝔒_{F} : R_{i}] = f_{E_{i}}. Put R′ = R_{1}R_{2} and f_{i} = [R′ : R_{i}]. Since R′ has conductor f = (f_{E_{1}}, f_{E_{2}}) by (37), we see that (f_{1}, f_{2}) = 1. Note that char(K) ∤ f_{i} = 1 by Proposition 33(b).

Next, consider the R_{i}-ideal I_{i} = f_{i}R′ and put π_{i} = π_{H(I_{i})} : E_{i} → E′_{i} = (E_{i})_{H(I_{i})}. Since by (45) we have E(H(I_{i})) ≃ R(I_{i}) = R(f_{i}R′) = R′, it follows from Remark 20 that there is an invertible R′-ideal I such that (E′_{1})_{H(I)} ≃ E′_{2}. Furthermore, by replacing I by cI if necessary (where c ∈ F^{×}), we can assume that (N(I), f_{1}) = 1 and that char(K) ∤ N(I).

Put H_{1} = Ker(π′_{1}) and H_{2} = Ker(π_{2} ∘ π_{H(I)}). Clearly (E′_{1})_{H_{i}} ≃ E_{i}, and so E(H_{i}) ≃ R_{i} ⊂ R′ ≃ End(E′_{1}). Furthermore, |H_{1}| = deg(π′_{1}) = deg(π_{1}) = f_{1}, and |H_{2}| = deg(π′_{2}) deg(π_{H(I)}) = f_{2}N(I), so (|H_{1}|, |H_{2}|) = 1. In addition, we have that char(K) ∤ |H_{i}|. Thus, by Proposition 29 we have E(H_{1} + H_{2}) = E(H_{1}) ∩ E(H_{2}) = R_{1} ∩ R_{2}, which has conductor lcm(f_{E_{1}}, f_{E_{2}}) by (37). Thus (E′_{1})_{H_{1}+H_{2}} has conductor lcm(f_{E_{1}}, f_{E_{2}}), and so the assertion follows because E ∼ E′_{1} ∼ (E′_{1})_{H_{1}+H_{2}}.

3.4 The quadratic form q_{E_{1},E_{2}}

Let E_{1}/K and E_{2}/K be any two isogenous elliptic curves, and put

q_{E_{1},E_{2}}(f) = deg(f), for f ∈ Hom(E_{1}, E_{2}).

Since Hom(E_{1}, E_{2}) ≃ ℤ^{r}, where r = rank(Hom(E_{1}, E_{2})) = dim_{ℚ}(End^{0}(E_{i})), we see that by fixing a basis of Hom(E_{1}, E_{2}), we obtain an explicit positive definite quadratic form in r variables. Thus, by varying over all bases of Hom(E_{1}, E_{2}), we obtain a GL_{r}(ℤ)-equivalence class of quadratic forms in r variables.

In the case that E_{i} is a CM elliptic curve, we have r = [F : ℚ] = 2, so q_{E_{1},E_{2}} defines an equivalence class of positive binary quadratic forms, i.e. q_{E_{1},E_{2}} ∼ ax^{2}+bxy+cy =: q, for some a, b, c ∈ ℤ with Δ(q) = b^{2} − 4ac < 0. Note that the discriminant Δ(q) and the content cont(q) = gcd(a, b, c) are invariants of the GL_{2}(ℤ)-equivalence class of q.

In order to determine q_{E_{1},E_{2}}, we introduce the following notation. Given a lattice L ∈ Lat_{F}, where F is an imaginary quadratic field, put

q_{L}(λ) = \frac{N(λ)}{N(L)}, for λ ∈ L,

where, as before, N(λ) = N_{F/ℚ}(λ) denotes the field norm. Note that q_{L}(λ) ∈ ℤ; cf. [3], §II.7. Thus, by choosing a basis {α, β} of L = ℤα + ℤβ, the map q_{L} defines an integral quadratic form

q_{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) Δ(q_{L}) = Δ(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_{E_{i}}, then

(79) q_{E_{1},E_{2}} ∼ cq_{L}, where L = I_{E}(E_{1})I_{E}(E_{2})^{−1} and c = \frac{lcm(f_{1}, f_{2})}{gcd(f_{1}, f_{2})}.

In particular, c = cont(q_{E_{1},E_{2}}) and

(80) Δ(q_{E_{1},E_{2}}) = −lcm(|Δ_{E_{1}}|, |Δ_{E_{2}}|) = lcm(f_{1}, f_{2})^{2}Δ_{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_{H_{1},H_{2}} and H = H(H_{1}, H_{2}). We first show that

(81) q_{E_{1},E_{2}}(h) = cq_{H}(Φ(h)), for all h ∈ Hom(E_{1}, E_{2});

here we used the identification E_{i} = E_{H_{i}}.

Since the H_{i}’s are ideal subgroups by (46), we have by (57) that N(L) = c\frac{n_{1}}{n_{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(\frac{1}{n_{2}} π_{2}hπ_{1}) = \frac{1}{n_{2}^{2}} deg(π_{2}hπ_{1}) = \frac{1}{n_{2}^{2}} deg(π_{2}) deg(h) deg(π_{1}) = \frac{n_{1}}{n_{2}} deg(h), and so

q_{E_{1},E_{2}}(h) = deg(h) = \frac{n_{2}}{n_{1}} N(Φ(h)) = c\frac{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})) = \frac{f_{1}}{f} I(H_{1})I(H_{2})^{−1} = \frac{f_{1}}{f} L, and so q_{H} ∼ q_{L}.

Since q_{L} is primitive by (78), we have from (79) that cont(q_{E_{1},E_{2}}) = c · cont(q_{L}) = c. Moreover, by (78) we have that Δ(q_{L}) = Δ(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 Δ(q_{L}) = f^{2}Δ_{F}. Thus we see that Δ(q_{E_{1},E_{2}}) = c^{2}Δ(q_{L}) = (cf)^{2}Δ(f) = lcm(f_{1}, f_{2})^{2}Δ_{F}, and so (80) follows.

Remark 38 If L ∈ Lat_{F} is any lattice, then we had seen above that q_{L} naturally defines a GL_{2}(ℤ)-equivalence class of positive binary quadratic forms. As is wellknown, one can also associate to L an SL_{2}(ℤ)-equivalence class of forms by restricting the set {q_{L,α,β}} to those forms that arise from oriented bases {α, β} of L, i.e. those for which Im(β/α) > 0 (where we view F ⊂ ℂ). Thus, if we write cq_{L}^{+} = {cq_{L,α,β} : L = ℤα + ℤβ, Im(β/α) > 0}, for c ∈ ℕ and L ∈ Lat_{F}, then it is well-known that the rule I ↦ q̃_{I}^{+} := [R(I) : R_{Δ}]q_{I}^{+} induces a bijection

q_{Δ} : Id(R_{Δ})/≃ →^{∼} Q_{Δ}/SL_{2}(ℤ)

between the set of isomorphism classes of non-zero ideals of the order R_{Δ} of discriminant Δ < 0 and the set of proper equivalence classes of positive binary quadratic forms of discriminant Δ.

Now if we combine this bijection with the bijection I_{E}^{+} defined in Corollary 19, then we obtain a bijection

q_{E}^{+} : Isog^{+}(E/K) →^{∼} Id(End(E))/≃ →^{∼} Q_{Δ_{E}}/SL_{2}(ℤ)

which is given by the formula

q_{E,E′}^{+} := q_{E}^{+}(E′) = q_{Δ_{E}}(I_{E}(E′)) = [R(I_{E}(E′)) : R_{E}]q_{I_{E}(E′)}^{+} = \frac{f_{E}}{f_{E′}} q_{I_{E}(E′)}^{+}, (82)

where the last equality follows from the second equation of (50).

On the other hand, if f_{E′}|f_{E}, then equation (79) tells us that

q_{E,E′} ∼ q_{E′,E} ∼ \frac{f_{E}}{f_{E′}} q_{I_{E}(E′)} and Δ(q_{E′,E}) = Δ_{E}.

Comparing this with (82), we therefore obtain the important relation that

q_{E,E′} ∼ q_{E,E′}^{+}, for E′ ∈ Isog^{+}(E/K), (83)

where, as before, the symbol ∼ (for quadratic forms) means GL_{2}(ℤ)-equivalence. Note, however, that q_{E,E′}^{+} denotes a proper (or SL_{2}(ℤ))-equivalence class of quadratic forms and hence is a finer invariant than the GL_{2}(ℤ)-equivalence class q_{E,E′}. In fact, it follows from the above that if E′, E′′ ∈ Isog^{+}(E/K), then we have that

q_{E,E′} ∼ q_{E,E′′} ⇔ I_{E}(E′′) ≃ I_{E}(E′) or I_{E}(E′′) ≃ I_{E}(E′)^{−1},

and so there are two non-isomorphic elliptic curves in Isog^{+}(E/K) which have the same form q, except when q is ambiguous, i.e. when I_{E}(E′) ≃ I_{E}(E′)^{−1}.

Corollary 39 Let E_{1}/K and E_{2}/K be two isogenous a CM elliptic curves with End^{0}(E_{i}) ≃ F. If f_{i} = f_{E_{i}}, then

(84) Δ(q_{E_{1},E_{2}}) = lcm(f_{1}, f_{2})^{2}Δ_{F} and cont(q_{E_{1},E_{2}}) = \frac{lcm(f_{1}, f_{2})}{gcd(f_{1}, f_{2})}.

Proof. By Proposition 36 there is a CM elliptic curve E such that E ∼ E_{i} and f_{E_{i}}|f_{E}, for i = 1, 2. Thus E_{i} ∈ Isog^{+}(E), for i = 1, 2, and so the assertion follows from Proposition 37.

4 Product abelian varieties

4.1 Kernel ideals and ideal subgroups of A^{n}

Let A = A_{1} × A_{2} × … × A_{n} be the product of the abelian varieties A_{1}, …, A_{n}/K, and let p_{i}^{A} : A → A_{i} denote the ith projection and e_{j}^{A} : A_{j} → A be the jth inclusion map. If A′ = A′_{1} × A′_{2} × … × A′_{m} is another product abelian variety, then (as is well-known) the group Hom(A, A′) can be identified with a set of m×n “matrices”. More precisely, we have the isomorphism

T_{A,A′} : Hom(A, A′) →^{∼} M(A, A′) := ⨁_{i=1}^{m} ⨁_{j=1}^{n} Hom(A_{j}, A′_{i})

given by the rule T_{A,A′}(h) = (h_{ij}), where h_{ij} = p_{i}^{A′} ∘ h ∘ e_{j}^{A} ∈ Hom(A_{j}, A′_{i}). We shall refer to the elements of M(A, A′) as “matrices”. Note that this identification is multiplicative in the sense that if A′′ = A′′_{1} × … × A′′_{t} is another abelian product, then we have the rule

(85) T_{A,A′′}(h′ ∘ h) = T_{A′,A′′}(h′) · T_{A,A′}(h), if h ∈ Hom(A, A′), h′ ∈ Hom(A′, A′′),

where the product on the right hand side is the product of “matrices” which is defined by the rule (h′_{ik})(h_{kj}) = (h′′_{ij}), where h′′_{ij} = ∑_{k} h′_{ik} ∘ h_{kj}. This follows easily from the identity ∑_{k=1}^{n} e_{k}^{A′}p_{k}^{A′} = 1_{A′}. In particular, if A = A_{1}^{n}, then T_{A,A} defines a ring isomorphism

T_{A_{1},n} = T_{A,A} : End(A_{1}^{n}) →^{∼} M(A_{1}^{n}, A_{1}^{n}) = M_{n}(End(A_{1}))

between End(A_{1}^{n}) and the ring of n × n matrices with coefficients in the ring End(A_{1}).

In order to study abelian varieties which are isogenous to A = A_{1}^{n}, we shall use the theory of kernel ideals and ideal subgroups of section 2. For this, we need to understand the ideals of M_{n}(R), where R = End(A_{1}). To construct such ideals, we shall use the following notation.

Notation. Let R be a ring. If α = (α_{ij}) ∈ M_{n}(R) is an n × n matrix, then we let α_{i} = (α_{i1}, …, α_{in}) ∈ R^{n} denote the ith row of α. Moreover, if M is subset of R^{n}, then we put

I_{n}(M) = I_{R,n}(M) = {α ∈ M_{n}(R) : α_{i} ∈ M, for 1 ≤ i ≤ n}.

Proposition 40 The rule M ↦ I_{n}(M) induces an inclusion preserving bijection between the set of left R-submodules of R^{n} and the set of left ideals of M_{n}(R). Furthermore, if M_{1} and M_{2} are two R-submodules of R^{n}, then

M_{1} ≃ M_{2} as R-modules ⇔ I_{n}(M_{1}) ≃ I_{n}(M_{2}) as M_{n}(R)-modules.

Proof. This follows almost immediately from the fact that {R, M_{n}(R), R^{n}, (R^{n})^{∗}} is a Morita context; cf. Curtis-Reiner[8], p. 64. Indeed, if P = R^{n}, with standard basis \underline{x} = \{x_{1}, …, x_{n}\}, then we have (as in [8]) the identification τ\underline{x} : End_{R}(P)^{op} →^{∼} M_{n}(R) (which is given by τ\underline{x}(α) = (a_{ij}), where x_{i}α = ∑_{j} a_{ij}x_{j}). Furthermore, by Morita (or otherwise) we have the canonical identification θ : P^{∗} ⊗_{R} P →^{∼} End_{R}(P)^{op} which is defined by θ(x^{∗} ⊗ y)z = x^{∗}(z)y for x^{∗} ∈ P^{∗} = Hom_{R}(P, R) and y, z ∈ P. Now if {x_{i}^{∗}} denotes the dual basis of P^{∗} (with respect to the basis {x_{i}}), then θ(x_{i}^{∗} ⊗ x_{j}) = ε_{ij}, where ε_{ij} = (δ_{ik}δ_{jl})_{kl} denotes the matrix whose (i, j)-th entry is 1 and is 0 otherwise. From this it is immediate that

θ(P^{∗} ⊗_{R} M) = I_{n}(M),

and so all the assertions (and more) follow from the Morita Theorem ([8], p. 60).

We now apply this to study kernel ideals associated to the product abelian variety A^{n}, where A is a fixed abelian variety. Recall from above that we have a canonical identification

T_{A,n} : End(A^{n}) →^{∼} M_{n}(R), where R := End(A).

For what follows, it is useful to introduce the following abbreviation. If I_{1}, …, I_{n} are left End(A)-ideals, then we write

(I_{1}| … |I_{n}) := T_{A,n}^{−1}(I_{n}(I_{1} ⊕ … ⊕ I_{n})),

which is a left End(A^{n})-ideal. Note that we have that

T_{A,n}((I_{1}|I_{2}| … |I_{n})) = {(α_{ij}) ∈ M_{n}(R) : α_{ij} ∈ I_{j}, for 1 ≤ i, j ≤ n}. (86)

Proposition 41 If H_{1}, …, H_{n} are finite subgroup schemes of A, then

I(H_{1} × H_{2} × … × H_{n}) = (I(H_{1})|I(H_{2})| … |I(H_{n})). (87)

Thus, if I_{1}, …, I_{n} are kernel ideals, then (I_{1}| … |I_{n}) is also a kernel ideal.

Proof. Write π_{i} = π_{H_{i}} : A → A_{i} := A_{H_{i}} and π = π_{1} × … × π_{n} : A^{n} → A_{1} × … × A_{n}. Since Ker(π) = H := H_{1} × … × H_{n}, we can identify π_{H_{1}×...×H_{n}} with π. Thus I(H) = Hom(A_{H}^{n}, A^{n})π.

Write T = T_{A,n}, T′ = T_{A^{n},A_{H}^{n}} and T′′ = T_{A_{H}^{n},A^{n}}. Since T′(π) = diag(π_{1}, …, π_{n}) is the diagonal “matrix” with entries π_{i} ∈ Hom(A, A_{i}), we see that if f ∈ End(A^{n}), then f ∈ I(H) ⇔ f = f′π, for some f′ ∈ Hom(A_{H}^{n}, A^{n}) ⇔ T (f) = T′′(f′)T′(π), for some T′′(f′) ∈ M(A_{H}^{n}, A^{n}). Thus, if we write T (f) = (α_{ij}) with α_{ij} ∈ End(A), then h ∈ I(H) ⇔ (α_{ij}) = (β_{ij})diag(π_{1}, …, π_{n}), for some β_{ij} ∈ Hom(A_{j}, A). Since the (i, j)-th entry of (β_{ij})diag(π_{1}, …, π_{n}) is β_{ij}π_{j}, we see that f ∈ I(H) ⇔ T (f) = (α_{ij}) with α_{ij} = β_{ij}π_{j} ∈ Hom(A_{j}, A)π_{j} = I(H_{j}), and so the first assertion follows; cf. (86).

The second assertion is clear: if the I_{j}’s are kernel ideals, then I_{j} = I(H_{j}), for some finite group scheme H_{j}, and so by the first assertion we have that (I_{1}| … |I_{n}) = I(H_{1} × … × H_{n}) is a kernel ideal; cf. Remark 7(b).

We also have the following result which is partially dual to Proposition 41.

Proposition 42 If I_{1}, …, I_{n} are left ideals of End(A), then

H((I_{1}|I_{2}| … |I_{n})) ≤ H(I_{1}) × H(I_{2}) × … × H(I_{n}), (88)

and equality holds if I_{1}, …, I_{n} are kernel subgroups. Thus, if H_{1}, …, H_{n} are ideal subgroups of A, then H_{1} × … × H_{n} is an ideal subgroup of A^{n}.

Proof. Let f_{i} ∈ I_{i}, where i = 1, …, n. Since T_{A,n}(f_{1} × … × f_{n}) = diag(f_{1}, …, f_{n}), we see that f_{1} × … × f_{n} ∈ (I_{1}| … |I_{n}). Thus

H((I_{1}|I_{2}| … |I_{n})) ≤ ⋂_{1≤i≤n} ⋂_{f_{i}∈I_{i}} Ker(f_{1}×…×f_{n}) = ( ⋂_{f_{1}∈I_{1}} Ker(f_{1}))×…×( ⋂_{f_{n}∈I_{n}} Ker(f_{n})) .

Since the right hand side equals H(I_{1}) × … × H(I_{n}), the first assertion follows.

To prove the second assertion, put H_{i} = H(I_{i}), so I(H_{i}) = I_{i} by hypothesis. Thus, by (88), (8), and (87) we obtain H((I_{1}| … |I_{n})) ≤ H_{1} × … H_{n} ≤ H(I(H_{1} × … × H_{n})) = H((I(H_{1})| … |I(H_{n}))) = H((I_{1}| … |I_{n})), and so we must have equality throughout. This proves the second assertion, and from this the last assertion follows immediately. Indeed, since each I_{i} := I(H_{i}) is a kernel ideal, we obtain from (87) that H(I_{1}| … |I_{n}) = H(I_{1}) × … × H(I_{n}) = H_{1} × … × H_{2}, the latter because each H_{i} is an ideal subgroup. But this means that H_{1} × … × H_{2} is an ideal subgroup; cf. Remark 7(b).

We can now put together what we proved so far to conclude the following result which partially generalizes Theorem 1 of the introduction.

Theorem 43 Let H_{1}, …, H_{n} and H′_{1}, …, H′_{n} be ideal subgroups of A. Then

A_{H_{1}} × … × A_{H_{n}} ≃ 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^{n} by Proposition 42, we have by (21) that A_{H}^{n} ≃ A_{H′}^{n} ⇔ I(H) ≃ I(H′). Now by (87) we have 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′) (as End(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 we shall now see.

Proof of Theorem 1. Let π_{i} : E → E_{i} and π′_{i} : E → E′_{i} be isogenies. Since f_{E_{i}}|f_{E} and f_{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_{H_{i}}, E′_{i} ≃ E_{H′_{i}} and I_{E}(E_{i}) ≃ I(H_{i}) and I_{E}(E′_{i}) ≃ I(H′_{i}).

4.2 The theorems of Steinitz and of Borevich and Faddeev

In order to derive further properties about abelian varieties which are isogenous to a product 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^{n} is isomorphic to a direct sum of R-ideals. Moreover, if I_{1}, …, I_{n} and J_{1}, … J_{m} are R-ideals, then

I_{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 already the first assertion of theorem may be false. As Borevich and Faddeev[2] observed, one needs the extra condition that the order R be cyclic in the sense that 𝔒_{F} /R is a cyclic R-module. In their papers, they prove the following generalization of Steinitz’s theorem.

Theorem 45 (Borevich/Faddeev) Let R be an order in a Dedekind domain 𝔒. Then:

(a) The order R is cyclic if and only if for all n ≥ 1 we have that every R-submodule of 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. Then there exist R-ideals I_{1}, …, I_{n} such that

M ≃ I_{1} ⊕ … ⊕ I_{n} and R(I_{1}) ⊂ R(I_{2}) ⊂ … R(I_{n}), (89)

and the orders R(I_{1}) ⊂ … ⊂ R(I_{n}) and the ideal class of the product I_{1} ⋯ I_{n} are uniquely determined by the isomorphism class of M. More precisely, if J_{1}, …, J_{m} are R-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 in Theorems 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 clear that Theorem 45 generalizes Theorem 44.

(b) Every order R_{Δ} = ℤ + ℤω_{Δ} in a quadratic field F = ℚ(\sqrt{Δ}) is cyclic because 𝔒_{F} = ℤ + ℤω_{F} = R_{Δ} + R_{Δ}ω_{F}. Thus, Theorem 45(b) applies to all orders in quadratic fields.

(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} ⊕ I

induces a bijection between the following sets:

(i) the set of lists (R_{1}, …, R_{n}; I) where R ⊂ R_{1} ⊂ … ⊂ R_{n} ⊂ 𝔒_{F} are orders containing R and I ∈ Pic(R_{n}) is a class of invertible R_{n}-ideals;

(ii) the set of isomorphism classes of finitely generated torsion-free R-modules of rank n.

Corollary 47 Let V be an n-dimensional F -vector space, where F ⊃ ℚ is a quadratic field, and let L be a lattice in V, i.e. L ⊂ V is a finitely generated subgroup which contains a basis of V. Then R_{F} (L) := (L : L)_{F} = {x ∈ F : xL ⊂ L} is an order of F, and L is an R_{F} (L)-module. Furthermore, there exists a sequence of orders R_{1} = R_{F} (L) ⊂ R_{2} ⊂ … ⊂ R_{n}, an invertible R_{n}-ideal I and a basis x_{1}, …, x_{n} of V such that L = R_{1}x_{1} + R_{2}x_{2} + … + R_{n−1}x_{n−1} + Ix_{n}. (91)

Proof. Since L is finitely generated, it follows easily that R_{F} (L) is a subring of 𝔒_{F}. Thus, R_{F} (L) is order of F provided that d𝔒_{F} ⊂ R_{F} (L), for some d. To see this, fix a basis {x_{1}, …, x_{n}} of V, and put L_{0} = ∑ O_{F} x_{i}. By the usual argument there is a n ∈ ℕ such that nL ⊂ L_{0}, and d := [L_{0} : nL] is finite. Since R_{F} (L_{0}) = 𝔒_{F}, we see that d𝔒_{F} ⊂ (nL : nL) = R_{F} (L).

Put R = R_{F} (L), so clearly L is an R-module. By Remark 46(b), (c) we see that there exist orders R_{1} ⊂ … ⊂ R_{n} and an invertible R_{n} ideal I such that L ≃ L′ := R_{1}y_{1} + … + R_{n−1}y_{n−1} + Iy_{n} (as R-modules), where {y_{1}, …, y_{n}} is any basis of V. But any such isomorphism extends to an isomorphism of F L = V to F L′ = V and hence is given by g ∈ Aut_{F} (L). Thus, L has the form (91) with respect to the basis x_{1} = g^{−1}(y_{1}), …, x_{n} = g^{−1}(y_{n}).

Finally, we observe that if L has the form (91), then R_{F} (L) = R_{1} ∩ … ∩ R_{n−1} ∩ R(I) = R_{1}, which proves the assertion that R_{F} (L) = R_{1}.

For later applications we note the following variant of the bijection mentioned in Remark 46(c).

Corollary 48 Let R be an order in quadratic field F and assume that n ≥ 2. If I is a non-zero R-ideal and if f_{1}, … f_{n−2} are positive integers with f_{R(I)}|f_{1}| … |f_{n−2}|f_{R}, and if R_{i} denotes the unique order of F of conductor f_{i}, then

M(I; f_{1}, …, f_{n−2}) := I ⊕ f_{R}R_{1} ⊕ … ⊕ f_{R}R_{n−2} ⊕ R

is an R-submodule of R^{n} with R_{F} (M) = R. Moreover, the map μ_{R,n} : (I; f_{1}, …, f_{n−2}) ↦ M(I; f_{1}, …, f_{n−2}) induces a bijection between:

(i) the set of sequences (I; f_{1}, …, f_{n−2}) where I is an isomorphism class of non-zero R-ideals and f_{R(I)}|f_{1}| … |f_{n−2}|f_{R};

(ii) the set of isomorphism classes of R-submodules M of R^{n} of rank n with R_{F} (M) = R.

Proof. If (I; f_{1}, …, f_{n−2}) is a tuple as in (i), put I_{k} = f_{R}R_{i}, for 1 ≤ k ≤ n − 2. Clearly R(I_{k}) = R_{k}, for 1 ≤ k ≤ n − 2, and so R(I) ⊃ R(I_{1}) ⊃ … ⊃ R(I_{n−2}) ⊃ R. Furthermore, since [R_{i} : R]|f_{R} = [𝔒_{F} : R], we see that I_{k} ⊂ R, and hence each I_{k} is an R-ideal. Thus M := M(I; f_{1}, …, f_{n−2}) is an R-submodule of R^{n}. Furthermore, since (M : M)_{F} = R(I) ∩ R(I_{1}) ∩ … ∩ R(I_{n−2}) ∩ R = R, we see that μ_{R,n} defines a map from the set described in (i) to the set described in (ii).

To see that μ_{R,n} is injective, suppose that M(I; f_{1}, …, f_{n−2}) ≃ M(I′; f′_{1}, …, f′_{n−2}), and put I_{0} = I, I_{k} = f_{R}R_{k}, 1 ≤ k ≤ n − 2 and I_{n−1} = R, and define I′_{k} similarly using (I′; f′_{1}, …, f′_{n−2}). Since R = R(I_{n−1}) ⊂ R(I_{n−2}) ⊂ … ⊂ R(I_{0}) and R = R(I′_{n−1}) ⊂ R(I′_{n−2}) ⊂ … ⊂ R(I′_{0}) are linearly ordered, if follows from Theorem 45(b) that R(_{k}) = R(I′_{k}) for 0 ≤ k ≤ n − 1 and that I_{0} ⋯ I_{n−1} ≃ I′_{0} ⋯ I′_{n−1}. Thus f_{k} = f′_{k}, for k = 1, …, n − 2. Moreover, since I_{0} ⋯ I_{n−1} = f_{R}^{n−2}I and I′_{0} ⋯ I′_{n−1} = f_{R}^{n−2}I′, we see that I ≃ I′, and so μ_{R,n} is injective.

To see that μ_{R,n} is surjective, let M be an R-submodule of R^{n} of rank n with (M : M) = R. Then by Corollary 47 we know that M ≃ R_{1} ⊕ … ⊕ R_{n−1} ⊕ I, where R_{1} ⊂ … ⊂ R_{n} are orders and I is an invertible R_{n}-ideal and that R_{1} = R because (M : M) = R by hypothesis. Put f_{i} = f_{R_{n−1−i}} for 1 ≤ i ≤ n − 2 Then clearly f_{R}I ⊂ f_{R}R_{n}, so f_{R}I is an R-ideal which is isomorphic to I, and so it is clear that M(f_{R}I; f_{1}, …, f_{n−2}) ≃ I ⊕ R_{n−2} ⊕ … ⊕ R_{1} ⊕ R ≃ M, and so μ_{R,n} is surjective.

One of the disadvantages of Theorem 45 is that it does not give a recipe for determining the orders R_{k} and ideals I_{k} such that (89) holds when M is given. For this we prove the following result (which is similar to Lemma 8 of [1], but without the restrictive hypothesis that R_{1} ⊂ R_{2}):

Proposition 49 Let L_{1} and L_{2} ∈ Lat_{F} be lattices in a quadratic field F, and put R = R(L_{1}) ∩ R(L_{2}). Then there is an R-module isomorphism

L_{1} ⊕ L_{2} ≃ R ⊕ (L_{1}L_{2}). (92)

Proof. Put R_{i} := R(L_{i}) and f_{i} := [R_{i} : R]. Then (f_{1}, f_{2}) = 1 by (37). We first claim:

(93) ∃ λ_{1}, λ_{2} ∈ F^{×} such that λ_{1}L_{1} + λ_{2}L_{2} = R.

Indeed, by replacing L_{i} by m_{i}L_{i} we may assume that the L_{i}’s are R-ideals, and so L_{i} ⊂ f_{i}R. Then L′_{i} := \frac{1}{f_{i}} L_{i} ⊂ R_{i} is an invertible R_{i}-ideal, and so ∃λ_{2} ∈ F^{×} such that λ_{2}L′_{2} + f_{1}R_{2} = R_{2} (cf. [18], p. 93) or, equivalently, that (N(λ_{2}L′_{2}), f_{1}) = 1; cf. [7], p. 143). Put (f_{1}, f_{2}) = 1, we thus have (N(L′′_{2}), f_{1}) = 1. Note also that m_{2} := [R : L′′_{2}] = f_{2}N(λ_{2}L′_{2}), so (m_{2}, f_{1}) = 1. Next choose λ_{1} ∈ F^{×} such that λ_{1}L′_{1} + m_{2}R_{1} = R_{1} and put L′′_{1} = λ_{1}L_{1} = f_{1}λ_{1}L′_{1} ⊂ f_{1}R_{1} ⊂ R. Then m_{1} := [R : L′′_{1}] = f_{1}N(λ_{1}L′_{1}) and so (m_{1}, m_{2}) = 1. From this it follows that L′′_{1} + L′′_{2} = R, which proves (93).

To prove (92), we may assume in view of (93) that L_{1} + L_{2} = R because L_{1} ⊕ L_{2} ≃ λ_{1}L_{1} ⊕ λL_{2}. Thus, there exist α_{i} ∈ L_{i} such that α_{1} − α_{2} = 1. Define the map

β : L_{1} ⊕ L_{2} → R ⊕ L_{1}L_{2} by β(λ_{1}, λ_{2}) = (λ_{1} + λ_{2}, λ_{1}α_{1} + λ_{2}α_{2})

Clearly, β is R-module homomorphism with β(L_{1} ⊕ L_{2}) ⊂ R ⊕ L_{1}L_{2}. It is clear that β is injective because det(\binom{1 α_{1}}{1 α_{2}}) = α_{2} − α_{1} = −1. Moreover, β is surjective because if r ∈ R, λ ∈ L_{1}L_{2} ⊂ L_{1} ∩ L_{2}, then β(α_{1}r − λ, λ − α_{2}r) = (r, λ). Thus β is an isomorphism, which proves (92).

Remark 50 It follows from the above result by induction that if L_{1}, …, L_{n} ∈ Lat_{F} and R = ∩_{i}R(L_{i}), then there is an R-module isomorphism

L_{1}⊕…⊕L_{n} ≃ R_{1}⊕…⊕R_{n−1}⊕(L_{1} ⋯ L_{n}) where R_{k} = R(L_{1} ⋯ L_{k})∩R(L_{k+1}) if k < n.

For later applications we also want to explain the connection between the (conductor of) the order R_{F} (L) defined above and the so-called central conductor of a suitable order in the matrix ring M_{n}(F). This central conductor is defined as follows.

Definition. Let R be an order in M_{n}(F), i.e. R ⊂ M_{n}(F) is a subring which is finitely generated as a ℤ-module and which contains an F -basis of M_{n}(F). For convenience, assume that F ⊃ ℚ is a quadratic field. Then the centre Z(R) of R is an order of F = Z(M_{n}(F)), and hence is uniquely determined by its conductor

f_{R} := f_{Z(R)} = [𝔒_{F} : Z(R)],

which we call the central conductor of R. Clearly, this is an invariant of the isomorphism class of R. Note that this term is closely related to that of [8], p. 604: there the central conductor is the ideal f_{R}𝔒_{F}.

Proposition 51 Let R be an order in a quadratic field F and let M be an R-submodule of R^{n} of rank n. Put

R(M) := (I_{n}(M) : I_{n}(M)) = {g ∈ M_{n}(F) : I_{n}(M)g ⊂ I_{n}(M)}.

Then we have R-ring isomorphisms

End(M)^{op} →^{∼} R(M) and R_{F} (M) →^{∼} Z(R(M)) (94)

and hence the central conductor of R(M) equals the conductor of R_{F} (M), i.e. f_{R(M)} = f_{R_{F} (M)}.

Proof. Put M := M_{n}(R), and view M as a left M-module. Then (as for any ring) we have the canonical identification ρ_{M} : M →^{∼} End_{M}(M) given by g ↦ ρ_{g}, where ρ_{g}(g′) = g′g denotes the right multiplication map. Combining this with the identification τ\underline{x} : End_{R}(R^{n})^{op} →^{∼} M defined in the proof of Proposition 40, we thus obtain an isomorphism ρ := ρ_{M} ∘ τ\underline{x} : End_{R}(R^{n})^{op} →^{∼} End_{M}(M), which extends to an isomorphism ρ̃: End_{F} (F^{n})^{op} →^{∼} EndM̃(M̃), where M̃ = M_{n}(F).

Since M is a lattice in F^{n}, every f ∈ End_{R}(M)^{op} extends uniquely to f̃ ∈ E := End_{F} (F^{n})^{op}, and so we can identify End_{R}(M)^{op} with the subring (M : M)_{E} = {f ∈ E : Mf ⊂ M} of E. It is then immediate from the definition of I_{n}(M) that

ρ̃((M : M)_{E}) = (I_{n}(M) : I_{n}(M)) = R(M).

This proves this first isomorphism of (94), and from this the second follows because Z((M : M)_{E}) = (M : M)_{E} ∩ Z(E) = (M : M)_{F} = R_{F} (M).

4.3 Products of CM elliptic curves

We now apply the results of the previous subsections to the case that A = E is a CM elliptic curve (in the sense of §3.1). The first result is the following.

Proposition 52 Let E/K be a CM elliptic curve and let I be a regular ideal of End(E^{n}), where n ≥ 1. Then I is a kernel ideal and there exist non-zero ideals I_{1}, …, I_{n} of End(E) such that I ≃ (I_{1}| … |I_{n}). Furthermore,

E_{H(I)}^{n} ≃ E_{H(I_{1})} × … × E_{H(I_{n})} and E(H(I)) ≃ End_{R}(I_{1} ⊕ … ⊕ I_{n})^{op}. (95)

Proof. Let I be an ideal of End(E^{n}), and put R = End(E). Since T = T_{E,n} : End(E^{n}) →^{∼} M_{n}(R) is an isomorphism, it follows from Proposition 40 that I = I_{n}(M), for some R-submodule M ⊂ R^{n}. Moreover, since I is regular, we see that M has finite index in R^{n}. Since F = End^{0}(E) is an (imaginary) quadratic field, we have by Corollary 47 that there exist orders R_{i} with R ⊂ R_{1} ⊂ … ⊂ R_{n} of R, an ideal I of R_{n} and a basis {x_{i}} of F^{n} such that M has the form (91). Put f = [R_{n} : R]. Then fR_{i} ⊂ R is an R-ideal for all i, and so

I = I_{1}x′_{1} + I_{2}x′_{2} + … + I_{n}x′_{n},

where I_{i} = fR_{i} for 1 ≤ i < n and I_{n} = fI and x′_{i} = \frac{1}{f}x_{i}. Thus, if g ∈ M_{n}(F) is the matrix that takes the standard basis of F^{n} to the basis {x′_{i}}, then we see that

I = (I_{1}| … |I_{n})h, (96)

where h = T_{E,n}(g)^{−1} ∈ End^{0}(E^{n}) and I_{1}, …, I_{n} are non-zero R-ideals. Thus I ≃ I′ := (I_{1}| … |I_{n}). Since each I_{k} is a kernel ideal of End(E) by Theorem 18, we have by Proposition 41 that I′ is a kernel ideal, and hence so is I by Remark 7(c). This proves the first two assertions.

Since the I_{k}’s are kernel ideals, we have H((I_{1}| … |I_{n}) = H(I_{1}) × … × H(I_{n}) by Proposition 42. Thus, by (19) we have E_{H(I)}^{n} ≃ E_{H((I_{1}|...|I_{n}))}^{n} = E_{H(I_{1})×...×H(I_{n})} ≃ E_{H(I_{1})} × … × E_{H(I_{n})}, which proves the first isomorphism of (95).

To prove the second, note first that since I ≃ I′ are kernel ideals of End(E^{n}), we have by (19) and Proposition 10 that E(H(I)) ≃ E(H(I′)) = (I′ : I′). On the other hand, since I′ = T_{E,n}^{−1}(I_{n}(I_{1} ⊕ … ⊕ I_{n})) by definition, it follows from (94) that (I′ : I′) ≃ End_{R}(I_{1} ⊕ … ⊕ I_{n})^{op}, and so the second isomorphism of (95) follows.

The above result allows us to work out the central conductor f_{A} of the abelian variety A = A_{H(I)}. This important invariant is defined as follows.

Definition. If A ∼ E^{n} is any abelian variety which is isogenous to E^{n}, then its central conductor is the central conductor f_{A} := f_{End(A)} of the order End(A) in End^{0}(A) ≃ End^{0}(E^{n}) = M_{n}(F). (Recall that f_{End(A)} was defined in §4.2).

Corollary 53 If A ≃ E_{1} × … × E_{n}, where E_{1}, …, E_{n} ∈ Isog(E/K), then

f_{A} = lcm(f_{E_{1}}, …, f_{E_{n}}).

Proof. By Proposition 36 there is an E_{0} ∈ Isog(E/K) such that f_{E_{0}} = lcm(f_{E_{1}}, …, f_{E_{n}}). Thus, if π_{i} : E_{0} → E_{i} is an isogeny, then H_{i} = Ker(π_{i}) is an ideal subgroup of E_{0} by (46) (because f_{E_{i}}|f_{E_{0}} by construction). Thus, H_{i} = H(I_{i}), for some R_{0}-ideal I_{i}, where R_{0} = End(E_{0}). Note that by (45) we have that End(E_{i}) ≃ E(H_{i}) = (I_{i} : I_{i}) = R(I_{i}), and so f_{E_{i}} = f_{R(I_{i})}.

Put H = H_{1} × … × H_{n}. Then A_{H} ≃ E_{H_{1}} × … × E_{H_{n}} ≃ E_{1} × … × E_{n} ≃ A. By Proposition 41 we have H = H(I), where I = (I_{1}| … |I_{n}). Thus, by (95) we have End(A) ≃ End(A_{H}) ≃ E(H) ≃ End_{R_{0}}(M)^{op}, where M = I_{1} ⊕ … ⊕ I_{n} ⊂ R_{0}^{n}, and so Z(End(A)) ≃ Z(End_{R_{0}}(M))^{op} = (M : M)_{F} = R_{F} (M) by (94). Now clearly (M : M)_{F} = R(I_{1}) ∩ … ∩ R(I_{n}). Since the latter has conductor lcm(f_{R(I_{1})}, …, f_{R(I_{n})}) by (37), the assertion follows.

We next want to characterize the ideal subgroups of E^{n}. The following characterization shows that the necessary condition of Corollary 11 is also sufficient. Moreover, this result can also be viewed as a generalization of the criterion (46) of Theorem 18.

Theorem 54 Let E/K is a CM elliptic curve, and let H be a finite subgroup scheme of E^{n}. Then:

(97) H is an ideal subgroup of E^{n} ⇔ Z(End(E^{n})) ⊂ E(H) ⇔ f_{E(H)} | f_{E}.

Before proving this, we note the following immediate application.

Corollary 55 If A ∼ E^{n} is an abelian variety which is isogenous to E^{n} with f_{A}|f_{E}, then there exist elliptic curves E_{1}, …, E_{n} ∈ Isog^{+}(E/K) such that A ≃ E_{1} × … × E_{n}.

Proof. Let π : E^{n} → A be an isogeny, and let H = Ker(π). Then E_{H}^{n} ≃ A and End(A) ≃ E(H), so f_{A} = f_{E(H)}. It thus follows from the hypothesis and (97) that H is an ideal subgroup. Thus, H = H(I), for some regular left ideal I of End(E^{n}), and hence by (95) we have A ≃ A_{H} ≃ E_{1} ×…×E_{n}, where E_{i} := E_{H(I_{i})}, for suitable ideals I_{i} of End(E). Since H(I_{i}) is an ideal subgroup of E, we have by (46) that f_{E_{i}}|f_{E}, and so E_{i} ∈ Isog^{+}(E/K), as claimed.

Another application of Theorem 54 is the following result which can be viewed as a solution of the isomorphism problem which was posed in the introduction of this paper. Note that it applies to an arbitrary tuple E_{1}, …, E_{n}/K of isogenous CM curves because by Proposition 36 we can always find an elliptic curve E/K such that E_{1}, …, E_{n} ∈ Isog^{+}(E/K).

Corollary 56 Let E/K be a CM elliptic curve and E_{1}, …, E_{n} ∈ Isog^{+}(E/K). If A ∼ E^{n} is an abelian variety isogenous to E^{n}, then the following conditions are equivalent:

(98)  A ≃ E_{1} × … × E_{n};
(99)  I_{E^{n}}(A) ≃ (I_{E}(E_{1}) : …: I_{E}(E_{n})) and f_{A}|f_{E};
(100) End(E)^{n} ⊗_{End(E^{n})} I_{E^{n}}(A) ≃ I_{E}(E_{1}) ⊕ … ⊕ I_{E}(E_{n}) and f_{A}|f_{E}.

Proof. Fix isogenies π : E^{n} → A and π_{i} : E → E_{i}, for 1 ≤ i ≤ n, and put H = Ker(π) and H_{i} = Ker(π_{i}). Also, put A′ = E_{1} × … × E_{n}. Since f_{E_{i}}|f_{E} by hypothesis, we know by (46) that H_{i} = H(I(H_{i})) is an ideal subgroup of E, so H′ := H_{1} × … × H_{n} is an ideal subgroup of E^{n} by Proposition 42. Furthermore, we have that I_{E^{n}}(A′) ≃ I(H′) = (I(H_{1}) : …: I(H_{n})) ≃ (I_{E}(E_{1}) : …: I_{E}(E_{n})) by (87).

Now suppose that (98) holds. Then by Corollary 53 we have that f_{A} = f_{E_{1}×...×E_{n}} = lcm(f_{E_{1}}, …, f_{E_{n}})|f_{E}. Moreover, I_{E^{n}}(A) ≃ I_{E^{n}}(A′) ≃ (I_{E}(E_{1}) : …: I_{E}(E_{n})), and so (99) holds. Conversely, if (99) holds, then H is an ideal subgroup of E^{n} by Theorem 54, and so A ≃ A′ by (21). Thus, conditions (98) and (99) are equivalent.

To prove the equivalence of conditions (99) and (100), consider the submodules M_{1} := End(E)^{n} ⊗_{End(E^{n})} I_{E^{n}}(A) and M_{2} = I_{E}(E_{1}) ⊕ … ⊕ I_{E}(E_{n}). Then by definition we have T_{E,n}^{−1}(I_{n}(M_{2})) = (I_{E}(E_{1}) : …: I_{E}(E_{n})) and by Morita equivalence (cf. the proof of Proposition 40) we have T_{E,n}^{−1}(I_{n}(M_{1})) ≃ I_{E^{n}}(A). Thus, since M_{1} ≃ M_{2} (as End(E)-modules) ⇔ T_{E,n}^{−1}I_{n}(M_{1}) ≃ T_{E,n}^{−1}I_{n}(M_{2}) (as End(E^{n})-modules) by Proposition 40, the equivalence of conditions (99) and (100) follows.

As first step towards establishing Theorem 54, we prove the following result.

Proposition 57 Let H be a finite subgroup scheme of E^{n}, where E/K is a CM elliptic curve. Then H is an ideal subgroup of E^{n} if and only if

(101) A_{H} ≃ E_{1} × … × E_{n}, where E_{i} ∈ Isog^{+}(E/K), for 1 ≤ i ≤ n.

Proof. Suppose first that H = H(I) is an ideal subgroup. Then by (95) we have E_{H}^{n} ≃ E_{1} × … × E_{n} where E_{i} = E_{H(I_{i})}. Since H(I_{i}) is an ideal subgroup of E, we have by (46) that f_{E_{i}}|f_{E}, i.e. that E_{i} ∈ Isog^{+}(E/K).

Conversely, suppose that (101) holds, and fix an isogeny π_{i} : E → E_{i} for 1 ≤ i ≤ n. Then by (46) we know that H_{i} := Ker(π_{i}) is an ideal subgroup of E, so H′ := H_{1} × … × H_{n} is an ideal subgroup of E^{n} by Proposition 42. Since E_{i} ≃ E_{H_{i}}, we have by the hypothesis (101) that E_{H}^{n} ≃ E_{H_{1}} × … × E_{H_{n}} ≃ E_{H′}^{n}, and so it follows from (22) that H is also an ideal subgroup of E^{n}.

As we shall see presently, Theorem 54 follows easily from the following result.

Theorem 58 Let E/K be a CM elliptic curve, where K is an algebraically closed field. If A ∼ E^{n} is an abelian variety which is isogenous to E^{n}, then there exist elliptic 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], using the results of Shioda and Mitani[24] (who proved the case n = 2). A different proof of this was given by Schoen; cf. [22], Satz 2.4. His proof is based on the above Theorem 45(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 Theorem 58 in the case K = ℂ. Suppose that A/ℂ is an abelian variety with the property that if B ∼ A, then B ≃ A_{1} × … × A_{n}, for some simple abelian varieties A_{i}. Then either A 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 equivalence of (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 by Corollary 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 have that End(E^{n}) = End(E^{n} ⊗ \overline{K}), and hence also that End(A) = End(A ⊗ \overline{K}). (To see this, note that End^{0}(E^{n}) = End^{0}(A) (because A ∼ E^{n}), and that End^{0}(E^{n}) = End^{0}(E^{n} ⊗ \overline{K}), and so the assertion follows because End(A ⊗ \overline{K})/End(A) is always torsionfree.) Thus, we have that E(H ⊗ \overline{K}) ≃ End(A ⊗ \overline{K}) = End(A) ≃ E(H), and so Z(End(E^{n} ⊗ \overline{K})) ⊂ Z(E(H ⊗ \overline{K})). Thus f_{A_{H⊗\overline{K}}} = f_{E(H⊗\overline{K})}|f_{E⊗\overline{K}} = f_{E}.

By Theorem 58 we have that A_{H⊗\overline{K}} ≃ E_{1} ×…×E_{n}, for some elliptic curves E_{i}/\overline{K}. By Corollary 53 we have f_{E_{i}}|f_{A_{H⊗\overline{K}}}|f_{E}, and so E_{i} ∈ Isog^{+}(E ⊗\overline{K}/\overline{K}). Thus, condition (101) holds for H ⊗ \overline{K}, and so H ⊗ \overline{K} is an ideal subgroup of (E ⊗ \overline{K})^{n} = E^{n} ⊗ \overline{K} by Proposition 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 lattice in 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 and the 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 because L′_{i} ∈ Lat_{F} (and hence ℂ/L′_{i} ∼ E_{L}).

Case 2. char(K) = 0. As was mentioned in the proof of Proposition 32, there is a CM elliptic curve E_{0}/\overline{ℚ} such that E_{0} ⊗ K ≃ E. Furthermore, if π : E^{n} → A is an isogeny with kernel H, then there is a subgroup scheme H_{0} of E_{0}^{n} such that H_{0} ⊗ K = H, and so A_{0} = (E_{0}^{n})_{H_{0}} is an abelian variety over \overline{ℚ} with A_{0} ⊗ K ≃ A. It is thus enough to verify the assertion for K = \overline{ℚ}.

Thus, let K = \overline{ℚ}. By Case 1 we know that there exist CM elliptic curves E_{1}, …, E_{n} ∈ Isog(E ⊗ ℂ/ℂ) such that A ⊗ ℂ ≃ E_{1} × … × E_{n}. As before, there exist CM elliptic curves E_{i0}/K such that E_{i0} ⊗ ℂ ≃ E_{i}. Put A′ = E_{10} × … × E_{n0}. As was just mentioned, every finite group scheme of A ⊗ ℂ is already defined over K, and so it follows that the condition A ⊗ ℂ ∼ A′ ⊗ ℂ implies that A ∼ A′. This proves the assertion in this case. Case 3. K = \overline{𝔽}_{p}. Let K^{#} denote the quotient field of the ring W (K) of Witt-vectors. For any ordinary abelian variety A/K, its Serre-Tate lift A^{#}/K^{#} exists and is uniquely characterized by the property that all endomorphisms of A/K lift to A^{#}; cf. [9], p. 238, or [13], p. 2368, and the references therein. Since E and hence A ∼ E^{n} are ordinary abelian varieties, their Serre-Tate lifts exist; note that (E^{#})^{n} ≃ (E^{n})^{#}. Since Hom((E^{n})^{#}, A^{#}) = Hom(E^{n}, A), it thus follows that A^{#} ∼ (E^{#})^{n}. By Case 2 we know that there is a finite extension K′/K^{#} and elliptic curves E′_{1}, …, E′_{n}/K′ such that A^{#}⊗K′ ≃ E′_{1}×…×E′_{n}. Since each E′_{i} ∼ E^{#} ⊗ K′ has good reduction E_{i}/K, we obtain that A ≃ E_{1} × … × E_{n} because A is the reduction of A^{#} ⊗ K′. Case 4. char(K) = p ≠ 0. This is partially similar to the proof of Case 2, but extra care has to taken here in descending A. To be precise, note first that there is a CM elliptic curve E_{0}/\overline{𝔽}_{p} such that E_{0} ⊗ K ≃ E; cf. the proof of Proposition 32. Furthermore, if π : E^{n} → A is an isogeny with kernel H, then by Lemma 60 below, there is a subgroup scheme H_{0} of E_{0}^{n} such that H_{0} ⊗ K = H, and so A_{0} = (E_{0}^{n})_{H_{0}} is an abelian variety over \overline{ℚ} with A_{0} ⊗ K ≃ A. By Case 3 there exist E_{1}, …, E_{n} ∼ E_{0} such that A_{0} ≃ E_{1} × … × E_{n}, and then A ≃ A_{0} ⊗ K ≃ E′_{1} × … × E′_{n}, where E′_{i} = E_{i} ⊗ K ∼ E, as desired.

In the above proof we made use of the following fact.

Lemma 60 Let E/K be a CM elliptic curve where K is an algebraically closed field, and let n ≥ 1. Then E^{n} has only finitely many subgroup schemes H of fixed rank N. Thus, if K′/K is any extension field, then every finite subgroup scheme of E^{n} ⊗ K′ is of the form H ⊗ K′, where H is a finite subgroup scheme of E^{n}.

Proof. If char(K) = 0, then every finite subgroup scheme is etale and the assertion is trivial. Thus, assume that char(K) = p ≠ 0. Since E is ordinary, we have for any r ≥ 1 that E[p^{r}] := Ker([p^{r}]_{E}) ≃ ℤ/p^{r}ℤ × μ_{p^{r}}, and hence that E^{n}[p^{r}] ≃ (ℤ/p^{r}ℤ)^{n} × (μ_{p^{r}})^{n}.

Thus E^{n} does not contain any local-local subgroup scheme (in the sense of [20], p. 136), and so by the structure of finite commutative group schemes ([20], p. 137) we have that any finite subgroup scheme H of E^{n} is isomorphic to H_{et} × (ℤ/p^{r_{1}}ℤ)^{t_{1}} × (μ_{p^{r_{2}}})^{t_{2}}, where H_{et} is an etale subgroup scheme of rank prime to p and r_{i}, t_{i} ≥ 0 are integers. Thus, there are only finitely many (isomorphism classes of) finite subgroup schemes of fixed rank N which are embeddable in E^{n}.

Now let H/K be a fixed finite subgroup scheme of rank N = p^{r}N′ with p ∤ N′ which is embeddable in E^{n}. Then Hom(H, E^{n}) = Hom(H, E^{n}[N)), and so Hom(H, E^{n}) = Hom(H_{et}, E[N′]) ⊕ Hom((ℤ/p^{r_{1}})^{t_{1}}, (ℤ/p^{r}ℤ)^{n}) ⊕ Hom((μ_{p^{r_{2}}})^{t_{2}}, (μ_{p^{r}})^{n}) is finite (since each piece of the decomposition is finite). In particular, there are only finitely many embeddings of H into E^{n}, and so it follows from the above that there are only finitely many finite subgroup schemes of fixed rank N.

To prove the last assertion, note that since A/K is projective, it follows from the theory of Hilbert schemes that the set of subgroup schemes of fixed rank N is represented by a quasi-projective scheme H_{N}. By what was just shown, H_{N}(K) is finite, and so it follows that dim H_{N} = 0. Thus H_{N}(K′) = H_{N}(K), for all K′/K, which yields the last assertion.

Remark 61 As the above proof shows, the assertions of Lemma 60 are true for any ordinary abelian variety A/K (in place of E^{n}/K). On the other hand, both statements are false for non-ordinary abelian varieties. In particular, if E is a supersingular elliptic curve, then already E^{2} has infinitely many subgroup schemes which are isomorphic to α_{p}, and their cardinality equals the cardinality of K (and hence increases when we enlarge K).

This concludes the proof of Theorem 58 and hence also of Theorem 54, as was proven above. We next prove Theorem 3 of the introduction, which is an easy consequence of Theorem 54.

Proof of Theorem 3. If A ≃ E_{1} × … × E_{n} with E_{i} ∼ E, then by Corollary 53 we have f_{A} = lcm(f_{E_{1}}, …, f_{E_{n}}). By Proposition 36 there exists an elliptic curve E_{0} ∼ E with f_{E_{0}} = lcm(f_{E_{1}}, …, f_{E_{n}}), and so f_{A} = f_{E_{0}}.

Conversely, suppose there exists E_{0} ∼ E such that f_{A}|f_{E_{0}}. Then A ∼ E^{n} ∼ E_{0}^{2}, and so there is an isogeny π : E_{0}^{n} → A. Put H = Ker(π). Then (E_{0}^{n})_{H} ≃ A and so f_{E(H)} = f_{A}|f_{E_{0}} by hypothesis. By Theorem 54 we therefore know that H is an ideal subgroup of E_{0}^{n} and so it follows from Proposition 57 that A ≃ (E_{0}^{n})_{H} ≃ E_{1} ×…×E_{n}, for some elliptic curves E_{1}, …, E_{n} ∼ E_{0}.

Next we shall use Theorem 3 to prove Theorem 2 of the introduction.

Proof of Theorem 2. If the field K is algebraically closed, then this is Theorem 58. Thus, assume that K is a finite field.

Let h_{A/K}(X) ∈ ℤ[X] (respectively, m_{A/K}(X) ∈ ℤ[X]) denote the characteristic polynomial (respectively, minimal polynomial) of the Frobenius endomorphism ϕ_{A} of A/K. Since A ∼ E^{n}, we have h_{A/K}(X) = h_{E/K}(X)^{n} and m_{A/K}(X) = m_{E/K}(X). Note that m_{E/K} = h_{E/K} is irreducible over ℚ since E/K is a CM elliptic curve.

Since ϕ_{A} ∈ Z(End(A)) ⊂ Z(End^{0}(A)) ≃ F = End^{0}(E), we see that ℤ[ϕ_{A}] is an order in F (because m_{A/K} is irreducible of degree 2). Thus, since ℤ[ϕ_{A}] ⊂ Z(End(A)), it follows that f_{A}|f_{ℤ[ϕ_{A}]}. Now by Waterhouse (cf. Proposition 34 above) there is an elliptic curve E_{0}/K with E_{0} ∼ E such that Δ_{E_{0}} = Δ(h_{E/K}). Thus, since the discriminant of the order ℤ[ϕ_{A}] is Δ(m_{A/K}) = Δ(h_{E/K}), we see that f_{ℤ[ϕ_{A}]} = f_{E_{0}} and that hence f_{A}|f_{E_{0}}. It therefore follows from Theorem 3 that A ≃ E_{1} × … × E_{n}, for some elliptic curves E_{1}, …, E_{n} ∼ E_{0} ∼ E.

We now turn to the proof of Theorem 5 of the introduction.

Proof of Theorem 5. We will construct bijections between the various sets defined by the conditions (i) – (v).

(i) ↔ (ii): Let (E′; f_{1}, …, f_{n−2}) be a tuple as in (i), and put φ_{1}(E′; f_{1}, …, f_{n}) = (I_{E}(E′); f_{1}, …, f_{n−2}). Since f_{E′}|f_{E}, we know by (46) that if π : E → E′ is any isogeny, then Ker(π) = H(I) for some ideal I of End(E). Thus I_{E}(E′) ≃ I because I is a kernel ideal by Theorem 18(a) and also End(E′) ≃ End(E_{H(I)}) ≃ R(I) by (49). Thus f_{E′} = f_{R(I)}, and so (I_{E}(E′); f_{1}, …, f_{n−2}) is in the set described by condition (ii). By Corollary 19 we thus see that φ_{1} defines a bijection between sets (i) and (ii).

(ii) ↔ (iii): If (I; f_{1}, …, f_{n−2}) is a tuple as in (ii), and put φ_{2}(I; f_{1}, …, f_{n−2}) = (q_{Δ}(I); f_{1}/f_{I}, …, f_{n−2}/f_{I}), where q_{Δ} is the bijection constructed in Remark 38 and f_{I} = f_{R(I)}. Here R_{Δ} ≃ End(E) (because E has discriminant Δ). By construction, q_{Δ}(I) has content cont(q_{Δ}(I)) = [R(I) : R_{Δ}] = f_{E}/f_{R(I)}, and so we see that (q_{Δ}(I); f_{1}/f_{I}, …, f_{n−2}/f_{I}) is in the set described by condition (iii) and that φ_{2} is a bijection.

(ii) ↔ (iv): This is Corollary 48.

(iv) ↔ (v): For an R-module M as in (iv), put φ_{4}(M) = T_{E,n}^{−1}(I_{n}(M)), which is a regular left End(E^{n})-ideal. By Proposition 40 and Proposition 51 we know that φ_{4} induces a bijection between the set described by (iv) and the following set:

(iv′) the set of isomorphism classes of regular left ideals I of End(E^{n}) ≃ M_{n}(R) with Z((I : I)) = R.

Next, if I is an ideal as in (iv′), then put φ_{5}(I) = A_{I} := E_{H(I)}^{n}. Since I is a kernel ideal by Proposition 52, we have that End(A_{I})) ≃ (I : I), and so Z(End(A_{I})) = Z((I : I)) = R = End(E), which means that f_{A_{I}} = f_{E}. Thus the isomorphism class of A_{I} is an element of the set described by (v). By (19) and the sentence before (21) we know that φ_{5} defines an injection from the set described by (iv′) into the set (v). Finally, this map is surjective. Indeed, if A ∼ E^{n} with f_{A} = f_{E}, and if π : E^{n} → A is any isogeny, then by Theorem 54 we know that H is an ideal subgroup, so Ker(π) = H(I) for some regular left ideal I of End(E^{n}). Thus A ≃ E_{H(I)}^{n}. Furthermore, by the above computation we have Z((I : I)) = R, and so I is in the set defined by condition (iv′).

Remark 62 (a) It is sometimes useful to have a direct description of the bijection between the sets defined by conditions (i) and (v) of Theorem 5. Indeed, if we unravel the bijections constructed in the proof of Theorem 5, then we obtain that this bijection is given by the rule (E′; f_{1}, …, f_{n−2}) ↦ A(E′; f_{1}, …, f_{n−2}; E), where

A(E′; f_{1}, …, f_{n−2}; E) := E′ × E_{1} × … × E_{n−2} × E, (102)

where E_{i} = E_{H(f_{E}R_{i})} and f_{R_{i}} = f_{i}. To verify this, note first that the bijection is given by φ_{5} ∘ φ_{4} ∘ μ_{R,n} ∘ φ_{1} in the notation of the proof of Theorem 5. Thus, if we put I_{1} = I_{E}(E′), I_{i+1} = f_{E}R_{i} for i = 1, …, n−2 and I_{n} = R, then μ_{R,n}(φ_{1}φ_{1}(E′, f_{1}, …, f_{n−2}) = I_{1} ⊕ … ⊕ I_{n}. On the other hand, φ_{5}(φ_{4}(I_{1} ⊕ … ⊕ I_{n})) = φ_{5}((I_{1} : …: I_{n})) = E_{H(I_{1})} × … × E_{H(I_{n})} by Proposition 41, and so the assertion follows.

(b) The bijections of Theorem 5 yield immediately a classification of all abelian varieties A ∼ E^{n} with f_{A}|f_{E}. More precisely, we have natural bijections between the follow sets:

(i) The set of sequences (E′; f_{1}, …, f_{n−1}) where E′ ∼ E is an isomorphism class of elliptic curves with f_{E′}|f_{E} and the f_{i}’s are positive integers with f_{E′}|f_{1}| … |f_{n−1}|f_{E}.

(ii) the set of sequences (I; f_{1}, …, f_{n−1}) where I is an isomorphism class of nonzero End(E)-ideals whose order R(I) has conductor f_{R(I)}|f_{1}| … |f_{n−1}|f_{E}.

(iii) the set of sequences (q; c_{1}, …, c_{n−2}) where q is a proper equivalence class of positive binary quadratic forms of discriminant Δ(q) = Δ/m^{2}, for some m ∈ ℕ, and c_{1}| … |c_{n−2}|cont(q).

(iv) the set of isomorphism classes of End(E)-submodules M of End(E)^{n} of rank n;

(v) the set of isomorphism classes of abelian varieties A ∼ E^{n} with central conductor f_{A}|f_{E}.

4.4 Abelian product surfaces

We now specialize the previous results to the case of abelian surfaces, i.e. to the case n = 2. In particular, we shall prove Theorem 4 and show how the results of Shioda and Mitani[24] follow from the above theorems.

We begin with the following refinement of Theorem 1 (in the case n = 2) which is closely related to Proposition 4.5 of [24].

Proposition 63 Let E_{1}, E_{2}, E′_{1}, E′_{2} ∈ Isog^{+}(E/K), where E/K is a CM elliptic curve, and put f_{i} = f_{E_{i}} and f′_{i} = f_{E′_{i}}. Then E_{1} × E_{2} ≃ E′_{1} × E′_{2} if and only if

lcm(f_{1}, f_{2}) = lcm(f′_{1}, f′_{2}) and I_{E}(E_{1})I_{E}(E_{2}) ≃ I_{E}(E′_{1})I_{E}(E′_{2}).

Proof. To prove this, we shall use the criterion of Theorem 1. Now by Proposition 49 we have that I_{E}(E_{1}) ⊕ I_{E}(E_{2}) ≃ I_{E}(E′_{1}) ⊕ I_{E}(E′_{2}) ⇔ R ⊕ I_{E}(E_{1})I_{E}(E_{2}) ≃ R′ ⊕ I_{E}(E′_{1})I_{E}(E′_{2}), where R = R(I_{E}(E_{1})) ∩ R(I_{E}(E_{2})) and R′ = R(I_{E}(E′_{1})) ∩ R(I_{E}(E′_{2})). Moreover, by Theorem (45) these modules are isomorphic if and only R = R′ and I_{E}(E_{1})I_{E}(E_{2}) ≃ I_{E}(E′_{1})I_{E}(E′_{2}). Since R(I_{E}(E_{i})) has conductor f_{i} by (50), we see that R has conductor f_{R} = lcm(f_{1}, f_{2}) by (37), and similarly f_{R′} = lcm(f′_{1}, f′_{2}). Thus R = R′ if and only if lcm(f_{1}, f_{2}) = lcm(f′_{1}, f′_{2}), and so the assertion follows from Theorem 1.

If K = ℂ, then above result can be restated in the following form which is essentially Proposition 4.5 of [24].

Corollary 64 Let L_{1}, L_{2}, L′_{1}, L′_{2} ∈ Lat_{F} be lattices in a quadratic field F and let f_{i} = f_{R(L_{i})} and f′_{i} = f_{R(L′_{i})} be the conductors of their associated orders. Then

E_{L_{1}} × E_{L_{2}} ≃ E_{L′_{1}} × E_{L′_{2}} ⇔ L_{1}L_{2} ≃ L′_{1}L′_{2} and lcm(f_{1}, f_{2}) = lcm(f′_{1}, f′_{2}).

Proof. Put E_{i} = E_{L_{i}} and E′_{i} = E_{L′_{i}}. Since End(E_{L_{i}}) ≃ R(L_{i}) by (64), we have f_{E_{i}} = f_{i} and similarly f_{E′_{i}} = f′_{i}. Let R = R(L_{1}) ∩ R(L_{2}) ∩ R(L′_{1}) ∩ R(L′_{2}), and choose n ∈ ℕ such that L := nR ⊂ L_{i} ∩ L′_{i}, for i = 1, 2. Put E = E_{L}. Since End(E) ≃ R ⊂ R(L_{i}) ≃ End(E_{L_{i}}), we see that E_{i} ∈ Isog^{+}(E/ℂ) and similarly E′_{i} ∈ Isog^{+}(E/ℂ). By Corollary 27 we have I_{E}(E_{i}) ≃ L_{i}^{−1}L and I_{E}(E′_{i}) ≃ (L′_{i})^{−1}L. Thus I_{E}(E_{1})I_{E}(E_{2}) ≃ I_{E}(E′_{1})I_{E}(E′_{2}) ⇔ L_{1}^{−1}L_{2}^{−1}L ≃ (L′_{1})^{−1}(L′_{2})^{−1}L ⇔ (L_{1}L_{2})^{−1} ≃ (L′_{1}L′_{2})^{−1}, the latter because R(L) ⊂ R(L_{1}^{−1}) ∩ R((L′_{1})^{−1}). Thus I_{E}(E_{1})I_{E}(E_{2}) ≃ I_{E}(E′_{1})I_{E}(E′_{2}) ⇔ L_{1}L_{2} ≃ L′_{1}L′_{2}, and hence it is clear that the corollary follows from Proposition 63.

We next prove Theorem 4, which is clearly a special case of the following more precise result.

Theorem 65 Let E/K be a CM elliptic curve of discriminant Δ = Δ_{E}. Then there exist bijections between the following sets:

(i) the set Isog^{+}(E/K) of isomorphism classes of elliptic curves E′/K with E′ ∼ E and f_{E′}|f_{E};

(ii) the set of non-zero ideal classes of End(E);

(iii) the set of proper equivalence classes of positive definite binary quadratic forms q with discriminant Δ(q) = Δ;

(iv) the set of End(R)-submodules M of End(E)^{2} of rank 2 with R_{F} (M) = R;

(v) the set of isomorphism classes of abelian surfaces A/K with A ∼ E^{2} and central conductor f_{A} = f_{E};

(vi) the set of isomorphism classes of abelian surfaces A/K with A ∼ E^{2} and discriminant Δ(A/K) = −Δ.

More precisely, the bijection between (i) and (ii) is given by the map I_{E}^{+} of Corollary 19, the bijection between (ii) and (iii) is the map q_{Δ} of Remark 38, the bijection between (iv) and (v) is given by the rule M ↦ (E^{2})_{H(T_{E,2}^{−1}(I_{n}(M)))}, and the sets (v) and (vi) are identical. In addition, the bijection between (i) and (v) and (vi) is induced by the rule E′ ↦ E × E′.

Remark 66 Note that the bijection between the sets described by conditions (iii) and (vi) of Theorem 65 (which is the same as that between the sets (i) and (ii) of Theorem 4) is not given explicitly. However, it can be described as follows: given q ∈ Q_{Δ_{E}}/SL_{2}(ℤ), let E′_{q} = E′_{E,q} ∈ Isog^{+}(E/K) be such that q_{E,E′_{q}}^{+} = q, where q_{E,E′}^{+} is as in Remark 38. Then the map q ↦ A_{q} = E × E′_{q} induces the bijection between the sets (iii) and (vi).

Proof. The equivalence of conditions (i) – (v) is just a restatement of Theorem 5 in the case n = 2. Moreover, by the proof of that theorem and by Remark 62(a) we know that the bijections are as indicated.

It thus remains to show that the sets (v) and (vi) are identical. For this, suppose first that condition (v) holds. Then by what was just said, it follows that A ≃ E′ × E, for some E′ ∈ Isog^{+}(E/K). Thus f_{A} = lcm(f_{E′}, f_{E}) = f_{E} by Corollary 53. It thus follows from Proposition 67 below that Δ(A/K) = −f_{A}^{2}Δ_{F} = −f_{E}^{2}Δ_{F} = −Δ, and so condition (vi) holds.

Conversely, suppose that condition (vi) holds. Since Δ(A/K) = −Δ_{E}, it follows from Proposition 67 below that Δ(A/K) = −f_{A}^{2}Δ_{F}, and so f_{A} = f_{E} because Δ_{E} = f_{E}^{2}Δ_{F}. Thus condition (v) holds.

In the above proof we had used the following fact.

Proposition 67 Let A/K be abelian surface which is isogenous to E ×E, where E/K is a CM elliptic curve. Then there is an integer m such that the discriminant of A/K is Δ(A/K) = −m^{2}f_{A}^{2}Δ_{F}, where F = End^{0}(E). Moreover, m = 1 if there exists an elliptic curve E_{0} ∼ E such that either f_{A}|f_{E_{0}} or such that Δ(A/K)|Δ_{E_{0}}.

Proof. Suppose first that A ≃ E_{1} × E_{2}, for some CM elliptic curves E_{i}/K which are isogenous. Then by [16], Proposition 22, we have NS(A ⊗ \overline{K}) ≃ ℤ^{2} ⊕ Hom(\overline{E}_{1}, \overline{E}_{2}), where \overline{E}_{i} = E_{i} ⊗ \overline{K}. Moreover, since Hom(E_{1}, E_{2}) = Hom(\overline{E}_{1}, \overline{E}_{2}), the explicit isomorphism shows that NS(A) = NS(A ⊗ \overline{K}); cf. [17], (proof of) Lemma 63. Let q_{A} be the intersection form on NS(A), i.e. q_{A}(D) = \frac{1}{2}(D.D), where (D.D) denotes the self-intersection number of a divisor (class) D ∈ NS(A). Then by [16], Proposition 22 (or [17], equation (6)) we have

q_{A} ∼ xy ⊥ q_{E_{1},E_{2}}, (103) where xy denotes the quadratic form associated to the hyperbolic plane and q_{E_{1},E_{2}} is as in §3.4. From (103) it follows immediately that

Δ(q_{A}) = −Δ(q_{E_{1},E_{2}}) = −lcm(f_{E_{1}}, f_{E_{2}})^{2}Δ_{F} = −f_{E_{1}×E_{2}}^{2}Δ_{F} , (104)

where the last two equations follow from equation (80) and Corollary 53, respectively. Thus, the asserted formula holds in this case (with m = 1).

Next, suppose that there exists an elliptic curve E_{0} such that f_{A}|f_{E_{0}}. Then by Corollary 16 there is an elliptic curve E_{1} ∼ E_{0} such that f_{A} = f_{E_{1}}, and then by Theorem 5 there exists E′ ∈ Isog^{+}(E_{1}/K) such that A ≃ E′ × E_{1}. Thus, the assertion follows from the previously discussed case.

Now suppose that A/K is any abelian surface with A ∼ E^{2}. Then NS(A) ⊗ ℚ ≃ NS(E^{2}) ⊗ ℚ = NS(E^{2} ⊗ \overline{K}) ⊗ ℚ, the latter by the discussion at the beginning of this proof. Thus NS(A) ⊗ ℚ = NS(\overline{A}) ⊗ ℚ, where \overline{A} := A ⊗ \overline{K}, and so m := [NS(\overline{A}) : NS(A)] < ∞. It thus follows that Δ(A/K) = m^{2}Δ(\overline{A}/\overline{K}). Now by Theorem 58 we have that \overline{A} ≃ \overline{E}_{1} × \overline{E}_{2} for some elliptic curves \overline{E}_{i} ∼ E ⊗ \overline{K}, and so by the previously discussed case we have Δ(\overline{A}/\overline{K}) = −f 2 ; \overline{A}Δ_{F}. Since f_{A} = f\overline{A} (as was explained in the proof of Theorem 54), we obtain that Δ(A/K) = −m^{2}f_{A}^{2}Δ_{F}, as claimed.

Finally, suppose that Δ(A/K)|Δ_{E_{0}} = f_{E_{0}}^{2}Δ_{F}, for some elliptic curve E_{0}/K. Since we have that Δ(A/K) = −m^{2}f_{A}^{2}Δ_{F} by what was just shown, we see that mf_{A} = f_{E_{0}} and so f_{A}|f_{E_{0}}. But in this case we showed above that we must have m = 1.

In view of the close relationship (103) that exists between the quadratic form q_{A} defined by the intersection pairing on NS(A) and the form q_{E,E′}^{+} which defines the one of the bijections of Theorem 65 (cf. Remark 66), it might be tempting to try to use q_{A} in place of q_{E,E′}^{+} to classify the abelian surfaces A/K with A ∼ E^{2}. This, however, does not lead to a bijection, as the following result shows.

Corollary 68 Let E/K be a CM elliptic curve with discriminant Δ = Δ_{E}, and let q ∈ Q_{Δ} be a positive binary quadratic form of discriminant Δ and content c. Then the number

N_{q} := #({A ∼ E × E : q_{A} ∼ xy ⊥ (−q)}/≃)

of isomorphism classes of abelian surfaces A/K which are isogenous to E^{2} and whose intersection form q_{A} is equivalent to xy ⊥ (−q) is equal to the number of primitive forms in the principal genus of primtive forms of discriminant Δ′ = Δ/c^{2}. Thus, if g(Δ′) denotes the number of genera of discriminant Δ′, then we have

(105) N_{q} = \frac{h(Δ′)}{g(D′)}, where Δ′ = Δ(q)/cont(q)^{2}.

Proof. Let q^{A} denote the (proper) equivalence class of binary forms of discriminant Δ which corresponds to A/K via the bijection of Theorem 65. Then by (103) and (83) we have q_{A} ∼ xy ⊥ (−q^{A}). Thus, q_{A} ∼ xy ⊥ (−q) ⇔ xy ⊥ (−q^{A}) ∼ xy ⊥ (−q) ⇔ q^{A} and q are in the same genus, the latter by Remark 27 of [17]. Since the number of such forms q^{A} equals the number of forms in the principal genus (and is given by the formula of (105)), the assertion follows.

We also note that the above Proposition 67 allows us to reformulate Theorem 3 in the case of surfaces as follows.

Theorem 69 Let A/K be an abelian suface which is isogenous to E × E, where E/K is a CM elliptic curve. Then there exist CM elliptic curves E_{1}, E_{2}/K such that A ≃ E_{1} × E_{2} if and only if Δ(A/K)|Δ_{E_{0}}, 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_{E_{0}}, for some E_{0} ∼ E. Thus, by (104) we have that Δ(A/K) = −f_{A}^{2}Δ_{F} |f_{E_{0}}^{2}Δ_{F} = Δ_{E_{0}}, as claimed.

Conversely, suppose that Δ(A/K)|Δ_{E_{0}}, for some E_{0} ∼ E. Then by Proposition 67 we have that Δ(A/K) = −f_{A}^{2}Δ_{F}, and so it follows that f_{A}|f_{E_{0}}. By Theorem 3 we thus 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 following result 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 is a bijection between the following sets:

(i) the set Q/SL_{2}(ℤ) of proper equivalence classes of positive definite binary quadratic forms;

(ii) the set of isomorphism classes of abelian surfaces A/K with Picard number ρ(A) := rank(NS(A)) = 4.

Remark 71 In the paper [24], the abelian surfaces A/ℂ with ρ(A) = 4 are called singular abelian surfaces; cf. [24], p. 459. This terminology unfortunately conflicts with classical terminology of singular abelian varieties used in the 19th century: these are 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 of simple abelian surfaces in characteristic 0 (cf. [20], p. 202) that if ρ(A) = 4, then A cannot be simple and so one sees easily that A ∼ E^{2}, where E/K is a CM elliptic curve. 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 some CM elliptic curve E/K.

To describe the bijection, fix for each discriminant Δ < 0 an elliptic curve E_{Δ} with Δ_{E_{Δ}} = Δ (which exists by Proposition 33(a)). Let

A(Δ) := {A/K : A ∼ E_{Δ}^{2} and Δ(A/K) = −Δ}/≃

denote the set of isomorphism classes of abelian surfaces A/K which are isogenous to E_{Δ} × E_{Δ} and have discriminant Δ(A/K) = −Δ. Now if A ∼ E^{2}, where E/K is some CM curve and if Δ(A/K) = −Δ, then it follows from Theorem 65 that E ∼ E_{Δ}, and so we see that the set A described by (ii′) is the disjoint union of the sets A(Δ), where Δ runs over all negative discriminants. We thus see from Remark 66 that the rule q ↦ E_{Δ} × E′_{E_{Δ},q} (notation as in Remark 66) induces the desired bijection

Q/SL_{2}(ℤ) = ˙ ⋃_{Δ<0} Q_{Δ}/SL_{2}(ℤ) →^{∼} ˙ ⋃_{Δ<0} A(Δ) = A.

Remark 72 The same argument as above shows that if K is an algebraically closed field of characteristic p ≠ 0, then we have a bijection between the following two sets:

(i) the set Q^{(p)}/SL_{2}(ℤ) of proper equivalence classes of positive definite binary quadratic forms whose discriminant Δ satisfies (\frac{Δ}{p}) = 1;

(ii) the set of isomorphism classes of abelian surfaces A/K such that A ∼ E^{2}, for some CM elliptic curve E/K.

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