Products of CM elliptic curves

Ernst Kani

Contents

1. Introduction  1
2. Isogenies, subgroups and ideals  5
2.1 Kernel ideals and ideal subgroups  5
2.2 The invariant I_{A}(B)  8
2.3 Homomorphisms  10
2.4 The quadratic case  15
3. CM elliptic curves  18
3.1 Kernel ideals and ideal subgroups of CM elliptic curves  18
3.2 The case K = ℂ  25
3.3 Endomorphism rings  27
3.4 The quadratic form q_{E_{1},E_{2}}  29
4. Product abelian varieties  32
4.1 Kernel ideals and ideal subgroups of A^{n}  32
4.2 The theorems of Steinitz and of Borevich and Faddeev  35
4.3 Products of CM elliptic curves  40
4.4 Abelian product surfaces  47
References  51

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[27] 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[24] 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 endomorphism ring conductor (or e-conductor) f_{E} = f_{End(E)} of E is a multiple of those of E_{i} and E′_{i} for all i. Note that Proposition 29 below guarantees that such a “suitable” elliptic curve E always exists. 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 which are isogenous to E and which satisfy the conditions 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 in view of Theorem 20 below, this result is actually a special case of a very general result about isomorphisms of product abelian varieties; cf. Theorem 46.

At first sight, the criterion of Theorem 1 does not seem to specialize to that of [27], Proposition 4.5. However, by using results due to Steinitz[29] and/or to Borevich/Faddeev[1] on the structure of R-modules when R is a quadratic order, one can easily deduce their result from that of Theorem 1; cf. subsection 4.4.

The following result shows that the general isomorphism problem can be reduced, at least in principle, to the previously considered subproblem.

Theorem 2 If A/K is an abelian variety which is isogenous to E^{n}, where E/K is a CM elliptic curve, then there exist CM elliptic curves E_{1}/K, …, E_{n}/K such that A ≃ E_{1} × … × E_{n}.

In the case that K = ℂ, this theorem was proved by Shioda and Mitani[27] when n = 2 (see also Ruppert[23]), and their work was extended to the general case by Lange[21]. Moreover, Schoen[24] gave a beautiful analysis of this theorem; cf. Remark 59 below. Here we give a variant of Schoen’s proof and show how to deduce the general case from the case K = ℂ; cf. §4.3.

Note that if we combine Theorem 2 with Theorem 1, then we get an indirect solution of the isomorphism problem mentioned at the beginning. However, in order to obtain a more intrinsic solution in terms of the ideal class I_{E^{n}}(A) = Hom(A, E^{n})π_{A} of End(E^{n}) ≃ M_{n}(End(E)), considerable further work is necessary.

Such a solution is presented in the following theorem (see also Theorem 63 below), which can be viewed as a generalization of Theorem 1. To state it in a compact form, we use here 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 the imaginary quadratic field F = Z(End^{0}(A)); cf. §4.3.

Theorem 3 Let E_{1}/K, …, E_{n}/K be a set of pairwise isogenous CM elliptic curves, and let A/K be an abelian variety which is isogenous to E_{1}^{n}. Then there exists an elliptic curve E/K which is isogenous to E_{1} such that f_{E}|f_{A} and f_{E}|f_{E_{i}}, for 1 ≤ i ≤ n. Moreover, for any such E we have that

A ≃ E_{1} × … × E_{n} ⇔ Hom(E^{n}, E) ⊗_{End(E^{n})} I_{E^{n}}(A) ≃ I_{E}(E_{1}) ⊕ … ⊕ I_{E}(E_{n})

as End(E)-modules.

In the case that n = 2 and K = ℂ, Shioda and Mitani[27] 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 of their result 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 endomorphism ring discriminant (or e-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 with e-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 (partial) generalization to abelian varieties of arbitrary dimension n.

Theorem 5 Let E/K be a CM elliptic curve with e-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 64(b) below.

The basic technique for proving these theorems is the method of Deuring[10], Shimura and Taniyama[26], and Waterhouse[30] 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 essential 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. The key result is the following theorem (see also Theorem 57 below) which classifies the ideal subgroups of A = E^{n}.

Theorem 6 Let E/K be 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 e-conductor f_{E} of E.

Note that the above theorem, which can be considered to be the main (technical) result of this paper, is already interesting in the case that n = 1; in this case it is essentially Theorem 20(b) below.

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 20, Corollary 21 and Proposition 29 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[27].

Acknowledgements. 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[30], §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[26], §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 [22], 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 ∩H_{i} 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 (since the finite subgroup schemes satisfy the descending chain condition (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 [26] and [30]), 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 [30], §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 that 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 that 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[30], 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 of kernel ideals 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 the quotient map f : A → A), 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 any 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 that 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 that 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 that

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} = β_{K′/K}^{A} : 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; cf. (20) below. For this, we first observe the following general rules concerning isomorphisms of quotients.

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. [30], p. 532) that

(17) A_{H_{1}} ≃ A_{H_{2}} ⇔ f^{−1}(H_{1}) = [n]^{−1}(H_{2}), for some f ∈ R ∩ R̃^{×} and n ∈ ℕ,
(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 that [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 that

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 that 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 the converses of (19) and (20) are false in general, 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 the I(H)- construction. Indeed, if we take H_{2} = 0, then we have

H(H, 0) = Hom(A_{H}, A)π_{H} = I(H) (26) 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}). (27)

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.

We also note that since Hom(A_{H_{1}}, A_{H_{2}}) is an (End(E_{H_{2}}), End(E_{H_{1}}))-bimodule, it follows from (27) that H(H_{1}, H_{2}) is an (E(H_{2}), E(H_{1}))-bimodule, i.e. we have that

E(H_{2})H(H_{1}, H_{2})E(H_{1}) = H(H_{1}, H_{2}).

(c) We have that 1_{A} ∈ H(H_{1}, H_{2}) if and only if H_{1} ≤ H_{2}. Indeed, if H_{1} ≤ H_{2}, then 1_{A} = Φ_{H_{1},H_{2}}(π_{H_{1},H_{2}}) ∈ H(H_{1}, H_{2}) because Φ_{H_{1},H_{2}}(π_{H_{1},H_{2}}) = \frac{1}{n_{H_{2}}} π′_{H_{2}}π_{H_{1},H_{2}}π_{H_{1}} = \frac{1}{n_{H_{2}}} π′_{H_{2}}π_{H_{2}} = 1_{A}. Conversely, if 1_{A} ∈ H(H_{1}, H_{2}), then 1_{A} = \frac{1}{n_{H_{2}}} π′_{H_{2}}hπ_{H_{1}}, for some h ∈ Hom(A_{H_{1}}, A_{H_{2}}) and then π_{H_{2}} = π_{H_{2}} \frac{1}{n_{H_{2}}} π′_{H_{2}}hπ_{H_{1}} = hπ_{H_{1}}, so H_{1} ≤ H_{2}.

In addition, we observe that if H_{1} ≥ H′_{1} and H_{2} ≤ H′_{2}, then it follows from (27) and the above identities Φ_{H′_{1},H_{1}}(π_{H′_{1},H_{1}}) = 1_{A} = Φ_{H_{2},H′_{2}}(π_{H_{2},H′_{2}}) 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)

(d) If H̄ ≤ A_{H} is a finite subgroup scheme of A_{H}, then we have that

Φ_{H}(I(H̄)) = H(π_{H}^{−1}(H̄), H). (29)

Indeed, if H_{1} = π_{H}^{−1}(H̄), then we have that π_{H_{1}} = πH̄π_{H}, and (A_{H})H̄ = A_{H_{1}}, and hence Φ_{H}(I(H̄)) = \frac{1}{n_{H}} π′_{H}I(H̄)π_{H} = \frac{1}{n_{H}} π′_{H}Hom((A_{H})H̄, A_{H})πH̄π_{H} = \frac{1}{n_{H}} π′_{H}Hom(A_{H_{1}}, A_{H})π_{H_{1}} = H(H_{1}, H) = H(π_{H}^{−1}(H̄), H).

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

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

Moreover, if H_{2} is an ideal subgroup, then equality holds in (30), and if I_{1} is a kernel ideal, then equality holds in (31).

Proof. To prove (30), 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 (30) follows.

In order to prove (31), 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})\} . (32)

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 (32) 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 there exists 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 (32).

Now we prove (31). 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 (32). This proves (31).

By combining (30) and (31) 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}))), (33)

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 (30). Similarly, combining (30) and (31) 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}) (34)

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

As an application, we can deduce the fact that an ideal subgroup H satisfies the extra condition that Z(R) ⊂ Z(E(H)), where Z(R) = {z ∈ R : xz = zx, ∀x ∈ R} denotes the centre of R, and Z(E(H)) is defined similarly. This condition is usually not true for arbitrary subgroup schemes of A, as we shall see in Remark 19(b) below.

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

Proof. By hypothesis, H = H(I), for some regular left R-ideal I, and so (I : I) ⊂ E(H) by (31). Since I is a left R-ideal, it follows that Z(R) ⊂ (I : I) (because z ∈ Z(R) ⇒ Iz = zI ⊂ I), and so Z(R) ⊂ E(H). Since R and E(H) are lattices in R̃, we have that Z(R) = R ∩ Z(R̃) and Z(E(H)) = E(H) ∩ Z(R̃), and so the assertion follows.

The following result, which is a variant of a result of [30], 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)}), (35)

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 (30), we have IJ ⊂ I(H)J ⊂ I(H) ⊂ R, and hence IJ is an R-ideal. Moreover, IJ is regular because by hypothesis there exist α ∈ I ∩R̃^{×} and β ∈ J ∩R̃^{×}, and so αβ ∈ IJ ∩R̃^{×}, which means that IJ is regular.

To prove (35), 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, (36)

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 (36), 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 (36) follows because [n] is faithfully flat.

Using (36), 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 (35).

We conclude this subsection by observing that under suitable hypotheses, the Hconstruction commutes with base-change; this generalizes the second formula of Remark 7(e).

Proposition 13 Let K′/K be a field extension, and suppose that dim End^{0}(A) = dim End^{0}(A ⊗ K′). Then the base-change maps

β_{K′/K} : End(A) →^{∼} End(A ⊗ K′) and β_{K′/K}^{0} : End^{0}(A) →^{∼} End^{0}(A ⊗ K′)

are isomorphisms, and we have for all finite subgroups H_{1}, H_{2} ≤ A that

β_{K′/K}^{0}(H(H_{1}, H_{2})) = H(H_{1} ⊗ K′, H_{2} ⊗ K′). (37)

To prove this, we first verify the following general facts.

Lemma 14 Let K′/K be a field extension.

(a) If H_{1} and H_{2} are subgroup schemes of A/K, then

H_{1} ≤ H_{2} ⇔ H_{1} ⊗ K′ ≤ H_{2} ⊗ K′. (38)

(b) If B/K is another abelian variety, then the cokernel of the base-change map

β_{K′/K}^{A,B} : Hom(A, B) → Hom(A ⊗ K′, B ⊗ K′)

is torsion-free, and so β_{K′/K}^{A,B} is surjective and hence is an isomorphism if and only if Hom(A, B) and Hom(A ⊗ K′, B ⊗ K′) have the same rank.

(c) If dim End^{0}(A) = dim End^{0}(A ⊗ K′), then β_{K′/K}^{A_{1},A_{2}} is an isomorphism for all abelian varieties A_{1} ∼ A_{2} ∼ A.

Proof. (a) If j_{H_{i}} : H_{i} ↪ A denotes the inclusion map, then j_{H_{i}⊗K′} = j_{H_{i}} ⊗K′. Thus, if H_{1} ≤ H_{2}, then j_{H_{1}} = j_{H_{2}} ∘ h for some h and so j_{H_{1}⊗K′} = j_{H_{2}⊗K′} ∘ h ⊗ K′, which means that H_{1} ⊗ K′ ≤ H_{2} ⊗ K′. Conversely, if this holds, then j_{H_{1}⊗K′} = j_{H_{2}⊗K′} ∘ h′ ⊗ K′, for some h′ : H_{1} ⊗ K′ → H_{2} ⊗ K′. Since j_{H_{2}} is a monomorphism, it follows that h′ satisfies the descent condition for K′/K (cf. [4], p. 136) and so h′ = h ⊗ K′, for some h : H_{1} → H_{2}. Thus H_{1} ≤ H_{2}, as claimed.

(b) Let h′ ∈ Hom(A_{K′}, B_{K′}), where A_{K′} = A ⊗ K′ and B_{K′} = B ⊗ K′, and suppose that there exists n > 0 such that nh′ ∈ β_{K′/K}(Hom(A, B)). Thus nh′ = h ⊗ K′ = h_{K′}, for some h ∈ Hom(A, B), and so A[n] ⊗ K′ = A_{K′}[n] ≤ Ker(h_{K′}) = Ker(h) ⊗ K′. By part (a) we have that A[n] ≤ Ker(h), and so h = ng, for some g ∈ Hom(A, B). Thus nh′ = ng_{K′}, and hence h′ = g_{K′} because [n] is an isogeny. Thus, h′ ∈ β_{K′/K}(Hom(A, B), which shows that Coker(β_{K′/K}) is torsionfree.

Now suppose that Hom(A, B) and Hom(A_{K′}, B_{K′}) have the same rank. Since β_{K′/K} is injective and Hom(A_{K′}, B_{K′}) is free of finite rank, it follows that Coker(β_{K′/K}^{A,B}) is a torsion group and hence equal to 0 by what was just shown. Thus β_{K′/K}^{A,B} is surjective and hence is an isomorphism.

(c) Since A_{1} ∼ A_{2} ∼ A, we see that Hom(A_{1}, A_{2}) has the same rank as End(A) and hence rank(Hom(A_{1}, A_{2})) = dim End^{0}(A). Similarly, rank(Hom(A_{1} ⊗ K′, A_{2} ⊗ K′)) = dim(End^{0}(A ⊗ K′)), and so the assertion follows from part (b).

Proof of Proposition 13. The first assertion follows from lemma 14(c). To verify (37), we first observe that we have that

Φ_{H_{1}⊗K′,H_{2}⊗K′} ∘ β_{K′/K}^{A_{H_{1}},A_{H_{2}}} = β_{K′/K}^{0} ∘ Φ_{H_{1},H_{2}}; (39)

this follows immediately from the definition (24) and the fact that π_{H_{i}⊗K′} = π_{H_{i}} ⊗ K′, π′_{H_{i}⊗K′} = π′_{H_{i}} ⊗ K′ and |H_{i} ⊗ K| = |H_{i}|. Since β_{K′/K}^{A_{H_{1}},A_{H_{2}}} is an isomorphism by Lemma 14(c), the formula (37) follows immediately from (39).

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 that

(40) 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 a unique order of conductor f. Moreover, if R_{1}, R_{2} are orders in F then we have that

(41) 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}, (42)

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}). (43)

In addition, we have the following formulae (44) and (45) which 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 (45) is stated without proof on p. 71 of [13].

Lemma 15 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}. (44)

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}. (45)

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 (44) 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 (44) 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 (44).

The formula (45) follows immediately from (44) 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})]. (46)

Indeed, multiplying (44) by L_{2}^{−1}, we obtain with (46) 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 (46). For this, we first note that since f = f_{R_{0}} by (43), we have that [R_{0} : R_{1}] = \frac{f_{1}}{f}, which is the second equality of (46). 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 (46).

Corollary 16 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 (45) to L_{1} = R and L_{2} = I yields

(R : I) = [R(I) : R]I^{−1} (47)

because here f_{2}|f_{1}, so f = f_{2} and \frac{f_{1}}{f} = [R(I) : R]. Next, apply (45) to L_{1} = R and L_{2} = (R : I). Since R(L_{2}) = R(I^{−1}) = R(I) by (47), we see that in this case (45) 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 17 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). (48)

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 16 that every non-zero ideal of R is divisorial and hence is 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 18 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′. Thus A′ := A_{H(I)} is isogenous to A and End(A′) ≃ (I : I) = R(I) = R′ by (48).

Remark 19 (a) Note that if we drop the hypothesis that R ⊂ R′ in Corollary 18, then the corresponding statement is in general no longer true; cf. §3.3 below.

(b) We can use the above Corollary 18 to construct an abelian variety A′ with a finite subgroup scheme H′ such that End(A′) = Z(End(A′)) ⊄ E(H′) = Z(E(H′)). In particular, H′ is not an ideal subgroup by Corollary 11.

Indeed, suppose that A/K is an abelian variety such that End^{0}(A) = F is a quadratic field but R := End(A) ≠ 𝔒_{F}. (For example, we can take A = E to be a CM elliptic curve over a sufficiently large ground field K; cf. §3.3.) Then by Corollary 18 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 that (A′)_{H′} ≃ A, so E(H′) ≃ End(A) = R. Thus 𝔒_{F} ≃ End(A′) ⊄ E(H′), as desired.

3 CM elliptic curves

3.1 Kernel ideals and ideal subgroups of CM elliptic curves

We now apply the theory of the previous section to the case that A = E is an 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 agrees with that of Serre/Tate[25], §6, but is slightly more restrictive than that of Lang[20] or of Silverman[28] 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 endomorphism ring conductor (or e-conductor) and endomorphism ring discriminant (or e-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[30]. In particular, part (b) seems to be new.

Theorem 20 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})}) →^{∼} H(H(I_{1}), H(I_{2})) = (I_{1} : I_{2}) (49)

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

(50) H is an ideal subgroup ⇔ R ⊂ E(H) ⇔ 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})), (51)

and we have E_{H_{1}} ≃ E_{H_{2}} ⇔ I(H_{1}) ≃ I(H_{2}). (52)

Proof. (a) This is a special case of Proposition 17.

(b) We first observe that it is enough to verify (50), for then (51) follows immediately from Proposition 10 and (52) follows from (21).

To prove (50), note first that the second equivalence follows from (41) (because E(H) ≃ End(E_{H})), and that one direction (⇒) of the first equivalence follows directly from Corollary 11. Conversely, suppose 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 (49) we have that End(R′) ≃ (I : I) = R′. Moreover, if π := π_{H(I)} : E → E′ and n := deg(π), then by (4) we have that 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 [30], 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 21 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} : Isog(E/K) → Id(R)/≃. We denote its restriction to Isog^{+}(E/K) by I_{E}^{+}.

To show that I_{E}^{+} is surjective, let I ∈ Id(R), and put E′ = E_{H(I)}. Then by (49) 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 (52) that E_{1} ≃ E_{2}, and so I_{E}^{+} is injective.

Remark 22 It follows from (49) that for any R-ideal I we have that

(53) End(E_{H(I)}) ≃ (I : I) = R(I) and so f_{E_{H(I)}} = f_{E}/[R(I) : R_{E}].

Moreover, by (51) we have that

(54) 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_{E}). (55)

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 with e-conductor f|f_{E} (which exists by Corollary 18), we see that Isog^{+}(E/K) is also finite set of cardinality

#(Isog^{+}(E/K)) = ∑_{f|f_{E}} h(Δ_{E}/f^{2}). (56)

On the other hand, the set Isog(E/K) is often infinite; for example, this is the case when K is algebraically closed; cf. Proposition 37 below.

We can also give a “numerical criterion” to detect ideal subgroups; cf. Proposition 27 below. For this we first prove the following result which is also of independent interest.

Proposition 23 If I is a non-zero ideal of R = End(E), then

|H(I)| = [R : I]. (57)

Remark 24 If I is not invertible (i.e. if R(I) ≠ R), then the above formula (57) 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).

The proof of Proposition 23 is based on the following simple fact.

Lemma 25 If H_{1}, H_{2} ≤ E are two subgroup schemes, then we have that

H(H_{2}, H_{1}) = \frac{|H_{2}|N(H(H_{1}, H_{2}))}{|H_{1}|}H(H_{1}, H_{2})^{−1}. (58)

In particular, for any subgroup scheme H ≤ E we have that

H(0, H) = \frac{N(I(H))}{|H|}I(H)^{−1}. (59)

Proof. In view of (42) we see that (58) is equivalent to the formula

n_{1}H(H_{2}, H_{1}) = n_{2}σ(H(H_{1}, H_{2})), (60)

where n_{i} := |H_{i}|. To prove this, we shall use the fact that for h ∈ End(E) we have that σ(h) = ĥ, where the dual isogeny ĥ is as in [28], §III.6, and that π′_{i} = π̂_{i}, where π_{i} = π_{H_{i}}. Thus, if h̃ = \frac{1}{n_{2}} π′_{2}hπ_{1} ∈ H(H_{1}, H_{2}), where h ∈ Hom(E_{H_{1}}, E_{H_{2}}), then n_{2}σ(h̃) = σ(π′_{2}hπ_{1}) = π̂_{1}ĥ(π′_{2})^{∧} = π′_{1}ĥπ_{2} ∈ n_{1}H(H_{2}, H_{1}) because ĥ ∈ Hom(E_{H_{2}}, E_{H_{1}}). This proves that n_{2}σ(H(H_{1}, H_{2})) ⊂ n_{1}H(H_{2}, H_{1}). By reversing the roles of H_{1} and H_{2} we also have that n_{1}σ(H(H_{2}, H_{1})) ⊂ n_{2}H(H_{1}, H_{2}), and hence by applying σ to both sides we obtain the opposite inclusion. This proves (60) and hence also (58). Finally,

(59) follows immediately from (58) (and (26)) by taking H_{2} = 0 and H_{1} = H in (58).

Proof of Proposition 23. Since I is a kernel ideal by Theorem 20(a), we have by (49) that (R : I) = H(H(R), H(I)) = H(0, H(I)). Since I(H(I)) = I, it thus follows from (47) and (59) that n[R(I) : R]I^{−1} = n(R : I) = nH(0, H(I)) = [R(I) : I]I^{−1}, where n = |H(I)|, and so nI^{−1} = [R : I]I^{−1}. Thus, by taking norms, we obtain that n^{2} = [R : I]^{2}, and so (57) follows.

As an application of (57), we prove the following formulae which will be used below.

Corollary 26 Let H_{1} and H_{2} be two ideal subgroups of E and put 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

(61) 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}| = |H(I(H_{i}))| and L_{i} = I(H_{i}). Since R(L_{i}) = E(H_{i}) ⊃ R by (50) and (51), we have that [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 (57). This proves the first equality of (61).

Now by (51) and (45) 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 (61), 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 (61) because lcm(f_{1}, f_{2}) = \frac{f_{1}f_{2}}{f}.

We now come to the following numerical criterion of ideal subgroups.

Proposition 27 For any finite subgroup scheme H ≤ E we have that

[R : I(H)] = [R(I(H)) : E(H)]|H|, (62)

and hence H is an ideal subgroup if and only if [R : I(H)] = |H|.

Remark 28 Note that formula (57) is a special case of (62). Indeed, since H := H(I) is an ideal subgroup, we have that R(I(H)) = E(H) by (51), so [R : I(H(I))] = |H(I)| by (62). Since I(H(I)) = I by Theorem 20(a), we see that this yields equation (57).

Proof of Proposition 27. Since H′ := H(I(H)) is an ideal subgroup and since I(H′) = I(H) by (9), it follows from Proposition 10 that

E(H′) = (I(H) : I(H)) =: R(I(H)) = H(H, H′). (63)

Since H ≤ H′, we have that π_{H′} = π_{0}π_{H}, where π_{0} = π_{H,H′}, and so H′ = π_{H}^{−1}(H_{0}), where H_{0} = Ker(π_{0}) ≤ E_{H}. We thus obtain from (29), (60) and (63) that

Φ_{H}(I(H_{0})) = H(H′, H) = n_{0}σ(H(H, H′)) = n_{0}σ(R(I(H))) = n_{0}R(I(H)), (64)

where n_{0} := \frac{|H′|}{|H|} = |H_{0}|. Since (E_{H})_{H_{0}} ≃ E_{H′}, it follows from this and (63) that E(H_{0}) ≃ E(H′) = R(I(H)). On the other hand, since I(H) is an E(H)-module by Remark 9(b), we have that

End(E_{H}) ≃ E(H) ⊂ E(H)R ⊂ R(I(H)), (65) and so End(E_{H}) ⊂ E(H_{0}) ≃ R(I(H)). It thus follows from the criterion (50) that H_{0} is an ideal subgroup of E_{H}. Thus H(I(H_{0})) = H_{0} and so by (57) we obtain that n_{0} = [End(E_{H}) : I(H_{0})] = [E(H) : Φ_{H}(I(H_{0}))]. This, together with (64), yields that [R(I(H)) : E(H)]n_{0} = [R(I(H)) : Φ_{H}(I(H_{0}))] = [R(I(H)) : n_{0}R(I(H))] = n_{0}^{2}, and hence [R(I(H)) : E(H)] = n_{0}. We thus obtain from (57) that [R : I(H)] = |H′| = n_{0}|H| = [R(I(H)) : E(H)]|H|, which proves (62).

It remains to prove the last statement. If H is an ideal subgroup, then H(I(H)) = H, and hence [R : I(H)] = |H| by (57). Conversely, if [R : I(H)] = |H|, then R(I(H)) = E(H) by (62), and so also E(H) = E(H)R by (65). Thus R ⊂ E(H), and hence H is an ideal subgroup by (50).

We can use the above numerical criterion to prove following existence result which plays an important role in the study of products of CM elliptic curves; cf. §4.3.

Proposition 29 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}}).

To prove this, we shall use the following (technical) facts.

Lemma 30 (a) If H is a finite subgroup of E, then Φ_{H}(I(Ker(π′_{H}))) = σ(I(H)), and hence R(I(Ker(π′_{H}))) ≃ R(I(H)).

(b) If E(H) ⊂ R, then R(I(H)) = R.

(c) If H_{1} and H_{2} are two finite subgroups of E with (|H_{1}|, |H_{2}|) = 1 and E(H_{i}) ⊂ R, for i = 1, 2, then E(H_{1} + H_{2}) = E(H_{1}) ∩ E(H_{2}).

Proof. (a) Put H̄ = Ker(π′_{H}). Since π′_{H}π_{H} = [n]_{E}, where n = |H|, we have that π_{H}^{−1}(H̄) = E[n], and so Φ_{H}(I(H̄)) = H(E[n], H) = nσ(H(H, E[n])) by (29) and (60). Since E[n] = H(Rn) is an ideal subgroup, we have by Proposition 10 that H(H, E[n]) = (I(H) : Rn) = \frac{1}{n}I(H), and so the first assertion follows. Moreover, since Φ_{H}^{0} is an isomorphism, we have that R(I(H̄)) ≃ R(Φ_{H}(I(H̄))) = R(σ(I(H))) = R(I(H)), and so the second assertion follows.

(b) Again, put H̄ = Ker(π′_{H}). Since (E_{H})H̄ ≃ E, we have that E(H̄) ≃ R ⊃ E(H) ≃ End(E_{H}), and so H̄ is an ideal subgroup of E_{H} by (50). Thus E(H̄) = R(I(H̄)) by (51), and so by part (a) we obtain that R(I(H)) ≃ R(I(H̄)) = E(H̄) ≃ R, and hence the assertion follows.

(c) First note that for any two finite subgroups H_{1}, H_{2} of E we have that

I(H_{1})I(H_{2}) ⊂ I(H_{1} + H_{2}) ⊂ I(H_{1}) ∩ I(H_{2}). (66)

Indeed, since H_{i} ≤ H_{1}+H_{2}, we have that I(H_{1}+H_{2}) ⊂ I(H_{i}) by (6) and so the second inclusion of (66) 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 that 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 (66) holds.

Put n_{i} = |H_{i}|, N_{i} = [R : I(H_{i})], and e_{i} = [R : E(H_{i})] = [R(I(H_{i})) : E(H_{i})], the latter by part (b). Thus, N_{i} = e_{i}n_{i} by (62). Since H_{i} ≤ Ker([n_{i}]), we have that 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.

Next we observe 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 (66). Using (65) and (43), we obtain for H := H_{1} + H_{2} that E(H) ⊂ R(I(H)) = R(I(H_{1})I(H_{2})) = R(I(H_{1}))R(I(H_{2})) = R · R = R, where the second last equality follows from part (b). Thus E(H) ⊂ R.

Next we note that H_{1}∩H_{2} = 0 (because |H_{1}∩H_{2}| | (n_{1}, n_{2}) = 1), and so |H| = n_{1}n_{2}. Moreover, since I(H) = I(H_{1})∩I(H_{2}), we see that [R : I(H)] = [R : I(H_{1})∩I(H_{2})] = N_{1}N_{2}, the latter because I(H_{1})+I(H_{2}) = R. Thus, by (62) and part (b) we obtain that N_{1}N_{2} = [R(I(H)) : E(H)]n_{1}n_{2} = [R : E(H)]n_{1}n_{2}, and so [R : E(H)] = \frac{N_{1}N_{2}}{n_{1}n_{2}} = 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 (41) that E(H) = E(H_{1}) ∩ E(H_{2}), as claimed.

Proof of Proposition 29. 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 (41), we see that (f_{1}, f_{2}) = 1.

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 (49) we have E(H(I_{i})) = R(I_{i}) = R(f_{i}R′) = R′, it follows from Remark 22 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^{×} is chosen suitably), we can assume that (N(I), f_{1}) = 1; cf. [7], p. 142, 143.

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, since |H_{1}| = deg(π′_{1}) = deg(π_{1}) = f_{1}, and since |H_{2}| = deg(π′_{2}) deg(π_{H(I)}) = f_{2}N(I), we see that (|H_{1}|, |H_{2}|) = 1. Thus, by Lemma 30(c) we have that 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 (41). Thus (E′_{1})_{H_{1}+H_{2}} has e-conductor lcm(f_{E_{1}}, f_{E_{2}}), and so the assertion follows because E ∼ E′_{1} ∼ (E′_{1})_{H_{1}+H_{2}}.

We can use the above result to give the following useful description of the ring R(I(H)).

Proposition 31 For any finite subgroup H of E we have that R(I(H)) = E(H)R.

Proof. By Proposition 29 there is an elliptic curve E′ ∼ E with f_{E′} = lcm(f_{E}, f_{E_{H}}). Fix an isogeny π : E′ → E, and put H_{0} = Ker(π) and H_{1} = π^{−1}(H). Since E′_{H_{0}} ≃ E and E′_{H_{1}} ≃ E_{H}, we have that E(H_{0}) ≃ End(E) = R and E(H_{1}) ≃ End(E_{H}) ≃ E(H), and so by (50) we see that H_{0} and H_{1} are ideal subgroups of E′ (because f_{E′_{H_{0}}} = f_{E}|f_{E′} and f_{E′_{H_{1}}} = f_{E_{H}}|f_{E′}). It thus follows from (29) and (51) that Φ_{H_{0}}(I(H)) = H(H_{1}, H_{0}) = (I(H_{1}) : I(H_{0})), and hence R(I(H)) ≃ R(Φ_{H_{0}}(I(H))) = R((I(H_{1}) : I(H_{0})) = R(I(H_{0}))R(I(H_{1})), the latter by (45) and (43). Since the H_{i} are ideal subgroups, we have by (51) that R(I(H_{i})) = E(H_{i}), so R(I(H)) ≃ E(H_{0})E(H_{1}) ≃ RE(H), and hence the assertion follows.

Remark 32 In view of Proposition 31, we can re-write the formula (62) as

[R : I(H)] = [RE(H) : E(H)]|H|. (67)

In the preprint [18], this formula, as well as Propositions 23 and 31, were derived (under an extra hypothesis) by a more complicated proof which used the complex theory (cf. Remark 35) and a specialization technique (not presented here).

It is interesting to note that the formula (67) implies immediately the non-trivial part of the criterion (50). Indeed, suppose that H ≤ E satisfies R ⊂ E(H), and put H′ = H(I(H)). Since H′ is an ideal subgroup, we also have that R ⊂ E(H′) by Corollary 11. Since I(H′) = I(H) by (9), it follows from (67) applied to H and H′ that |H| = [R : I(H)] = |H′|, and so H = H′ is an ideal subgroup.

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}) (68)

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. [20], §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 see now. Via this identification we can determine I(H) and H(I) as follows.

Proposition 33 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]. (69)

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

(70)  I(H) = Ψ_{L}((L : L_{H})),
(71)  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 (69).

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 (70).

To prove (71), 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 (71) follows.

Corollary 34 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 that I_{E_{L}}(E_{L′}) ≃ I(H) = ψ_{L}((L : L′)) ≃ (L : L′) by (70). This proves the assertion because (L : L′) ≃ L(L′)^{−1} by (45).

Remark 35 (a) We can use the above Proposition 33 to give quick proofs of Propositions 23, 27 and 31 in the case that K = ℂ. For example, since 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}, we have by (70), (45), (43) and (69) that R(I(H)) = Ψ_{L}^{0}(R((L : L_{H})) = Ψ_{L}^{0}(R(L)R(L_{H})) = RE(H), which proves Proposition 31. (Here Ψ_{L}^{0} : F →^{∼} End^{0}(E_{L}) is the canonical extension of Ψ_{L} to F.) The other results are proved similarly; cf. [18].

(b) The above formula (70) can be generalized to give a similar interpretation of the lattice H(H_{1}, H_{2}). Indeed, if L ∈ Lat_{F}, where F is an imaginary quadratic field, and if H_{i} = ρ_{L}(L_{H_{i}}/L) are two finite subgroups of E_{L}, then we have that

H(H_{1}, H_{2}) = Ψ_{L}^{0}((L_{H_{2}} : L_{H_{1}})); (72)

cf. [18], where this and other similar formulae are proved.

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 e-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 21: 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 e-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 36 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. [20], 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. [20], p. 184. By the Deuring Lifting Theorem ([20], 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 (due to Deuring[10]) classifies the possible e-conductors of elliptic curves in Isog(E/K) when K is algebraically closed.

Proposition 37 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 36); 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}}.

On the other hand, for arithmetic fields we have the following situation.

Proposition 38 If K is a finitely generated field, then the set of e-discriminants Δ_{E} of all CM elliptic curves E/K is finite. In particular, Isog(E/K) is a finite set, and hence there is a unique integer f_{E/K}^{min} ≥ 1 with the property that for any f ≥ 1, there is an elliptic curve E′ ∼ E with f_{E′} = f if and only if f | f_{E/K}^{min}.

Proof. To prove the first assertion, let K_{0} be the algebraic closure of the prime subfield of K in K. Suppose first that char(K) ≠ 0; then K_{0} ≃ 𝔽_{q} is a finite field. If E/K is a CM elliptic curve, then its j-invariant j_{E} ∈ K ∩ \overline{K}_{0} = K_{0}, and so there are at most q such j_{E}’s. Thus, there are only finitely many \overline{K}-isomorphism classes of CMelliptic curves over K, and hence only finitely many endomorphism rings (because End(E) = End(E ⊗ \overline{K})). This proves the first assertion when char(K) ≠ 0.

Now suppose that char(K) = 0. Then K_{0} is a number field and we may assume without loss of generality that K ⊂ ℂ. If E/K is a CM elliptic curve, then as before j_{E} ∈ K ∩ \overline{K}_{0} = K_{0}. By §3.2 we know that E ⊗ ℂ ≃ E_{L}, for some lattice L (of some quadratic field F), and so j_{E} = j(L) (where j denotes the j-function on the upper half-plane). Moreover, we have that End(E) ≃ End(E_{L}) ≃ R(L) by (69), and so Δ_{E} = Δ(R(L)). By the theory of complex multiplication (cf. [7], Theorem 11.1) we have that [j(L) : ℚ] = h(Δ(R(L))), and so it follows that h(Δ_{E}) ≤ [K_{0} : ℚ]. Since there are only finitely many discriminants Δ with a given class number h(Δ) = h (Heilbronn[12]), the first assertion follows.

Since the set of e-conductors in Isog(E/K) is finite (by what was just proved), and since the set of isomorphism classes with given e-conductor is finite by (55), it follows that Isog(E/K) is finite.

Put f_{E/K}^{min} = lcm{f_{E′} : E′ ∈ Isog(E/K)}. Then by definition we have that f_{E′}|f_{E/K}^{min}, for all E′ ∈ Isog(E/K), and that f_{E/K}^{min} is the smallest number with this property. Moreover, if f|f_{E/K}^{min}, then there is an elliptic curve E′ ∈ Isog(E/K) such that f_{E′} = f. To see this, we first use Proposition 29 to construct an E_{0} ∈ Isog(E/K) with f_{E_{0}} = f_{E/K}^{min}, and then apply Corollary 18 to A = E_{0} to find an elliptic curve E′ ∈ Isog(E/K) such that f_{E′} = f. This proves the last statement.

Remark 39 If E/K is a CM curve over a finite field K = 𝔽_{q}, then it follows from Theorem 4.2 of Waterhouse[30] that we have that (f_{E/K}^{min})^{2}Δ_{F} = a_{E/K}^{2} − 4q, where

a_{E/K} = 1 + q − |E(𝔽_{q})| and F = ℚ(\sqrt{a_{E/K}^{2} − 4q}); cf. [18]. In particular, we see that

|Δ_{E}| ≤ (f_{E/K}^{min})^{2}|Δ_{F} | = |a_{E/K}^{2} − 4q| ≤ 4q.

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 deg is a positive definite quadratic form on Hom(E_{1}, E_{2}) ≃ ℤ^{r}, where r = rank(Hom(E_{1}, E_{2})) = dim_{ℚ}(End^{0}(E_{i})) (cf. [28], p. 88), 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 that 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 binary 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

(73) Δ(q_{L}) = Δ(R(L)) and cont(q_{L}) = 1.

We now prove:

Proposition 40 If E/K is a CM elliptic curve and E_{1}, E_{2} ∈ Isog^{+}(E), then

(74) 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})},

where f_{i} := f_{E_{i}}. In particular, c = cont(q_{E_{1},E_{2}}) and

(75) 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

(76) 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 (50), we have by (61) that N(L) = c\frac{n_{1}}{n_{2}} . Moreover, since we have by (57) that deg(f) = [R : Rf] = 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 (76) and hence also (74) because by (51) and (45) 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 (73), we have from (74) that cont(q_{E_{1},E_{2}}) = c · cont(q_{L}) = c. Moreover, by (73) 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 (43) and (41) 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 (75) follows.

Remark 41 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 21, 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′)}^{+}, (77)

where the last equality follows from the second equation of (54).

On the other hand, if f_{E′}|f_{E}, then equation (74) 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 (77), we therefore obtain the important relation that

q_{E,E′} ∼ q_{E,E′}^{+}, for E′ ∈ Isog^{+}(E/K), (78)

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}. (Here and above, I_{E}(E′)^{−1} denotes the inverse of the lattice I_{E}(E′).)

Corollary 42 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

(79) 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 29 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 40.

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

(80) 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 43 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

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

Similarly, the rule I ↦ M(I) := R^{n} ⊗_{M_{n}(R)} I induces an inclusion preserving bijection between the set of left ideals of M_{n}(R) and the set of left R-submodules of R^{n}, and for any two left M_{n}(R)-ideals I_{1}, I_{2} we have that

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

In addition, these rules are inverse to each other in the sense that

M(I_{n}(M)) ≃ M and I_{n}(M(I)) ≃ I. (83)

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) = (α_{ij}), where x_{i}α = α_{i} = ∑_{j} α_{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

τ\underline{x}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}. (84)

Proposition 44 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})). (85)

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. (84).

The second assertion clearly follows from the first. Indeed, 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 44.

Proposition 45 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}), (86)

and equality holds if I_{1}, …, I_{n} are kernel ideals. 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_{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 (86), (8), and (85) we obtain that 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 (85) that H(I_{1}| … |I_{n}) = H(I_{1}) × … × H(I_{n}) = H_{1} × … × H_{n}, the latter because each H_{i} is an ideal subgroup. But this means that H_{1} × … × H_{n} is an ideal subgroup; cf. Remark 7(b).

We can now put together what we proved so far to deduce the following result which (in view of (50)) generalizes Theorem 1 of the introduction.

Theorem 46 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 45, we have by (21) that A_{H}^{n} ≃ A_{H′}^{n} ⇔ I(H) ≃ I(H′). Now by (85) we have that 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 43. 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 (50) that H_{i} = Ker(π_{i}) and H′_{i} = Ker(π_{i}) are ideal subgroups, and so the assertion follows from Theorem 46 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[29] and to Borevich and Faddeev[1] about the R-module structure of the submodules of R^{n}.

Theorem 47 (Steinitz) If R is 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 has a cyclic index 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 48 (Borevich/Faddeev) Let R be an order in a Dedekind domain 𝔒. Then:

(a) The order R has cyclic index 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 an order which has cyclic index, 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}), (87)

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

(88) 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 (87) 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 (88) is the content of [1], Theorem 7.

Remark 49 (a) Since a Dedekind domain trivially has cyclic index, it is clear that Theorem 48 generalizes Theorem 47.

(b) Every order R_{Δ} = ℤ + ℤω_{Δ} in a quadratic field F = ℚ(\sqrt{Δ}) has cyclic index because 𝔒_{F} = ℤ + ℤω_{Δ_{F}} = R_{Δ} + R_{Δ}ω_{Δ_{F}}. Thus, Theorem 48(b) applies to all orders in quadratic fields.

(c) It follows from the above Theorem 48(b) (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 50 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}. (89)

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)_{F} = R_{F} (L), as claimed.

Put R = R_{F} (L), so clearly L is an R-module. By Remark 49(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 (89) 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 (89), 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 49(c).

Corollary 51 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}, for 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 48(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)_{F} = R. Then by Corollary 50 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)_{F} = 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. Thus μ_{R,n} is surjective.

One of the disadvantages of Theorem 48 is that it does not give a recipe for determining the orders R_{k} and ideals I_{k} such that (87) holds when M is given. To remedy 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 52 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}). (90)

Proof. Put R_{i} := R(L_{i}) and f_{i} := [R_{i} : R]. Then (f_{1}, f_{2}) = 1 by (41). We first claim:

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

Indeed, by replacing L_{i} by n_{i}L_{i} (for some n_{i} ∈ ℕ), 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 there exists λ_{2} ∈ F^{×} such that λ_{2}L′_{2} + f_{1}R_{2} = R_{2} (cf. [20], p. 93) and hence (N(λ_{2}L′_{2}), f_{1}) = 1; cf. [7], p. 143. Put L′′_{2} := λ_{2}L_{2} = λ_{2}f_{2}L′_{2} ⊂ f_{2}R_{2} ⊂ R; thus N(L′′_{2}) = f_{2}^{2}N(λ_{2}L′_{2}) and hence (N(L′′_{2}), f_{1}) = 1. Moreover, since m_{2} := [R : L′′_{2}] = \frac{1}{f_{2}} [R_{2} : L′′_{2}] = \frac{1}{f_{2}} N(L′′_{2}), we have that (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}] = \frac{1}{f_{1}} [R_{1} : 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 (91).

To prove (90), we may assume in view of (91) that L_{1} + L_{2} = R because L_{1} ⊕ L_{2} ≃ λ_{1}L_{1} ⊕ λ_{2}L_{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}, α_{2}λ_{1} + α_{1}λ_{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 1}{α_{2} α_{1}}) = α_{1} − α_{2} = 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 (90).

Remark 53 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 defined as the ideal f_{R}𝔒_{F}.

Proposition 54 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)) (92)

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 43, 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 (92), 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 55 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}. (93)

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 43 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 50 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 (89). 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, (94)

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 20, we have by Proposition 44 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 that H((I_{1}| … |I_{n})) = H(I_{1})×…×H(I_{n}) by Proposition 45. Thus, by (19) we have that 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 (93).

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 (92) that (I′ : I′) ≃ End_{R}(I_{1} ⊕ … ⊕ I_{n})^{op}, and so the second isomorphism of (93) follows.

This leads to the following (partial) characterization of the ideal subgroups of E^{n}.

Corollary 56 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

(95) 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 (93) we have that 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 (50) that f_{E_{i}}|f_{E}, i.e. that E_{i} ∈ Isog^{+}(E/K).

Conversely, suppose that (95) holds, and fix an isogeny π_{i} : E → E_{i} for 1 ≤ i ≤ n. Then by (50) 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 45. Since E_{i} ≃ E_{H_{i}}, we have by the hypothesis (95) 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}.

The above result represents only a first step towards an (intrinsic) characterization of ideal subgroups H of E^{n} in terms of the structure of End(A_{H}) and/or of its centre Z(End(A_{H})) ⊂ 𝔒_{F}, where F = End^{0}(E). This centre is determined by the central conductor f_{A} of the abelian variety A = A_{H}, which 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).

The following characterization of the ideal subgroups of E^{n} shows that the necessary condition of Corollary 11 is also sufficient. Moreover, this result can also be viewed as a generalization of the criterion (50) of Theorem 20.

Theorem 57 Let E/K is a CM elliptic curve, and let H be a finite subgroup scheme of E^{n}. Then:

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

This theorem follows easily from Corollary 56 and Theorem 2 of the introduction, as we shall see now.

Proof of Theorem 57 (using Theorem 2). First note that the second equivalence of (96) follows from (41) and that the one direction (⇒) of the first equivalence of (96) is true by Corollary 11. It thus remains to prove the other direction.

Thus, suppose that f_{E}|f_{A_{H}}. By Theorem 2 there exist elliptic curves E_{i}/K such that A_{H} ≃ E_{1} × … × E_{n}; in particular, E_{i} ∼ E. By Lemma 58 below we have that f_{A_{H}}|f_{E_{i}}, and so E_{i} ∈ Isog^{+}(E/K). Thus H satisfies condition (95) and hence H is an ideal subgroup of E^{n} by Corollary 56.

In the above proof we had used the following simple fact.

Lemma 58 If E_{1}, …, E_{n} ∈ Isog(E/K), then f_{E_{1}×...×E_{n}} = lcm(f_{E_{1}}, …, f_{E_{n}}).

Proof. By Proposition 29 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 (50) (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 (49) 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 E_{H}^{n} ≃ E_{H_{1}} × … × E_{H_{n}} ≃ E_{1} × … × E_{n} =: A. By Proposition 44 we have that H = H(I), where I = (I_{1}| … |I_{n}). Thus, by (93) we have that End(A) ≃ End(E_{H}^{n}) ≃ 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 (92). 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 (41), the assertion follows.

To finish the proof of Theorem 57, it remains to prove Theorem 2 of the introduction. Before giving the proof, we insert here the following remark.

Remark 59 In the case that K = ℂ, Theorem 2 was first proven by Lange[21], using the results of Shioda and Mitani[27] (who proved the case n = 2). A different proof of this was given by Schoen; cf. [24], Satz 2.4. His proof is based on the above Theorem 48(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[24] also proves the following interesting “converse” to Theorem 2 in the case that 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.

Proof of Theorem 2. 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 50 we know that L′ has the form (89). 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 (89). 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. First note that there exists a finitely generated subfield K_{0} ⊂ K such that there is a CM elliptic curve E_{0}/K_{0} with E_{0}⊗K ≃ E, an abelian variety A_{0}/K_{0} with A_{0}⊗K ≃ A and A_{0} ∼ E_{0}^{n}. Fix an embedding K_{0} ⊂ ℂ. Then by Case 1 and Lemma 60 below, there exist elliptic curves E_{0,i}/K_{0} with A_{0} ≃ E_{0,1}×…×E_{0,n}, and so A ≃ E_{1}×…×E_{n}, where E_{i} = E_{0,i} ⊗ K. Case 3. K = \overline{𝔽}_{p}.

Let K^{#} denote the quotient field of the ring W (K) of Witt-vectors over K. 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 [14], 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 exist elliptic curves E′_{1}, …, E′_{n}/K^{#} such that A^{#} ≃ E′_{1} × … × E′_{n}. Since each E′_{i} ∼ E^{#} has good reduction E_{i}/K, we obtain that A ≃ E_{1} × … × E_{n} because A^{#} has reduction A. Case 4. char(K) = p ≠ 0. By Lemma 60 below we may assume without loss of generality that K is algebraically closed. Then there is a CM elliptic curve E_{0}/\overline{𝔽}_{p} such that E_{0} ⊗ K ≃ E; cf. the proof of Proposition 36. Furthermore, if π : E^{n} → A is an isogeny with kernel H, then by Lemma 61 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 two “descent” facts.

Lemma 60 Let A/K be an abelian variety with A ∼ E^{n}, where E/K is a CM elliptic curve, and suppose that there is a field extension K′/K such that A⊗K′ ≃ E′_{1}×…×E′_{n} for some elliptic curves E′_{i}/K′. Then A ≃ E_{1}×…×E_{n}, where each E_{i}/K is an elliptic curve with E_{i} ⊗ K′ ≃ E′_{i}.

Proof. Fix an isomorphism φ : A ⊗ K′ →^{∼} B′ := E′_{1} × … × E′_{n}, and consider h′_{i} = φ^{−1} ∘ e_{i}^{B′} ∘ p_{i}^{B′} ∘ φ ∈ End(A ⊗ K′). Since the base-change map β_{K′/K} : End(A) → End(A ⊗ K′) is an isomorphism by Lemma 14(c) (because dim End^{0}(E^{n}) = 2n^{2} = dim End^{0}(E^{n} ⊗ K′)), we have that h′_{i} = h_{i} ⊗ K′, for some h_{i} ∈ End(A). Then E_{i} := Im(h_{i}) is an abelian subvariety of A/K such that E_{i} ⊗ K′ ≃ Im(h′_{i}) ≃ E′_{i}. Thus, E_{i}/K is an elliptic curve on A ∼ E^{n}, and hence E_{i} ∼ E (by Poincaré’s reducibility theorem). Put B := E_{1} × … × E_{n} ∼ E^{n}. By hypothesis, there is an isomorphism ψ′ : A ⊗ K′ →^{∼} B ⊗ K′, and by Lemma 14(c) we have that ψ′ = ψ ⊗ K′ for some ψ ∈ Hom(A, B). Since ψ′ is an isomorphism, so is ψ, and hence A ≃ B, as claimed.

Lemma 61 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 [22], p. 136), and so by the structure of finite commutative group schemes ([22], 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 above, 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 62 As the above proof shows, the assertions of Lemma 61 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 2 and hence also of Theorem 57, as was explained above. We now turn to some applications of Theorem 57. The first of these is the following refinement of Theorem 3 of the introduction.

Theorem 63 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:

(97)  A ≃ E_{1} × … × E_{n};
(98)  I_{E^{n}}(A) ≃ (I_{E}(E_{1})| … |I_{E}(E_{n})) and f_{A}|f_{E};
(99) Hom(E^{n}, E) ⊗_{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 (50) 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 45. 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 (85).

Now suppose that (97) holds. Then by Lemma 58 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 (98) holds. Conversely, if (98) holds, then H is an ideal subgroup of E^{n} by Theorem 57, and so A ≃ A′ by (21). Thus, conditions (97) and (98) are equivalent.

To prove the equivalence of conditions (98) and (99), note first that Hom(E^{n}, E) is naturally an (End(E), End(E^{n}))-bimodule, and that the rule h ↦ (h∘e_{1}^{E^{n}}, …, h∘e_{n}^{E^{n}}) defines a bijection e_{∗} : Hom(E^{n}, E) →^{∼} End(E)^{n} which, via the identification T_{E,n} : End(E^{n}) →^{∼} M_{n}(R), carries the (End(E), End(E^{n}))-bimodule structure on Hom(E^{n}, E) into the (R, M_{n}(R))-bimodule structure on R^{n}, where R = End(E). Thus, for any left ideal I of End(E^{n}), we have an R-module identification

e_{∗}(Hom(E^{n}, E) ⊗_{End(E^{n})} I) = R^{n} ⊗_{M_{n}(R)} T_{E,n}(I)) =: M(T_{E,n}(I)),

and hence it follows from (83) and the definition of (I_{1}| … |I_{n}) that for any R-ideals I_{1}, … I_{n} we have an R-module isomorphism

Hom(E^{n}, E) ⊗_{End(E^{n})} (I_{1}| … |I_{n}) ≃ I_{1} ⊕ … ⊕ I_{n}.

In view of these identifications, it is clear from (82) that conditions (98) and (99) are equivalent.

Proof of Theorem 3. By Theorem 2 we know that there exist elliptic curves E′_{i} ∼ E_{1} such that A ≃ E′_{1}×…×E′_{n}, and so by Lemma 58 we have that f_{A} = lcm(f_{E′_{1}}, …, f_{E′_{n}}). Now by Proposition 29 there exists an elliptic curve E ∼ E_{1} such that f_{E} = lcm(f_{E_{1}}, …, f_{E_{n}}, f_{E′_{1}}, …, f_{E′_{n}}), and so the first statement follows. The second is a special case of Theorem 63.

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 (50) 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 20(a) and also End(E′) ≃ End(E_{H(I)}) ≃ R(I) by (53). 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 21 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 41 and f_{I} = f_{R(I)}. Here R_{Δ} ≃ End(E) (because E has e-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 51.

(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 43 and Proposition 54 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.

Now 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 55, we have that End(A_{I}) ≃ (I : I) by (31), 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 57 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}, and so I is in the set defined by condition (iv′) because the above computation shows that Z((I : I)) = R.

Remark 64 (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, (100)

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 44, 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 following 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[27] 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 [27].

Proposition 65 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 52 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 48 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 (54), we see that R has conductor f_{R} = lcm(f_{1}, f_{2}) by (41), 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 [27].

Corollary 66 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 (68), 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 34 we have that 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 65.

We next prove Theorem 4, which is clearly a special case of the following more precise result. Recall from the introduction that the discriminant Δ(A/K) of an abelian surface A/K is the discriminant of the intersection form q_{A} on the Néron-Severi group NS(A) of A/K.

Theorem 67 Let E/K be a CM elliptic curve with e-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 Q_{Δ}/SL_{2}(ℤ) 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 21, the bijection between (ii) and (iii) is the map q_{Δ} of Remark 41, 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 68 Note that the bijection between the sets described by conditions (iii) and (vi) of Theorem 67 (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 41. 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 64(a) we know that the bijections are as indicated. On the other hand, the fact that the sets (v) and (vi) are identical follows immediately from the following general fact of Proposition 69 because Δ_{E} = f_{E}^{2}Δ_{F}.

Proposition 69 Let E/K be a CM elliptic curve, and let F = End^{0}(E). If A/K is an abelian surface which is isogenous to E^{2}, then its discriminant is Δ(A/K) = −f_{A}^{2}Δ_{F}.

Proof. By Theorem 2 we know that that A ≃ E_{1}×E_{2}, for some CM elliptic curves E_{i}/K which are isogenous to E/K. Then by [17], 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}), by Lemma 14(c), the explicit isomorphism shows that NS(A) = NS(A ⊗ \overline{K}); cf. [19], (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 [17], Proposition 22, or by [19], equation (6), we have that

q_{A} ∼ xy ⊥ q_{E_{1},E_{2}}, (101)

where xy denotes the quadratic form associated to the hyperbolic plane and q_{E_{1},E_{2}} is as in §3.4. From (101) 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} , (102)

where the last two equations follow from equation (75) and Lemma 58, respectively. This proves the asserted formula.

In view of the close relationship (101) that exists between the quadratic form q_{A} defined by the intersection pairing on NS(A) and the form q_{E,E′}^{+} which defines one of the bijections of Theorem 67 (cf. Remark 68), it might be tempting to try to use q_{A} in place of q_{E,E′}^{+} in order to classify the abelian surfaces A/K with A ∼ E^{2}. This, however, does not lead to a bijection because several non-isomorphic abelian surfaces may have equivalent forms q_{A}, as the following result shows.

Corollary 70 Let E/K be a CM elliptic curve with e-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 primitive forms of discriminant Δ′ = Δ/c^{2}. Thus, if g(Δ′) denotes the number of genera of discriminant Δ′, then we have

(103) 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 67. Then by (101) and (78) we have that 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 [19]. Since the number of such forms q^{A} equals the number of forms in the principal genus (and is given by the formula of (103)), the assertion follows.

From the above Theorem 4 (or from Theorem 67) we can deduce the following result which is essentially the same as Theorem 3.1 of [27]:

Theorem 71 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 72 In the paper [27], the abelian surfaces A/ℂ with ρ(A) = 4 are called singular abelian surfaces; cf. [27], 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[16], p. 167, 187 (and the references therein) and Humbert [15].

Proof of Theorem 71. It follows from the classification theory of endomorphisms of simple abelian surfaces in characteristic 0 (cf. [22], 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 37(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 67 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 68 that the rule q ↦ E_{Δ} × E′_{E_{Δ},q} (notation as in Remark 68) induces the desired bijection

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

Remark 73 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.

References

[1] Z. Borevich, D. Faddeev, Representations of orders with cyclic index. Trudy Mat. Inst. Steklov = Proc. Steklov Inst. Math. 80 (1965), 51–65.

[2] Z. Borevich, D. Faddeev, A note on orders with cyclic index. Dokl. Akad. Nauk SSSR 164 (1965), 727–728 = Soviet Math. Doklady 6 (1965), 1273–1274.

[3] Z. Borevich, I. Shafarevich, Number Theory. Academic Press, New York, 1966.

[4] S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models. Springer-Verlag, Berlin, 1990.

[5] N. Bourbaki, Commutative Algebra, Ch. I-VII. Addison-Wesley, Reading, 1972.

[6] J. Buchmann, U. Vollmer, Binary Quadratic Forms. Springer, Berlin, 2007.

[7] D. Cox, Primes of the Form x^{2} + ny^{2}: Fermat, Class Field Theory and Complex Multiplication. Wiley, New York, 1989.

[8] C. Curtis, I. Reiner, Methods of Representation Theory I. J. Wiley & Sons, New York, 1981.

[9] P. Deligne, Variétés abéliennes ordinaires sur un corps fini. Invent. Math. 6 (1969), 238-243.

[10] M. Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Hamburg 14 (1941), 197–272.

[11] R. Hartshorne, Algebraic Geometry. Springer-Verlag, New York, 1977.

[12] H. Heilbronn, On the class-number of imaginary quadratic fields. Quart. J. Math. 5 (1934), 150–160 = Collected Papers, Wiley, New York, 1988, pp. 177–187.

[13] F. Hirzebruch, D. Zagier, Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus. Invent. math. 36 (1976), 57–113 = Gesammelte Abh./Coll. Papers II, Springer-Verlag, Berlin, 1987, pp. 409–465.

[14] E. W. Howe, Principally polarized ordinary abelian varieties over finite fields. Trans. Amer. Math. Soc. 347 (1995), 2361-2401.

[15] G. Humbert, Sur les fonctions abéliennes singulières. I. J. de Math. (ser. 5) 5 (1899), 233–350 = Œuvres, Gauthier-Villars et Cie., Paris, 1929, pp. 297–401.

[16] A. Hurwitz, Über algebraische Korrespondenzen und das verallgemeinerte Korrespondenzprinzip. Math. Ann. 28 (1887), 561–585 = Math. Werke I, Birhäuser, Basel, 1932, pp. 163 – 188.

[17] E. Kani, The moduli spaces of Jacobians isomorphic to a product of two elliptic curves. Preprint, 39 pages.

[18] E. Kani, Products of CM elliptic curves. Inst. Exp. Math., Essen, Universität Duisburg-Essen, IEM Preprint No. 2–2009, 54 pages.

[19] E. Kani, The existence of Jacobians isomorphic to a product of two elliptic curves. Inst. Exp. Math., Essen, Universität Duisburg-Essen, IEM Preprint No. 3–2009, 36 pages.

[20] S. Lang, Elliptic Functions. Addison-Wesley, Reading, MA, 1972.

[21] H. Lange, Produkte elliptischer Kurven. Nachr. Akad. Wiss. Göttingen Math.- Phys. Kl. II (1975), 95–108.

[22] D. Mumford, Abelian Varieties. Oxford U Press, Oxford, 1970.

[23] W. Ruppert, When is an abelian surface isomorphic or isogenous to a product of elliptic curves? Math. Z. 203 (1990), 293–299.

[24] C. Schoen, Produkte abelscher Varietäten und Moduln über Ordnungen. J. reine angew. Math. 429 (1992), 115–123.

[25] J.-P. Serre, J. Tate, Good reduction of abelian varieties. Ann. Math 88 (1968), 492–517 = Œuvres/Collected Papers II, Springer-Verlag, Berlin, 1985, pp. 472– 497.

[26] G. Shimura, Y. Taniyama, Complex Multiplication of Abelian Varieties. Math. Soc. Japan, Tokyo, 1961.

[27] T. Shioda, N. Mitani, Singular abelian surfaces and binary quadratic forms. In: Classification of algebraic varieties and compact complex manifolds. Lect. Notes Math. 412 (1974), 259–287.

[28] J. H. Silverman, The arithmetic of elliptic curves. Graduate Texts in Mathematics, Vol. 106, Springer-Verlag New York, 1986.

[29] E. Steinitz, Zur Theorie der Moduln. Math. Ann. 52 (1899), 1–57.

[30] W.C. Waterhouse, Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4), 2 (1969), 521-560.