Jacobians isomorphic to a product of two elliptic curves

Ernst Kani

1 Introduction

In 1965 Hayashida and Nishi initiated the study of genus 2 curves C whose Jacobian J_{C} is isomorphic to a product A = E_{1} ×E_{2} of two elliptic curves. In their papers [12],

[14] and [13], they determined the number of curves C with J_{C} ≃ A for a fixed A in many cases, thereby exhibiting the existence of such curves. A similar count was done for supersingular curves by Ibukiyama, Katsura and Oort[16].

Recently there has been renewed interest in such curves, particularly in connection with moduli problems; cf. Earle[6], Lange[26], and McMullen[28], [29].

The purpose of this article is determine how such curves are distributed in the moduli space M_{2} of genus 2 curves over an algebraically closed field K. By a result of Lange[25] it is known that these lie on infinitely many curves in M_{2}; see also [6]. Here we want to make the nature of these curves precise.

To this end, let us say that a curve C has type d if J_{C} ≃ E_{1} × E_{2}, where E_{1} and E_{2} are connected by a cyclic isogeny of degree d. (If E_{1} has CM or is supersingular, then this definition has to slightly modified; see §4 below.) Since every curve C with J_{C} ≃ E_{1} × E_{2} has some type d ≥ 1 (cf. Proposition 25), the following result describes the set of all such curves in M_{2}:

Theorem 1 The set T (d) ⊂ M_{2} of curves of type d is a closed subset of M_{2}. If T (d) is non-empty, then T (d) is a finite union of irreducible curves. Moreover, if char(K) ∤ d, then each such component is birationally isomorphic either to the modular curve X_{0}(d)^{+} or to a degree 2 quotient thereof.

Here X_{0}(d)^{+} = X_{0}(d)/⟨w_{d}⟩ is (as in [27], p. 145) the quotient of the usual modular curve X_{0}(d) by the Fricke involution w_{d}.

The key tool for proving this and other related results is the concept of a “generalized Humbert variety” which is introduced here. This generalizes the Humbert surfaces of Humbert and is based on a refinement of the usual Humbert invariant (cf. [35]) that was suggested in [19]. There it was observed that each curve C comes equipped with a canonically defined positive definite quadratic form q_{C} which can be used to define the Humbert invariant (and hence Humbert surfaces).

It turns out that the curves C of type d can be characterized by a property of their associated refined Humbert invariant q_{C} as defined in §2. To formulate this property in a convenient manner, let us say that a positive definite binary quadratic form q has type d if it has discriminant −16d and is either primitive and lies in the principal genus (but q is not equivalent to the principal (norm) form) or else q = 4q_{1}, where q_{1} is primitive (of discriminant −d) and lies in the principal genus. (Such quadratic forms are studied in detail in §5.) We then have:

Theorem 2 If C is a curve of genus 2, then C has type d if and only if its refined Humbert invariant q_{C} primitively represents a form of type d.

In view of this, we might expect the various forms q of type d to give us the components of the curve T (d), and this is indeed the case. To make this precise, let H(q) denote the set of isomorphism classes of curves C in the moduli space M_{2} such that q_{C} represents q primitively; we call H(q) the generalized Humbert variety associated to q; cf. §3. Thus, Theorem 2 can be restated in terms of the H(q)’s; cf. Theorem 13 (which is a refinement of Theorem 2). If Q̄_{d}^{∗} denotes the set of GL_{2}(ℤ)- equivalence classes of forms of type d, then we prove in §8:

Theorem 3 If char(K) ∤ d, then the H(q), where q ∈ Q̄_{d}^{∗}, are precisely the irreducible components of T (d). Thus T (d) has precisely t^{∗}(d) := #Q̄_{d}^{∗} irreducible components.

The precise birational structure of the curves H(q) depends on whether or not q is an ambiguous form, i.e. on whether or not q has order 2 in the group Q̄_{−16d} of equivalence classes of primitive forms of discriminant −16d. (In the case that q′ = \frac{1}{4}q is primitive of discriminant −d, then this means that q′ has order 2 in Q̄_{−d}.)

Theorem 4 Let q ∈ Q̄_{d}^{∗}. If q is not an ambiguous class, then H(q) ∼ X_{0}(d)^{+}; otherwise H(q) ∼ X_{0}(d)^{+}/⟨α_{q}⟩, where α_{q} is a suitable Atkin-Lehner involution.

This result is made more precise in §10, where an explicit recipe for the Atkin-Lehner involution α_{q} is given; cf. Proposition 53 and Theorem 55. Note that it can happen in certain cases that α_{q} acts trivially on X_{0}(d)^{+}; these cases are analyzed there as well.

An interesting but difficult question is to characterize the d′s for which there is no curve of type d, i.e. to determine the d’s for which T (d) is empty or, equivalently, for which t^{∗}(d) = 0. Now from its definition one might expect that t^{∗}(d) could be expressed in terms of suitable class numbers of binary quadratic forms, or more precisely, in terms of the number h̄(D) = h(D)/g(D) of (proper) equivalence classes of forms in the principal genus of discriminate D = −16d. This is essentially correct, but the formula is complicated by the fact that we need to count forms up to GL_{2}(ℤ)- equivalence instead of the more usual SL_{2}(ℤ)-equivalence, and so one also needs to know the number of spinor genera of discriminant −16d; cf. Remark 35.

Nevertheless, one has that the condition t^{∗}(d) = 0 is essentially equivalent to the condition that h̄(−16d) = 1 (cf. Corollary 34), and hence the precise determination of the exceptional d’s hinges on the solution of a classical problem in number theory which was first raised by Gauss. Indeed, Gauss[10], Art. 303, conjectured not only that there are only finitely many d’s with h(−4d) = 1 but also that the same is true for h̄(−4d) = 1, and this was later proven by Chowla[3] in 1934. Moreover, Gauss also conjectured that such d’s satisfy d ≤ 1848, but this does not seem to have been proved unconditionally yet (even though his class-number 1 conjecture has been settled). Nevertheless, Weinberger[37] has shown that Gauss’s conjecture follows from the Generalized Riemann Hypothesis (GRH). We thus prove in §7:

Corollary 5 T (d) is empty for the following 21 values of d ≥ 1:

d = 1, 2, 4, 6, 10, 12, 18, 22, 28, 30, 42, 58, 60, 70, 78, 102, 130, 190, 210, 330, 462. (1)

If Gauss’s Conjecture is true, then these are all the d’s for which T (d) = ∅. In particular, there are only finitely many d’s for which T (d) = ∅, and these are all given by (1) if (GRH) holds.

Note that the above result can also be viewed as an existence theorem, and hence as a refinement of the work of Hayashida[12]; cf. Remark 42.

Finally, it should be mentioned that there is a close connection between the results obtained here and the study of elliptic subcovers f : C → E of genus 2 curves, as is explained in [8], §6.

Acknowledgements. I would like to thank Gerd Frey for the many stimulating discussions which we had about this work. In addition, I would like to gratefully acknowledge receipt of funding from the from the Natural Sciences and Engineering Research Council of Canada (NSERC).

2 The refined Humbert invariant

Let A be an abelian surface over an algebraically closed field K of arbitrary characteristic, and assume that A has a principal polarization θ ∈ NS(A) = Div(A)/ ≡, where ≡ denotes numerical equivalence. Thus, θ = cl(D), where D ∈ Div(A) is an ample divisor with self-intersection number (D.D) = 2. Put

q_{(A,θ)}(D) = (D.θ)^{2} − 2(D.D), for D ∈ NS(A), (2)

where (.) denotes the intersection number of divisors. From the Hodge index theorem it follows easily that q_{(A,θ)} defines a positive definite quadratic form on the quotient group NS(A, θ) = NS(A)/ℤθ; cf. [19], §3. Since NS(A, θ) ≃ ℤ^{ρ−1}, where ρ = rk(NS) is the Picard number, we see that q_{(A,θ)} defines an (equivalence class of) integral, positive definite quadratic form(s) in ρ − 1 variables, which will be called the refined Humbert invariant of the principally polarized abelian variety (A, θ).

As was explained in [19], §5, q_{(A,θ)} is closely related to the classical Humbert invariant attached to an abelian surface A/ℂ: indeed, any number Δ which is primitively represented by q_{(A,θ)} is a (classical) Humbert invariant of the principally polarized abelian surface (A, θ). It thus follows that the subset

H_{Δ} = {⟨A, θ⟩ ∈ A_{2}(K) : q_{(A,θ)} primitively represents Δ}

of the moduli space A_{2} (which classifies isomorphism classes ⟨A, θ⟩ of principally polarized abelian surfaces) is precisely the Humbert surface of discriminant (or invariant) Δ as defined by Humbert[15] or [35], §IX.2. By Humbert, this defines an irreducible surface in A_{2}(ℂ) whenever Δ ≡ 0, 1 (mod 4), and is empty otherwise.

As was indicated in the introduction, we are primarily interested in the principally polarized abelian varieties that arise as Jacobians of (irreducible) genus 2 curves. Now if M_{2} denotes the moduli space of smooth, irreducible genus 2 curves, then we have Jacobi morphism j_{2} : M_{2} → A_{2} which takes a curve C to its principally polarized Jacobian (J_{C}, θ_{C}) in A_{2}(K). (Note that θ_{C} is the class of a curve isomorphic to C.)

Over ℂ, it is a classical fact (cf. Humbert[15], §17, or Krazer[23], p. 485) that the complement A_{2} \ j_{2}(M_{2}) is the Humbert surface H_{1} of invariant 1. By a result of Weil[36], this is true over an arbitrary field, as we now show:

Proposition 6 Let ⟨A, θ⟩ ∈ A_{2}(K). Then ⟨A, θ⟩ ∉ j_{2}(M_{2}(K)) if and only if q_{(A,θ)} represents 1, i.e. q_{(A,θ)}(D) = 1, for some D ∈ NS(A). Thus

A_{2} \ j_{2}(M_{2}) = H_{1}.

Proof. By Weil[36], Satz 2, we have that ⟨A, θ⟩ ∉ j_{2}(M_{2}) if and only if (A, θ) ≃ (E_{1} × E_{2}, θ_{1} + θ_{2}) is a product of two elliptic curves and θ = θ_{1} + θ_{2} is the product polarization (where θ_{i} = cl(pr_{i}^{∗}(0_{E_{i}})), for i = 1, 2).

Now if (A, θ) ≃ (E_{1} × E_{2}, θ_{1} + θ_{2}), then (θ.θ_{i}) = 1, (θ_{i}.θ_{i}) = 0 and so q_{(A,θ)}(θ_{i}) = 1. Conversely, suppose q_{(A,θ)}(D) = 1 for some D. Then D is necessarily primitive,

for if D = mD′ with D′ ∈ NS(A), then 1 = q_{θ}(D) = m^{2}q_{θ}(D′), and so m = ±1, i.e. D is primitive. Thus, by [19], Theorem 3.1, there exists an elliptic curve E on A with (E.θ) = 1. Put θ_{1} = θ − cl(E). Then θ_{1}^{2} = θ^{2} − 2(θ.E) + E^{2} = 0, and so θ_{1} = cl(mE′), for some elliptic curve E′ on A and some m ∈ ℤ; cf. [19], Prop. 2.3. But since θ = cl(E + mE′), we have 2 = θ^{2} = 2m(E.E′), so m = 1. Thus θ = cl(E + E′). By Weil[36], Satz 2, we have that (A, θ) ≃ (E_{1} × E_{2}, θ_{1} + θ_{2}) with θ_{i} = pr_{i}^{∗}(0_{E_{i}}), i = 1, 2, and so the assertion follows.

Remark 7 The above shows that the rule (E_{1}, E_{2}) ↦ ⟨E_{1} × E_{2}, pr_{1}^{∗}0_{E_{1}} + pr_{2}^{∗}0_{E_{2}}⟩ defines a surjection A_{1} × A_{1} → H_{1}, where A_{1} denotes the moduli space of elliptic curves. It not difficult to see (by an argument similar to that of the proof of Proposition 43) that this map is a proper morphism, and so j_{2}(A_{2}) is an open subset of A_{2}. Since j_{2} is birational, it thus follows (by Zariski’s Main Theorem) that j_{2} is an open immersion. Note that by Oort/Steenbrink[34], the Torelli map j_{g} : M_{g} → A_{g} need not be an immersion if g ≥ 5 and char(K) ≠ 0.

3 Generalized Humbert varieties

The definition of a Humbert surface can be generalized as follows. Given any integral positive definite quadratic form q, let

H(q) = {⟨A, θ⟩ ∈ A_{2}(K) : q_{(A,θ)} primitively represents q}.

Since clearly H(Δx^{2}) = H_{Δ} and since it can be shown (cf. [21]) that H(q) is always an algebraic subset of M_{2}, we shall call H(q) a generalized Humbert variety of A_{2}.

The H(q)’s can be used to describe intersections of Humbert surfaces:

Proposition 8 If m and n are distinct positive integers, then

H_{m} ∩ H_{n} = ⋃_{q} H(q), (3)

where the union runs over all equivalence classes of positive definite binary quadratic forms q which primitively represent both m and n.

Proof. Let q be such a form, and let ⟨A, θ⟩ ∈ H(q). Then q_{(A,θ)} primitively represents q. Since m is primitively represented by q, it follows that m also primitively represented by q_{(A,θ)}, so ⟨A, θ⟩ ∈ H_{m}. Thus H(q) ⊂ H_{m}, and similarly, H(q) ⊂ H_{n}, so H(q) ⊂ H_{n} ∩ H_{m}. This shows that the right side of (3) is contained in the left side.

Conversely, suppose ⟨A, θ⟩ ∈ H_{m} ∩ H_{n}. Then there exist primitive vectors v, w ∈ M := NS(A, θ) such that q_{(A,θ)}(v) = m and q_{(A,θ)}(w) = n. If v and w were linearly dependent, then v = ±w and hence q_{(A,θ)}(v) = q_{(A,θ)}(w), contrary to the hypothesis. Thus, v and w are linearly independent and hence M_{0} := ℤv + ℤw has rank 2. Let M_{1} be the saturation of M_{0} in M. Then the restriction q of q_{(A,θ)} to M_{1} is a positive definite, binary quadratic form which is primitively represented by q_{(A,θ)}, and so ⟨A, θ⟩ ∈ H(q). Moreover, m = q(v) is primitively represented by q (because v is primitive in M, hence also in M_{1}. Similarly, n = q(w) is primitively represented by q. Thus q is one of the forms of the right side of (3), so ⟨A, θ⟩ ∈ ∪H(q).

Remark 9 (a) Note that there are only finitely many equivalence classes of forms q satisfying the conditions of Proposition 8 because their discriminants are bounded: |disc(q)| ≤ 4mn.

(b) The above proposition and Humbert’s results imply that dim \overline{H(q)} ≤ 1, for all binary positive-definite quadratic forms q.

(c) The above proposition can be viewed as giving a partial answer to a question raised by McMullen[28], p. 96.

In the sequel we shall need to work out the refined Humbert invariant in many cases, and for this it is useful to know its discriminant/determinant. (Here, as usual, the determinant det(M, β) of a bilinear module (M, β) is the determinant of any Gram matrix (β(x_{i}, x_{j})) associated to a basis {x_{i}} of M, and the determinant det(M, q) of a quadratic module (M, q) is the determinant of the associated bilinear module (M, β_{q}), where β_{q} is the bilinear form associated to q.) It turns out that it is closely related to that of the Néron-Severi group, viewed as bilinear module via the intersection pairing:

Proposition 10 Let ρ = rank(NS(A)). Then the determinant of the quadratic module (NS(A, θ), q_{(A,θ)}) is related to that of the Néron-Severi group by the formula

det(NS(A, θ), q_{(A,θ)}) = \frac{1}{2}(−4)^{ρ−1} det(NS(A), (.)).

Proof. Let β = β_{A} denote the intersection pairing on NS(A), and let M_{0} = {(x.θ)θ − 2x : x ∈ NS(A)}. Clearly, (y.θ) = 0, if y ∈ M_{0}, i.e. M_{0} ⊥ ℤθ. Thus, if we put M = M_{0} + ℤθ, then det(β_{|M}) = 2 det(β_{|M_{0}}), where β_{|M} = β_{|M×M} (and β_{|M_{0}} = β_{|M_{0}×M_{0}}). Moreover, since M ⊃ 2NS(A), we see that M has finite index in NS(A), and so det(β_{|M}) = n^{2} det(β), where n = [NS(A) : M]. Similarly, if we put M̄ = M/ℤθ, then [NS(A, θ) : M̄] = [NS(A) : M] = n, and so det((βq̄)_{|M̄}) = n^{2} det(βq̄), where q̄ = q_{θ}. Now for y_{i} ∈ M_{0} we have βq̄(y_{1}, y_{1}) = −4β(y_{1}, y_{2}), and hence det((βq̄)_{|M̄}) = (−4)^{s} det(β_{|M_{0}}), where s = rank(M_{0}). (Note that if x_{1}, …, x_{s} form a basis of M_{0}, then their images in M̄ form a basis of M̄.) Since s = ρ − 1, we thus obtain

det(βq̄) = \frac{1}{n^{2}} det((βq̄)_{|M̄}) = \frac{(−4)^{ρ−1}}{n^{2}} det(β_{|M_{0}}) = \frac{(−4)^{ρ−1}}{2n^{2}} det(β_{|M}) = \frac{(−4)^{ρ−1}}{2} det(β).

4 Curves of type d

We now focus our attention to those curves C of genus 2 whose Jacobian J_{C} is isomorphic to a product of two elliptic curves. As we shall see below (cf. Proposition 25), these can be classified by an integer d called its type, which is defined as follows.

Definition. Let d ≥ 1 be an integer. A curve C is said to have type d if there exist two elliptic curves E_{1}, E_{2}, a cyclic isogeny h : E_{1} → E_{2} of degree d = deg(h) and an isomorphism α : J_{C} →^{∼} E_{1} × E_{2} such that

(4) θ_{C} ≡ α^{∗}(aθ_{1} + bθ_{2} + cΓ_{h}), for some a, b, c ∈ ℤ,

where θ_{i} = pr_{i}^{∗}(0_{E_{i}}), for i = 1, 2, and Γ_{h} ⊂ E_{1} × E_{2} denotes the graph of h. We denote the set of isomorphism classes ⟨C⟩ of curves C of type d by T (d) ⊂ M_{2}(K).

Remark 11 Suppose that J_{C} ≃ E_{1} × E_{2}. If E_{1} has no complex multiplication (i.e. if End(E_{1}) = ℤ), then its type d is uniquely determined by C by the formula det(NS(J_{C})) = 2d, as we shall see below (cf. Corollary 26). In the other cases, however, C may have several types associated to it.

The first main result is that curves of type d can be characterized by a property of the refined Humbert invariant q_{C} := q_{(J_{C},θ_{C})} associated to C. To formulate this in a simple manner, we first introduce the following class of binary quadratic forms.

Definition. Let d ≥ 1 be an integer. An integral quadratic form q is said to be of type d if it is binary, positive-definite of discriminant −16d, and if:

• either q is primitive and in the principal genus (i.e. q ∼ q_{1}^{2}, for some primitive form q_{1} of discriminant −16d) but not principal (i.e q ≁ x^{2} + 4dy^{2}),

• or q = 4q_{1}, for some primitive form q_{1} of discriminant −d ≡ 1 (mod 4) which is in the principal genus.

Remark 12 Since NS(J_{C}, θ_{C}) does not come with an explicit basis, the quadratic form q_{C} is only defined up to GL_{n}(ℤ)-equivalence, where n = rk(NS(J_{C}, θ_{C})). However, when dealing with binary quadratic forms, it is better to use proper (or SL_{2}(ℤ))- equivalence since the proper equivalence classes (of fixed discriminant) form a group. We shall denote proper equivalence throughout by the symbol ∼, and use ≈ for GL_{n}(ℤ)-equivalence. Note that for primitive binary quadratic forms we have q_{1} ≈ q_{2} ⇔ q_{1} ∼ q_{2} or q_{1} ∼ q_{2}^{−1}, and so the above conditions do not depend on the choice of the representative q of the GL_{2}(ℤ)-equivalence class.

The following basic result is a restatement of Theorem 2 of the introduction; it relates curves of type d to forms of type d.

Theorem 13 A curve C has type d if and only ⟨C⟩ ∈ H(q), for some quadratic form q of type d. Thus

T (d) = ⋃_{q∈Q̄_{d}^{∗}} H(q),

where Q̄_{d}^{∗} denotes the set of GL_{2}(ℤ)-equivalence classes of forms of type d.

The proof of this theorem will be deferred until section 6 since it requires some basic facts about forms of type d which will be presented in the next section. In section 7 we shall also prove an existence theorem which shows that H(q) is nonempty whenever q is a form of type d; cf. Theorem 30.

5 Quadratic forms of type d

This section is devoted to a detailed study of the (binary) quadratic forms of type d which were introduced in the previous section. In particular, it will be shown that each proper equivalence class of such forms can be represented by a “standard prototype” q_{s} which is associated to a solution s = (n_{1}, n_{2}, k) of the equation

n_{1}n_{2} − k^{2}d = 1. (5)

To define these prototypes, we first introduce some notation.

Notation. Fix an integer d ≥ 1, and let

P (d) = {(n_{1}, n_{2}, k) ∈ ℤ^{3} : n_{1} > 0, n_{2} > 0, n_{1}n_{2} − k^{2}d = 1}

denote the set of solutions of (5) with positive n_{i}’s. It is convenient to split P (d) into two parts: P (d) = P (d)^{odd} ˙∪ P (d)^{even}, where P (d)^{even} = {(n_{1}, n_{2}, k) ∈ P (d) : 2|n_{1}, 2|n_{2}}.

For a given discriminant D ≡ 0, 1 (mod 4) and an integer n, let

Q_{D}^{(n)} = {[a, b, c] ∈ ℤ^{3} : a > 0, b^{2} − 4ac = D, gcd(a, b, c)|n}

denote the set of binary quadratic forms of discriminant D whose content gcd(a, b, c) divides n. Here, as usual, we identify [a, b, c] with the quadratic form ax^{2} + bxy + cy^{2}.

We first note that the set P (d) of solutions of (5) can be identified with a suitable set of quadratic forms of discriminant −4d:

Lemma 14 The assignment (n_{1}, n_{2}, k) ↦ [n_{1}d, 2kd, n_{2}] induces a bijection

f_{d} : P (d) →^{∼} Q_{−4d}^{(2)}(d) := {[a, b, c] ∈ Q_{−4d}^{(2)} : d|a, 2d|b}.

Moreover, f_{d}(n_{1}, n_{2}, k) is primitive if and only if gcd(n_{1}, n_{2}, 2) = 1.

Proof. If s = (n_{1}, n_{2}, k) ∈ P (d), then disc(f_{d}(s)) = (2dk)^{2} − 4(n_{1}d)n_{2} = 4d(dk^{2} − n_{1}n_{2}) = −4d. Furthermore, since gcd(n_{1}n_{2}, k^{2}d) = 1 by (5), we have gcd(n_{1}d, 2kd, n_{2}) = gcd(n_{1}, n_{2}, 2)|2, so f_{d}(s) ∈ Q_{−4d}^{(2)}(d). (In particular, f_{d}(s) is primitive if and only if gcd(n_{1}, n_{2}, 2) = 1.) Conversely, if disc([n_{1}d, 2dk, n_{2}]) = −4d, then n_{1}n_{2} − k^{2}d = 1, so (n_{1}, n_{2}, k) ∈ P (d).

We now study quadratic forms of the following type. For s = (n_{1}, n_{2}, k) ∈ P (d), put q_{s}(x, y) = n_{2}^{2}x^{2} − 2k(t − d)xy + n_{1}^{2}ty^{2}, where t = d(n_{1}n_{2} + 3). (6)

Using (5) and the definition of t, we see that

k^{2}(t − d)^{2} + 4d = n_{1}^{2}n_{2}^{2}t, (7) and so disc(q_{s}) = −16d. As we shall see presently, q_{s} is always a form of type d, provided that q_{s} is not in the principal class. The converse is also true (up to proper equivalence), but is harder to prove:

Proposition 15 If s = (n_{1}, n_{2}, k) ∈ P (d), then q_{s} has type d, provided that q_{s} is not equivalent to the principal form. Conversely, if q is any form of type d, then there exists s ∈ P (d) such that q is properly equivalent to q_{s}.

The easy direction of this result is contained in the following more precise assertion.

Lemma 16 Let s = (n_{1}, n_{2}, k) ∈ P (d) and put t = d(n_{1}n_{2} + 3).

(a) If n_{2} is odd, then q̃_{s} := [n_{2}, 2k(t − d), n_{2}n_{1}^{2}t] ∈ Q_{−16d}^{(1)} and

q_{s} ∼ q̃_{s} ∘ q̃_{s} and q̃_{s} ∘ 1_{−4d} ∼ f_{d}(s). (8)

Here 1_{−4d} = [1, 0, d] denotes the principal form of discriminant −4d, ∘ denotes the composition of binary forms, and ∼ denotes proper equivalence.

(b) If n_{1} and n_{2} are even, then q_{s} = 4q′(s) with q′_{s} ∈ Q_{−d}^{(1)}. Moreover, f_{d}(s) = 2f′_{d}(s) with f′_{d}(s) ∈ Q_{−d}^{(1)} and we have

q′_{s} ∼ f′_{d}(s) ∘ f′_{d}(s). (9)

Proof. (a) From (7) we see that disc(q̃_{s}) = −16d, and so q̃_{s} ∈ Q_{−16d}^{(1)} because gcd(n_{2}, −16d) = gcd(n_{2}, 2) = 1. By the proof of [7], Lemma 1, it thus follows that q̃_{s}∘q̃_{s} ∼ q_{s}. The second formula of (8) follows directly from the composition formula of Arndt applied to q̃_{s} and [d, 0, 1] ∼ 1_{−4d}; cf. [2], p. 129. (Note that B = 2kd satisfies the required congruences.)

(b) Here k, d and hence t are odd, so t − d = 2t_{1} is even; More precisely, t_{1} = d(2ab+1), where a = \frac{n_{2}}{2} and b = \frac{n_{1}}{2} . Thus q_{s} = 4q′_{s} where q′_{s} = [a^{2}, −kt_{1}, tb^{2}]. Clearly, disc(q′_{s}) = \frac{1}{16}disc(q_{s}) = −d, so in particular −d ≡ 1 (mod 4). Since gcd(a, −d) = 1, we see that q′_{s} ∈ ℚ_{−d}^{(1)}. Similarly, f_{d}(s) = 2f′_{d}(s) with f′_{d}(s) = [bd, kd, a] ∈ Q_{−d}^{(1)} (because gcd(a, −d) = 1).

To prove (9), put q̃′_{s} = [a, −kt_{1}, abt] ∈ Q_{−d}^{(1)}. Since gcd(a, d) = 1, we have again by [7], Lemma 1, that q̃′_{s} ∘ q̃′_{s} ∼ q′_{s}. Now q̃′_{s} ∼ f′_{d}(s) because if we let y = −kdb, then the matrix g = \binom{1 y}{0 1} ∈ SL_{2}(ℤ) transforms f′_{d}(s) into q̃′_{s}; cf. formula (15) below. Thus, f′_{d}(s) ∘ f′_{d}(s) ∼ q̃′_{s} ∘ q̃′_{s} ∼ q′_{s}, which proves (9).

In order to prove the converse, we shall first interpret the relation (8) in terms of a natural map π′_{d}. To construct this map, recall that for any discriminant D, the set Q̄_{D} = Q_{D}^{(1)}/SL_{2}(ℤ) of proper equivalence classes of primitive forms of discriminant D form an abelian group under the composition of forms; cf. e.g. [2], p. 61. In addition, for any n ≥ 1 we have a natural group homomorphism π_{D,n} : Q̄_{n^{2}D} → Q̄_{D} given by q ↦ q ∘ 1_{D}; cf. [2], p. 132. We now prove:

Lemma 17 If d > 1, then there is a unique homomorphism π′_{d} : Q̄_{−4d} → Q̄_{−16d} such that π′_{d}(π_{−4d,2}(q)) ∼ q ∘ q, for all q ∈ Q̄_{−16d}. (10)

Furthermore, the image of π_{d} is (Q̄_{−16d})^{2}, the principal genus of discriminant −16d.

Proof. First note that π_{D,n} is always surjective. Indeed, by using the well-known identification of Q̄_{D} with Pic(𝔒_{D}), where 𝔒_{D} = ℤ + \frac{1}{2}(1 + \sqrt{D})ℤ is the order of discriminant D, the map π_{D,n} corresponds to the canonical map Pic(𝔒_{D}) → Pic(𝔒_{n^{2}D}) induced by the inclusion 𝔒_{n^{2}D} ⊂ 𝔒_{D}, which is known to be surjective; cf. Lang[24], p. 94.

From the explicit formula for h(D) := |Q̄_{D}| = |Pic(𝔒_{D})| (cf. [24], p. 95), we see that |Ker(π_{D,2})| = 2, if D = −4d and d > 1; in fact, we have

Ker(π_{−4d,2}) = {1_{−16d}, q_{d}}, (11)

where q_{d} = [4, 0, d], if d ≡ 1 (2), and q_{d} = [4, 4, d + 1], if d ≡ 0 (2), as is easy to verify. Thus, if S(q) = q ∘ q denotes the squaring homomorphism on Q̄_{D}, then Ker(π_{D,2}) ≤ Ker(S), and so by the universal property of quotients, there is a unique homomorphism π′_{d} : Q̄_{4D} → Q̄_{D} such that S = π′_{d} ∘ π_{D,2}. This proves the first assertion, and the second follows because (Q̄_{4D})^{2} is the image of S.

Corollary 18 If s = (n_{1}, n_{2}, k) ∈ P (d)^{odd}, i.e. if gcd(n_{1}, n_{2}, 2) = 1, then q_{s} ∼ π′_{d}(f_{d}(s)), and if s ∈ P (d)^{even}, then q′_{s} ∼ f′_{d}(s)^{2}.

Proof. If n_{2} is odd, then this follows directly from (8) and (10), and if n_{1} and n_{2} are both even, then q′_{s} ∼ f′_{d}(s)^{2} by (9).

There remains the case that n_{1} is odd (and n_{2} even). Here we observe that

f_{d}(n_{1}, n_{2}, k) ∼ f_{d}(n_{2}, n_{1}, −k), for all s = (n_{1}, n_{2}, k) ∈ P (d),

because the matrix g = \binom{n_{2} −k}{−kd n_{1}} ∈ SL_{2}(ℤ) transforms f_{d}(s) into f_{d}(s′), where s′ = (n_{2}, n_{1}, −k). Similarly, we have

q_{s} ∼ q_{s′},

because the matrix g′ = \binom{n_{1} y}{k n_{2}} ∈ SL_{2}(ℤ), where y = (n_{1}n_{2} + 1)kd, transforms q_{s} into q_{s′}. Thus, since n_{1} is odd, we have by (8) that q_{s′} ∼ π′_{d}(f_{d}(s′)), and so q_{s} ∼ q_{s′} ∼ π′_{d}(f_{d}(s′)) ∼ π′_{d}(f_{d}(s)), as claimed.

Corollary 19 For d > 1 we have

|Ker(π′_{d})| = \frac{1}{2}g(−16d) = 2^{ω(d)−1}, (12)

where g(D) = |Q̄_{D}/Q̄_{D}^{2}| denotes the number of genera of discriminant D and ω(d) the number of distinct prime divisors of d. Thus

(13) q ∈ Ker(π′_{d}) ⇔ q ∼ [d_{1}, 0, d_{2}], where d_{1}d_{2} = d, d_{1} ≤ d_{2}, and gcd(d_{1}, d_{2}) = 1.

Proof. Since π′_{d} ∘ π_{D,2} = S by (10), and |Ker(π_{D,2})| = 2 (cf. the proof of Lemma 17), we see that |Ker(π′_{d})| = \frac{1}{2}|Ker(S)| = \frac{1}{2}|Coker(S)| = \frac{1}{2}g(4D). This proves the first equality of (12). To prove the second, recall that Gauss’s genus theory yields

g(D) = 2^{ω(D)−1+ε(D)}, (14)

where ε(D) = 1 if D ≡ 0 (mod 32), ε = −1 if D ≡ 4 (mod 16) and ε(D) = 0 otherwise; cf. [18], p. 170. From this, the formula (12) follows easily.

Let d_{1}, d_{2} be as indicated. If d_{1} is odd, then [d_{1}, 0, 4d_{2}] ∈ Ker(S) and so [d_{1}, 0, d_{2}] ∼ [d_{1}, 0, 4d_{2}] ∘ 1_{D} ∼ π_{D,2}([d_{1}, 0, 4d_{2}]) ∈ Ker(π′_{d}). Similarly, if d_{1} is even, then d_{2} is odd, and then [d_{1}, 0, d_{2}] ∼ π_{D,2}([4d_{1}, 0, d_{2}]) ∈ Ker(π′_{d}). Since the forms [d_{1}, 0, d_{2}] are all reduced, they yield 2^{ω(d)−1} distinct equivalence classes in Ker(π′_{d}). By (12) we have thus found all the classes in Ker(π′_{d}) and so (13) follows.

Lemma 20 The inclusion Q_{−4d}^{(2)}(d) ⊂ Q_{−4d}^{(2)} induces a bijection

Q_{−4d}^{(2)}(d)/Γ_{0}(d) →^{∼} Q_{−4d}^{(2)}/SL_{2}(ℤ),

where Γ_{0}(d) = \{g = \binom{x y}{z w} ∈ SL_{2}(ℤ) : d|z\}.

Proof. Recall that the action of g = \binom{x y}{z w} ∈ SL_{2}(ℤ) on Q_{D}^{(n)} is given by

[a, b, c]g = [ax^{2} + bxz + cz^{2}, b(xw + yz) + 2(axy + czw), ay^{2} + byw + cw^{2}]; (15)

cf. [2], p. 4. In other words, we have M(qg) = g^{t}M(q)q, where M(q) = \binom{a b}{b c} denotes the matrix associated to q = [a, 2b, c]. From this we see easily that Γ_{0}(d) acts on Q_{D}^{(2)}(d), where D = −4d, and so we have a map j : Q_{D}^{(2)}(d)/Γ_{0}(d) → Q_{D}^{(2)}/SL_{2}(ℤ).

To see that j is injective, suppose that q_{i} = [a_{i}d, 2b_{i}d, c_{i}] ∈ Q_{D}^{(2)}(d), are such that j(q_{1}) = j(q_{2}). Then there is a g = \binom{x y}{z w} ∈ GL_{2}(ℤ) such that m_{2} = m_{1}^{g} := g^{t}m_{1}g. Then a_{2}d = a_{1}dx^{2} + 2b_{1}dxz + c_{1}z^{2} and b_{2}d = b_{1}d(xw + yz) + (a_{1}dxy + c_{1}zw), so d| gcd(c_{1}z^{2}, c_{1}zw) = c_{1}z, and hence d|z because gcd(c_{1}, d) = 1. (Recall that (a_{i}, c_{i}, b_{i}) ∈ P (d); cf. Lemma 14.) Thus, g ∈ Γ_{0}(d), and so j is injective.

We now prove that j is surjective. Let q = [a, 2b, c] ∈ Q_{−4d}^{(2)}. We first note that by replacing q by qg with a suitable g ∈ SL_{2}(ℤ) we may assume gcd(a, d) = 1. Indeed, if q ∈ Q_{−4d}^{(1)}, then this is well-known; cf. [2], pp. 49-50. In the other case we have q = 2q_{1}, where q_{1} ∈ Q_{−d}^{(1)} and −d ≡ 1 (mod 4), and so the assertion follows by same argument applied to q_{1}. Thus, gcd(a, d) = 1 and hence also gcd(a, b) = 1 because ac − b^{2} = d. Thus, there exist x, y ∈ ℤ such that g = \binom{−b x}{a y} ∈ SL_{2}(ℤ). Then qg = [ad, 2yd, ∗], and so we see that q ∈ Im(j). This proves that j is bijective.

Proof of Proposition 15. Let s ∈ P (d). If s ∈ P (d)^{odd}, then f_{d}(s) is primitive of discriminant −4d and hence by Corollary 18 and Lemma 17 we see that q_{s} ∼ π′_{d}(f_{d}(s)) is in the principal genus of discriminant −16d (and is primitive). Thus, q_{s} is of type d, provided that q_{s} is not in the principal class.

On the other hand, if s ∈ P (d)^{even}, then by Lemma 16(b) we know that q_{s} = 4q′_{s}, where q′_{s} is primitive of discriminant −d. Moreover, (9) shows that q′_{s} is in the principal genus, so q_{s} has type d also in this case.

Conversely, suppose q is a form of type d. Assume first that q is primitive. Since q lies in the principal genus, we have by Lemma 17 that q ∼ π′_{d}(q_{1}), for some q_{1} ∈ Q_{−4d}^{(1)}. By Lemma 20 (and Lemma 14) we have q_{1} ∼ f_{d}(s), for some s ∈ P (d)^{odd}. Thus, q ∼ π′_{d}(f_{d}(s)) ∼ q_{s}, the latter by Corollary 18.

Next, suppose that q is not primitive, so by definition q = 4q′, where q′ ∼ q′′ ∘ q′′ for some q′′ ∈ Q_{−d}^{(1)}. Then 2q′′ ∈ Q_{−4d}^{(2)}, and so by Lemma 20 (and Lemma 14) there is an s ∈ P (d)^{even} such that 2q′′ ∼ f_{d}(s). Thus, q′′ ∼ f′_{d}(s) and so by (9) we obtain q′_{s} ∼ f′_{d}(s) ∘ f_{d}(s) ∼ q′′ ∘ q′′ ∼ q′. We therefore have q_{s} = 4q′_{s} ∼ 4q′ = q, as claimed.

We now derive some properties of modules endowed with forms of prototype q_{s}, where s ∈ P (d). These will be used in the next section.

Lemma 21 Let (M, q) be a quadratic module of rank 2, and suppose that M has a basis {v_{1}, v_{2}} such that for some s = (n_{1}, n_{2}, k) ∈ P (d) we have

q(xv_{1} + yv_{2}) = q_{s}(x, y), for all x, y ∈ ℤ.

Put w_{1} = v_{1} and w_{2} = −n_{1}^{2}v_{1} − kv_{2}. Then

q(xw_{1} + yw_{2}) = n_{2}^{2}x^{2} + 2(n_{1}n_{2} − 2)xy + n_{1}^{2}y^{2}. (16)

Moreover, for w_{3} = n_{1}kdv_{1} + n_{2}v_{2} we have q(w_{3}) = 4dn_{1}n_{2}, provided that k ≠ 0.

Proof. The relation (16) is a straight-forward computation, using the transformation law (15) applied to g = \binom{1 −n_{1}^{2}}{0 −k} and the relation (5).

To compute q(w_{3}), note first that by (5) we have kw_{3} = −(n_{1}w_{1} + n_{2}w_{2}). Thus, by (16) we obtain k^{2}q(w_{3}) = q(n_{1}w_{1} + n_{2}w_{2}) = 4n_{1}n_{2}(n_{1}n_{2} − 1) = 4n_{1}n_{2}k^{2}d, and so q(w_{3}) = 4n_{1}n_{2}d because k ≠ 0.

6 The product surface E_{1} × E_{2}

The aim of this section is to prove the basic classification Theorem 13. For this, it is useful to use the following “presentation” of the Néron-Severi group NS(A) of a product surface A = E_{1} × E_{2} of two elliptic curves E_{1} and E_{2}.

Proposition 22 For a, b ∈ ℤ and h ∈ Hom(E_{1}, E_{2}) put

(17) D(a, b, h) = (a − deg(h))θ_{1} + (b − 1)θ_{2} + Γ_{−h} ∈ Div(A), where θ_{i} = p_{i}^{∗}(0_{E_{i}}), and Γ_{f} ∈ Div(A) is the graph of f = −h. Then the rule (a, b, h) ↦ cl(D(a, b, h)) defines a group isomorphism

D = D_{E_{1},E_{2}} : ℤ ⊕ ℤ ⊕ Hom(E_{1}, E_{2}) →^{∼} NS(E_{1} × E_{2}),

and we have (D(a, b, f).D(a′, b′, f′)) = ab′ + ba′ − β_{d}(f, f′), (18)

where β_{d} is the bilinear form associated to the degree quadratic form on Hom(E_{1}, E_{2}), i.e. β_{d}(f, f′) = deg(f + f′) − deg(f) − deg(f′). In addition, the homomorphism ϕ_{D} : A → Â associated to D = D(a, b, f) is given by

ϕ_{D(a,b,f)} = λ_{1} ⊗ λ_{2} ∘ ( \begin{matrix}[a]_{E_{1}} & f^{t} \\ f & [b]_{E_{2}} \end{matrix} ) , (19)

where λ_{1} ⊗ λ_{2} denotes the product polarization associated the canonical polarizations λ_{i} : E_{i} →^{∼} Ê_{i}, for i = 1, 2, and f^{t} = λ_{1}^{−1}f̂λ_{2} is the dual map.

Proof. Most of this is well-known; for example, the fact that D is an isomorphism is a special case of the basic relation between correspondences of curves and homomorphisms of their Jacobians; cf. [31], p. 185. In the appendix below we derive this in Proposition 61 as a special case of a more general construction (based on (19)) which has the advantage of being more functorial.

Corollary 23 The determinant of the Néron-Severi group of E_{1} × E_{2} with respect to the intersection form is given by

(20) det(NS(E_{1} × E_{2})) = (−1)^{ρ−1} det(Hom(E_{1}, E_{2}), β_{d}),

where ρ = rank(NS(E_{1} × E_{2})) = rank(Hom(E_{1}, E_{2})) + 2.

Proof. Put Γ_{f}^{∗} = D(0, 0, f). If f_{1}, …, f_{r} is a basis of Hom(E_{1}, E_{2}), then by Proposition 22 we have that θ_{1}, θ_{2}, Γ_{f_{1}}^{∗}, …, Γ_{f_{r}}^{∗} is a basis of NS(E_{1} × E_{2}) and so by (18) we see that the Gram matrix G(θ_{1}, θ_{2}, Γ_{1}^{∗}, …, Γ_{f_{r}}^{∗}) of the intersection form with respect to this basis is given by the block diagonal matrix

G(θ_{1}, θ_{2}, Γ_{f_{1}}^{∗}, …, Γ_{f_{r}}^{∗}) = diag (\binom{0 1}{1 0}, −G(f_{1}, …, f_{r})) ,

where G(f_{1}, …, f_{r}) is the Gram matrix of β_{d} with respect to the basis f_{1}, …, f_{r}. From this, formula (20) follows by taking the determinant of both sides.

In the sequel we shall be particularly interested in the set P(A) consisting of those divisors D ∈ NS(A) which define principal polarizations on A. These can characterized by using the set P (d) introduced in the previous section.

Corollary 24 Let D = D(a, b, h) ∈ NS(A). Then D defines a principal polarization (i.e. D ∈ P(A)) if and only if a > 0 and ab − deg(h) = 1. Thus, every principal polarization of A has the form

(21) D_{s,h} = D(n_{1}, n_{2}, kh) with h ∈ Hom(E_{1}, E_{2}) and s = (n_{1}, n_{2}, k) ∈ P (deg(h)).

Proof. By the Riemann-Roch Theorem (cf. [30], p. 127), D ∈ P(A) if and only if D is ample and D^{2} = 2, and this holds if and only if D^{2} = 2 and (D.θ_{2}) > 0; cf. [19], Corollary 2.2b). Thus, the first assertion follows in view of (18). The second follows from this and the fact that deg(kh) = k^{2} deg(h).

We now turn to the study of curves C of type d. As promised, we first verify that every curve whose Jacobian is isomorphic to a product of two elliptic curves has a type d, for some d ≥ 1.

Proposition 25 Let C be a curve such that its Jacobian J_{C} has an isomorphism α : J_{C} →^{∼} E_{1} × E_{2} to a product of two elliptic curves. Then there exists a cyclic isogeny h : E_{1} → E_{2} of some degree d ≥ 1 such that

(22) θ_{C} ≡ α^{∗}(D_{s,h}), for some s = (n_{1}, n_{2}, k) ∈ P (d) with k ≠ 0.

In particular, E_{1} is isogenous to E_{2} and C has type d.

Proof. Put D ≡ (α^{−1})^{∗}(θ_{C}) ∈ P(E_{1} × E_{2}). By Proposition 22 and Corollary 24, D = D(n_{1}, n_{2}, h_{1}), for some integers n_{1}, n_{2} and homomorphism h_{1} ∈ Hom(E_{1}, E_{2}) satisfying n_{1}n_{2}−deg(h_{1}) = 1 and n_{1} > 0. Note that h_{1} ≠ 0, for otherwise n_{1} = n_{2} = 1 which means D ≡ θ_{1} + θ_{2}. But then q_{C}(α^{∗}θ_{1}) = 1, which contradicts Proposition 6.

Thus, we can write h_{1} = kh, where h is a cyclic isogeny and k ≠ 0, and so we see that (22) holds with s = (n_{1}, n_{2}, k) ∈ P (d). Note that this means that C has type d because D ≡ kD(n_{1}, n_{2}, h) = k(n_{1} − d)θ_{1} + k(n_{2} − 1)θ_{2} + kΓ_{−h}.

Corollary 26 If J_{C} ≃ E_{1} × E_{2}, where End(E_{1}) = ℤ, then C is a curve of unique type d = \frac{1}{2} det(NS(J_{C})).

Proof. Since E_{2} ∼ E_{1} by Proposition 25, it follows that Hom(E_{1}, E_{2}) = ℤh, and so by (20) we have det(NS(J_{C})) = (−1)^{2} det(NS(E_{1} × E_{2})) = β_{d}(h, h) = 2d, where d = deg(h).

Note that h is necessarily cyclic, and that the only cyclic isogenies in Hom(E_{1}, E_{2}) are ±h. Thus, if α : J_{C} →^{∼} E_{1} × E_{2} is any isomorphism, then θ_{C} ≡ α^{∗}(D_{s,h}), for some s ∈ P (d), and so C has (unique) type d = \frac{1}{2} det(NS(J_{C})).

Remark 27 (a) Although the type d is uniquely determined by the curve C in the above situation, the elliptic curves E_{1} and E_{2} are not unique (up to isomorphism).

Indeed, if d has more than one prime factor, then we can have an isomorphism E_{1} × E_{2} ≃ E′_{1} × E′_{2} with E′_{1} ≄ E_{1}, E_{2}; cf. Proposition 49 below.

(b) If C is any curve of type d satisfying (22) with (n_{1}, n_{2}, k) ∈ P (d), then ⟨C⟩ ∈ H_{n_{1}^{2}} ∩ H_{n_{2}^{2}} because q_{C}(α^{∗}(θ_{1})) = n_{2}^{2} and q_{C}(α^{∗}(θ_{1})) = n_{1}^{2} by (18) (and (2)). (Note that since α^{∗}(θ_{i}) is an elliptic curve, its image in NS(J_{C}, θ_{C}) is primitive by [19], Theorem 2.8.) Thus, if n_{1} ≠ n_{2}, then we see by Proposition 8 that ⟨C⟩ ∈ H(q), for some binary quadratic form q.

We now turn to the proof of Theorem 13. One direction is contained in the following more precise result.

Proposition 28 Let A = E_{1} × E_{2} and θ = D_{s,h} ∈ P(A), where h is a cyclic isogeny of degree d and s = (n_{1}, n_{2}, k) ∈ P (d). Let θ̄_{1}, θ̄_{2} and Γ̄_{h}^{∗} denote the images of θ_{1}, θ_{2}, and Γ_{h}^{∗} = D(0, 0, h) in NS(A, θ), respectively.

(a) M̄ := ⟨θ̄_{1}, θ̄_{2}, Γ̄_{h}^{∗}⟩ is a primitive submodule of NS(A, θ), and so ⟨A, θ⟩ ∈ H(qM̄), where q_{|M} denotes the restriction of q_{θ} = q_{(A,θ)} to M̄.

(b) Let D̄ = kdθ̄_{2} + n_{1}Γ̄_{h}^{∗}. Then {θ̄_{1}, D̄} is a basis of M̄, and we have

(23) q_{θ}(xθ̄_{1} + yD̄) = q_{s}(x, y) ∀x, y, ∈ ℤ, where q_{s} is defined by (6).

Proof. (a) Since h is a cyclic isogeny, it is a primitive element in Hom(E_{1}, E_{2}), and so we can extend h to a basis h_{1} = h, h_{2}, …, h_{r} of Hom(E_{1}, E_{2}). Then {cl(θ_{1}), cl(θ_{2}), cl(Γ_{h_{1}}^{∗}), …, cl(Γ_{h_{r}}^{∗})} is a basis of NS(E_{1} × E_{2}); cf. Corollary 23. Thus, M := ⟨cl(θ_{1}), cl(θ_{2}), cl(Γ_{h}^{∗})⟩ is a primitive submodule of NS(E_{1} × E_{2}), and so we see that M̄ = M/(ℤθ) is a primitive submodule of NS(A, θ). This means that q_{θ} primitively represents qM̄, and so ⟨A, θ⟩ ∈ H(qM̄).

(b) Put D = D(0, kd, n_{1}h) ∈ NS(A); thus, D̄ is the image of D in NS(A, θ). Using (5), we see that cl(θ_{2}) = n_{1}θ − kD − n_{1}^{2}cl(θ_{1}), and cl(Γ_{h}^{∗}) = n_{2}D − kdθ + n_{1}kdθ_{1}, so {θ, cl(θ_{1}), D} is a basis of M, and hence {θ̄_{1}, D̄} is a basis of M̄.

Put D_{1} = xθ_{1} + yD. Then by computing intersection numbers we find that (θ.D_{1}) = n_{2}x − n_{1}kdy and D_{1}^{2} = 2(kdxy − n_{1}^{2}dy^{2}), and so q_{θ}(D_{1}) = (θ.D_{1})^{2} − 2D_{1}^{2} = n_{2}^{2}x^{2} − 2kd(n_{1}n_{2} + 2)xy + n_{1}^{2}d(k^{2}d + 4)y^{2} = q_{s}(x, y); here we used the fact that k^{2}d + 4 = n_{1}n_{2} + 3 by (5).

For the other direction we shall use the following elementary fact.

Lemma 29 Let \overline{cl} : NS(A) → NS(A, θ) = NS(A)/ℤθ denote the quotient map, and let D̄ ∈ NS(A, θ). If n ∈ ℤ, then there exists D ∈ NS(A) with

(D.θ) = n and \overline{cl}(D) = D̄ (24)

if and only if n ≡ q_{(A,θ)}(D̄) (mod 2).

Proof. If D exists, then q_{θ}(D̄) = q_{θ}(D) = n^{2} − 2D^{2} ≡ n^{2} ≡ n (mod 2). Conversely, suppose that n ≡ q_{C}(D̄) (mod 2), and let D_{0} ∈ NS(A) be any class with \overline{cl}(D_{0}) = D̄. Put n_{0} = (D.θ). Then, by what was just shown, n_{0} ≡ q_{C}(D̄) ≡ n (mod 2), and so D = \frac{1}{2}(n − n_{0})θ + D_{0} satisfies (24).

Proof of Theorem 13. If C is a curve of type d, then (22) holds for some s = (n_{1}, n_{2}, k) ∈ P (d) by Proposition 25, and so Proposition 28 shows that the form q_{s} is primitively represented by q_{C}. Note that q_{s} cannot represent 1 by Proposition 6, so q_{s} cannot be in the principal class. Thus, q_{s} is a form of type d by Proposition 15.

Conversely, suppose that ⟨C⟩ ∈ H(q), where q is a form of type d. Then by Proposition 15 we know q ∼ q_{s} for some s = (n_{1}, n_{2}, k) ∈ P (d). (Note that k ≠ 0 for otherwise q_{s} represents 1 = n_{2}^{2}.) Thus, there exist D̄′_{1}, D̄′_{2} ∈ NS(J_{C}, θ_{C}) (which generate a primitive submodule M̄ of NS(J_{C}, θ_{C})) such that

q_{C}(xD̄′_{1} + yD̄′_{2}) = q_{s}(x, y), for all x, y ∈ ℤ.

Put D̄_{1} = D̄′_{1} and D̄_{2} = −n_{1}^{2}D̄′_{1} − kD̄′_{2}; note that D̄_{1} and D̄_{2} are primitive in M̄ and hence in NS(J_{C}, θ_{C}) because gcd(−n_{1}^{2}, k) = 1. Applying Lemma 21 to M = M̄ and v_{i} = D̄′_{i}, we see from (16) that q_{C}(D̄_{1}) = n_{2}^{2} and q_{C}(D̄_{2}) = n_{1}^{2}. Thus, by Theorem 3.2 of [19] we know that there are unique elliptic subgroups E_{i} ≤ J_{C} such that \overline{cl}(E_{i}) = D̄_{i}, for i = 1, 2, and that we have (E_{1}.θ_{C}) = n_{2} and (E_{2}.θ_{C}) = n_{1}. Furthermore, since E_{i}^{2} = 0, we have 4(E_{1}.E_{2}) = 2(E_{1} + E_{2})^{2} = ((E_{1} + E_{2}).θ_{C})^{2} − q_{C}(E_{1} + E_{2}) = (n_{1} +n_{2})^{2} −q_{C}(D̄_{1} +D̄_{2}). By (16) we know that q_{C}(D̄_{1} +D̄_{2}) = n_{2}^{2} +2(n_{1}n_{2} −2)+n_{1}^{2}, and so (E_{1}.E_{2}) = 1. Thus, there is an isomorphism α : J_{C} →^{∼} E_{1} × E_{2} such that α^{∗}θ_{1} = E_{2} and α^{∗}θ_{2} = E_{1}.

It remains to show that C has type d = −\frac{1}{16}disc(q). For this, put D = α_{∗}θ_{C} ∈ P(E_{1} × E_{2}), and write D = D(a, b, ch), where a, b, c ∈ ℤ and h ∈ Hom(E_{1}, E_{2}) is cyclic. Then n_{1} = (θ_{C}.E_{2}) = (D.θ_{2}) = a, so a = n_{1} and similarly b = n_{2}.

To prove that d = deg(h), consider D̄_{3} := n_{1}kdD̄′_{1} +n_{2}D̄′_{2}. Since q_{C}(D̄_{3}) = 4dn_{1}n_{2} by Lemma 21, we know by Lemma 29 that there exists D_{3} ∈ NS(J_{C}) such that (D_{3}.θ_{C}) = −2kd and \overline{cl}(D_{3}) = D̄_{3}. We now observe that

θ_{C} ≡ n_{1}E_{1} + n_{2}E_{2} + kD_{3}. (25)

Indeed, since kD̄_{3} = −(n_{1}D̄_{1} + n_{2}D̄_{2}) (cf. the proof of Lemma 21), it follows that θ′ := n_{1}E_{1} + n_{2}E_{2} + kD_{3} = mθ_{C}, for some m ∈ ℤ. But then 2m = mθ_{C}^{2} = (θ′.θ_{C}) = n_{1}n_{2}+n_{2}n_{1}+k(−2kd) = 2, so m = 1. Thus (25) holds, and so we obtain kα_{∗}D_{3} = cΓ_{h}^{∗}. Since Γ_{h}^{∗} is primitive in NS(E_{1} × E_{2}), it follows that α_{∗}D_{3} = c′Γ_{h}^{∗}, where c′ = \frac{c}{k} ∈ ℤ. Thus, D_{3} = c′D′_{3}, where D′_{3} := α^{∗}(Γ_{h}^{∗}), and so D̄′_{3} = \overline{cl}(D′_{3}) ∈ M̄ = ℤD′_{1} + ℤD′_{2} because M̄ is a primitive submodule of NS(J_{C}, θ_{C}). Now c′D̄′_{3} = D̄_{3} = n_{1}kdD̄′_{1}+n_{2}D̄′_{2}, so c′| gcd(n_{1}kd, n_{2}) = gcd(n_{1}, n_{2}). But gcd(c′, n_{1}n_{2}) = 1 because n_{1}n_{2} − c^{2} deg(h) = 1 (since D ∈ P(E_{1} × E_{2})), and so c′ = ±1, i.e. deg(h) = d. Thus C has type d.

7 The existence theorem

We now show that H(q) is non-empty, whenever q is a form of type d. This follows from the following more precise assertion:

Theorem 30 Suppose that q is a binary quadratic form of type d. Let E_{1} be any elliptic curve with End(E_{1}) = ℤ, and let E_{2} = E_{1}/H, where H ≤ E_{1} is any cyclic subgroup (scheme) of degree d. Then there exists a curve C with J_{C} ≃ E_{1} × E_{2} such that q_{C} is equivalent to q, i.e. q_{C} ≈ q; in particular, ⟨C⟩ ∈ H(q).

To prove this, we shall use the following refinement of Corollary 24.

Proposition 31 Suppose Hom(E_{1}, E_{2}) = ℤh, and let d = deg(h). Then the map s ↦ D_{s,h} defines a bijection between the set P (d) and the set P(A) of principal polarizations on A := E_{1} × E_{2}. Furthermore, θ ∈ P(A) is represented by a smooth curve C of genus 2 if and only if q_{(A,θ)} is not in the principal class.

Proof. The first assertion follows immediately from Corollary 24 since here every D ∈ NS(E_{1} ×E_{2}) has the form D(a, b, ch), and since deg(ch) = c^{2} deg(h). The second follows immediately from Proposition 6 because a binary quadratic form represents 1 if and only if it is in the principal class.

Proof of Theorem 30. By Proposition 15 there exists s ∈ P (d) such that q ∼ q_{s}. Put θ = D_{s,h} ∈ NS(E_{1} × E_{2}), where h : E_{1} → E_{2} = E_{1}/H_{1} denotes the quotient map. (Note that h is cyclic, and so Hom(E_{1}, E_{2}) = ℤh.) By Proposition 31 we see that θ ∈ P(A), and Proposition 28 shows that q_{(A,θ)} ≈ q_{s} ∼ q. (Note that M̄ = NS(A, θ) because M̄ is primitive in NS(A, θ) and rk(NS(A, θ)) = 2.) Since q is not in the principal class by hypothesis, Proposition 31 shows that (A, θ) ≃ (J_{C}, θ_{C}), for some curve C of genus 2. By construction, q_{C} ≈ q.

We now consider some applications of the Existence Theorem 30. The first is the following useful fact.

Corollary 32 If q_{i} is a quadratic form of type d_{i}, for i = 1, 2, then H(q_{1}) = H(q_{2}) if and only if q_{1} ≈ q_{2}.

Proof. If q_{1} ≈ q_{2}, then H(q_{1}) = H(q_{2}) by definition. Conversely, suppose H(q_{1}) = H(q_{2}). By Theorem 30 there exists ⟨C⟩ ∈ H(q_{1}) such that q_{C} ≈ q_{1}. Since ⟨C⟩ ∈ H(q_{2}), this means that q_{C} primitively represents q_{2}, and so q_{2} ≈ q_{C} because both have rank 2. Thus q_{1} ≈ q_{2}, as asserted.

Remark 33 The above proof also shows that if q_{1} ≉ q_{2}, then H(q_{1}) ∩ H(q_{2}) consists only of CM-points, i.e. of points ⟨A, θ⟩ such that A ≃ E_{1} × E_{2}, where E_{1} ∼ E_{2} are elliptic curves which have complex multiplication (or are supersingular).

Corollary 34 We have for d ≥ 1 that T (d) = ∅ ⇔ Q̄_{d}^{∗} = ∅, and hence

T (d) = ∅ ⇔ h̄(−16d) = 1 and d ≢ 3 (4) ⇔ h̄(−16d) = 1 and d ≠ 3, 7, 15. (26)

where h̄(D) = h(D)/g(D) denotes the number of forms in the principal genus of primitive binary quadratic forms of discriminant D = −16d.

Proof. The first assertion follows directly from Theorems 13 and 30. To prove the first equivalence of (26), note first that it follows from the definitions that the number t(d) of SL_{2}(ℤ)-equivalence classes of forms of type d is given by

(27) t(d) = \{ \begin{matrix}h̄(−16d) − 1, & if d ≢ 3 (mod 4) \\ h̄(−16d) − 1 + h̄(−d), & if d ≡ 3 (mod 4). \end{matrix}

In view of Remark 12 we see that Q̄_{d}^{∗} = ∅ ⇔ t(d) = 0, and so the first equivalence follows. To prove the second, it is enough to show that if d ≡ 3(4), then h̄(−16d) > 1 when d ≠ 3, 7, 15. For this we observe that (14) implies that g(−16d) = 2g(−4d), when d ≡ 3(4) and that hence h̄(−16d) = h̄(−4d) because h(−16d) = 2h(−4d) (cf. Lemma 17). Now by Hall[11], Theorem I, we have h̄(−4d) > 1 when d ≡ 3(4) and d ≠ 3, 7, 15, and so the second equivalence follows.

Remark 35 It is clear that the above number t(d) is closely connected to the number t^{∗}(d) = #Q̄_{d}^{∗} of GL_{2}(ℤ)-equivalence classes of forms of type d. To make this connection precise, however, we require another invariant of forms of discriminant D: the number s̄(D) = |Q̄_{D}^{2}[2]| of ambiguous classes in the principal genus. This number is closely related to the number s(D) = [Q̄_{D} : Q̄_{D}^{4}] of spinor genera of (primitive) forms of discriminant D as defined by Estes/Pall [7], for we have s̄(D) = s(D)/g(D). Now by Remark 12 we have

h̄^{∗}(D) := #(Q̄_{D}^{2}/GL_{2}(ℤ)) = \frac{1}{2}(h̄(D) + s̄(D)), (28)

and so we see that

(29) t^{∗}(d) := #Q̄_{d}^{∗} = \{ \begin{matrix}h̄^{∗}(−16d) − 1, & if d ≢ 3 (mod 4) \\ h̄^{∗}(−16d) − 1 + h̄^{∗}(−d), & if d ≡ 3 (mod 4). \end{matrix}

Proof of Corollary 5: From Gauss[10], Art. 303, we know that h̄(−16d) = 1 when d is one of the values of (1); cf. also Dickson[5], p. 89. Thus, by Corollary 34 we see that T (d) = ∅ for those values of d. Moreover, if we look at the list of exceptional discriminants which are of the form −16d with d ≠ 3, 7, 15, then we obtain the list (1). Finally, the finiteness assertion follows from Chowla[3], and the fact that (GRH) implies Gauss’s Conjecture was proved by Weinberger[37].

For later applications it is useful to refine the above existence theorem by determining the number of isomorphism classes of curves C on E_{1} × E_{2} such that q_{C} ≈ q.

Theorem 36 Let A = E_{1} × E_{2}, where Hom(E_{1}, E_{2}) = ℤh, and let q be a quadratic form of type d := deg(h). Then the number N_{A}(q) of isomorphism classes of smooth genus 2 curves C on A with q_{C} ≈ q is given by:

(30) N_{A}(q) = \{ \begin{matrix}2^{ω(d)−2} & if q ∈ Q̄_{−16d}^{2}[2] \ \{q_{d}\} or if \frac{1}{4}q ∈ Q̄_{−d}^{2}[2] \ \{1_{−d}\}, \\ 2^{ω(d)−1} & otherwise, \end{matrix}

where q_{d} = 4x^{2} + dy^{2}, if d ≡ 0(2), and q_{d} = 4x^{2} + 4xy + (d + 1)y^{2}, if d ≡ 1(2).

Remark 37 Note that q_{d} is not necessarily in Q̄_{−16d}^{2}[2]. In fact, this is the case if and only if d ≡ 0, 1, 5 (8), as can be verified by checking the generic characters of q_{d}.

As we shall see presently, the above theorem follows easily from the following fact which is interesting in itself.

Proposition 38 Let A = E_{1} × E_{2}, where Hom(E_{1}, E_{2}) = ℤh, and let d = deg(h). If C is a smooth genus 2 curve on A, then C ≡ D_{s,h}, for some s ∈ P (d) with f_{d}(s) ∉ Ker(π′_{d}), and the isomorphism class of C is uniquely determined by the GL_{2}(ℤ)-equivalence class of the binary quadratic form f_{d}(s). Furthermore, q_{C} ≈ q_{s}.

Before proving this, let us see how Theorem 36 follows from it.

Proof of Theorem 36: Suppose first that q is primitive, so q ≈ q_{1}^{2}, where q_{1} ∈ Q̄_{−16d}. If C is any curve on A, then by Proposition 38 we have that C ≡ D_{s,h} with s ∈ P (d), and that q_{C} ≈ q_{s}. We thus see from Corollary 18 and Proposition 38 that

N_{A}(q) = #(π′_{d}^{−1}(q) ∪ π′_{d}^{−1}(q^{−1}))/GL_{2}(ℤ).

Now if q ≁ q^{−1}, then the sets π′_{d}^{−1}(q) and π′_{d}^{−1}(q^{−1}) are interchanged under the GL_{2}(ℤ)-action, and so N_{A}(q) = #(π′_{d}^{−1}(q)) = |Ker(π′_{d})| = 2^{ω(d)−1} by Corollary 19. (Note that we can assume d > 1 for otherwise Q̄_{d}^{∗} is empty by (26).) This proves (30) in this case. Next, suppose q ∼ q^{−1}, i.e. q ∈ Q̄_{−16d}^{2}[2]. Now if q ∈ Ker(π_{−4d,2}), i.e. if q ∼ q_{d} by (11), then π′_{d}^{−1}(q) ⊂ Q̄_{−4d}[2] (cf. Lemma 17), and so N_{A}(q) = #(π′_{d}^{−1}(q)) = |Ker(π′_{d})| = 2^{ω(d)−1} again. On the other hand, if q ∈ Q̄_{−16d}^{2}[2] \ {q_{d}}, then π′_{d}^{−1}(q) ∩ Q̄_{−4d}[2] = ∅, and so the GL_{2}(ℤ)-action has no fixed points on π′_{d}^{−1}(q), and hence N_{A}(q) = \frac{1}{2}#(π′_{d}^{−1}(q)) = \frac{1}{2}|Ker(π′_{d})| = 2^{ω(d)−2} by Corollary 19.

Finally, suppose that q is not primitive. Then q ≈ 4q_{1} with q_{1} ∈ Q̄_{−d}^{2} and d ≡ 3 (mod 4). In this case we have by the same reasoning as above that

N_{A}(q) = #(S_{d}^{−1}(q_{1}) ∪ S_{d}^{−1}(q_{1}^{−1}))/GL_{2}(ℤ),

where S_{d} : Q̄_{−d} → Q̄_{−d}^{2} is the squaring map. Since |Ker(S_{d})| = g(−d) = 2^{ω(d)−1} (cf. (14)), a similar analysis as above yields (30).

We now turn to the proof of Proposition 38. For this, we require the following information about the functorial behaviour of the divisor D_{s,f}.

Proposition 39 If g = \binom{a b}{cd e} ∈ Γ_{0}^{±}(d) := Γ_{0}(d)∪\binom{−1 0}{0 1}Γ_{0}(d), and if f ∈ Hom(E_{1}, E_{2}) has degree d, then

α_{g,f} = ( \begin{matrix}[a]_{E_{1}} & bf^{t} \\ cf & [e]_{E_{2}} \end{matrix} ) ∈ Aut(E_{1} × E_{2}),

and we have α_{g,f}^{∗}(D_{s,f}) := D_{sg,f}, for all s ∈ P (d), (31)

where sg ∈ P (d) is defined by the rule f_{d}(sg) = f_{d}(s)g.

Proof. We first observe that if [g]_{E_{2}} ∈ End(E_{2} × E_{2}) denotes the endomorphism induced by the matrix g ∈ M_{2}(ℤ), then we have

α_{g,f} = (f^{t} × 1_{E_{2}})^{−1} ∘ [g]_{E_{2}} ∘ (f^{t} × 1_{E_{2}}), (32)

and so α_{g,f} ∈ Aut(A) as deg(α_{g,f}) = deg([g]_{E_{2}}) = (det(g))^{2} = 1; cf. Corollary 63.

Although we could deduce (31) directly from the pullback formula (70) by a tedious calculation, it is easier to apply formula (60) to the map Ψ_{f} := Φ_{λ_{1}⊗λ_{2},f^{t}×1} : NS(A) → End(E_{2} ×E_{2}) which is introduced in Proposition 56 of the appendix. In our situation (60) becomes

(33) Ψ_{f}(α_{g,f}^{∗}D) = [g^{t}]_{E_{2}}Ψ_{f}(D)[g]_{E_{2}}, for all D ∈ NS(A),

because by (32) and (63) we have

c_{f^{t}×1}(α_{g,f}) = [g]_{E_{2}} and r_{λ_{2}⊗λ_{2}}(c_{h}(α_{g,f})) = [g^{t}]_{E_{2}}. (34)

Next we observe that by (19) we have

Ψ_{f}(D(a, b, cf)) = ( \begin{matrix}f & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}[a] & cf^{t} \\ cf & [b] \end{matrix} ) ( \begin{matrix}f^{t} & 0 \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}[ad] & [cd] \\ [cd] & [b] \end{matrix} ) , ∀a, b, c ∈ ℤ, (35)

and so Ψ_{f}(D_{s,f}) = [M(f_{d}(s))]_{E_{2}}, where (as before) M(q) = \binom{a b}{b c} ∈ M_{2}(ℤ) denotes the matrix associated to the quadratic form q = [a, 2b, c].

Since the action of g on quadratic forms is given by the formula M(f_{d}(gs)) := M(f_{d}(s)g) = g^{t}M(f_{d}(s))g, we thus obtain from (33) that

Ψ_{f}(α_{g,f}^{∗}D_{s,f}) = [g^{t}]_{E_{2}}Ψ_{f}(D_{s,f})[g]_{E_{2}} = [g^{t}M(f_{d}(s))g]_{E_{2}} = [M(f_{d}(sg))]_{E_{2}} = Ψ_{f}(D_{sg,f}),

and so (31) follows because Ψ_{f} is injective (cf. Corollary 58).

Corollary 40 If A = E_{1} × E_{2} and Hom(E_{1}, E_{2}) = ℤh, where deg(h) = d, then the map g ↦ α_{g,h} defines a group isomorphism Γ_{0}^{±}(d) →^{∼} Aut(A), and hence the rule D_{s,f} ↦ f_{d}(s) induces bijections

f̄_{A} : P(A)/Aut(A) →^{∼} Q_{−4d}^{(2)}(d)/Γ_{0}^{±}(d) →^{∼} Q_{−4d}^{(2)}/GL_{2}(ℤ). (36)

Proof. By Proposition 39 we know that g ↦ α_{g,h} defines an (injective) map Γ_{0}^{±}(d) → Aut(A). Now since Hom(E_{1}, E_{2}) = ℤh and hence Hom(E_{2}, E_{1}) = ℤh^{t}, we see that every α ∈ Aut(A) has the form α = \binom{a bh^{t}}{ch e}, for some a, b, c, e ∈ ℤ. But since 1 = deg(α) = (ae − bcd)^{2} by (69) (cf. proof of Proposition 39), we see that g := \binom{a b}{cd e} ∈ Γ_{0}^{±}(d) and so α = α_{g,h}. Thus, the map g ↦ α_{g,h} is bijective. Moreover, since c_{h^{t}×1} is a ring homomorphism, it follows from (34) that this bijection is an isomorphism of groups.

By combining Proposition 31 with Lemma 14 we see that the map D_{s,h} ↦ s ↦ f_{d}(s) defines a bijection f_{A} : P(A) →^{∼} Q_{−4d}^{(2)}(d). By (31) this is Γ_{0}^{±}(d)-equivariant, and so the first bijection of (36) follows. The second follows from Lemma 20.

Proof of Proposition 38. If C is a smooth curve of genus 2 on A, then it defines a principal polarization on A (cf. Weil[36] or [19]), and so C ≡ D_{s,h} with s ∈ P (d) by Proposition 31. Moreover, q_{C} ≈ q_{s} by (the proof of) Theorem 30, so f_{d}(s) ∉ Ker(π′_{d}) by Corollary 18 because q_{C} ≉ 1_{−16d} by Proposition 31.

If C′ is another curve on A which is isomorphic to C, then by Torelli’s theorem there exists an automorphism α ∈ Aut(A) with α(C) = C′, and so it follows from by Corollary 40 that the isomorphism class is uniquely determined by the GL_{2}(ℤ)- equivalence class of f_{d}(s).

Corollary 41 Let A = E_{1}×E_{2}, where Hom(E_{1}, E_{2}) = ℤh, and let d = deg(h). Then the number N_{A} of isomorphism classes of smooth genus 2 curves on A is

N_{A} = #(Q_{−4d}^{(2)}/GL_{2}(ℤ)) − 2^{ω(d)−1} = \{ \begin{matrix}\frac{1}{2}h(−4d) & if d ≡ 0, 1, 5 (mod 8) \\ \frac{1}{2}(h(−4d) − 2^{ω(d)−1}) & if d ≡ 2, 4, 6 (mod 8) \\ \frac{1}{2}(h(−4d) + h(−d)) & if d ≡ 3, 7 (mod 8) \end{matrix} ,

except when d = 1; in that case N_{A} = 0.

Proof. By Corollary 40 the total number of isomorphism classes of principal polarizations on A is #(Q_{−4d}^{(2)}/GL_{2}(ℤ)). By Proposition 38 we know that f_{d}(s) ∈ Q_{−4d}^{(2)} corresponds to a smooth curve if and only if f_{d}(s) ∉ Ker(π′_{d}), and so the first formula for N_{A} follows from (12). The second formula follows from this and (14) because #(Q_{D}^{(1)}/GL_{2}(ℤ)) = \frac{1}{2}(h(D) + g(D)).

Remark 42 The number N_{A} was also determined by Hayashida [12], §7-8, but his formula for N_{A} is much more complicated than the one above since he gives the result in terms of the class number h_{K} of the associated imaginary quadratic field K = ℚ(\sqrt{−d}). However, by using the well-known relation between h(−16d) and h_{K} (cf. Lang[24], p. 95), a somewhat tedious calculation shows that the two formulae give the same result.

8 The irreducibility of H(q)

The next task is to show that the generalized Humbert variety H(q) is a closed and irreducible subset of A_{2} when q is quadratic form of type d. This will be done by exhibiting H(q) as the image of the modular curve X_{0}(d) by a suitable morphism μ_{s}.

To define this morphism, recall that X_{0}(d) classifies cyclic isogenies of degree d of elliptic curves, i.e. X_{0}(d)(K) can be identified with the set of isomorphism classes ⟨f : E → E′⟩, where f is a cyclic isogeny of degree d; cf. [4], p. 283, or [22], p. 100.

Proposition 43 Let s ∈ P (d). Then the rule

⟨f : E → E′⟩ ↦ ⟨E × E′, D_{s,f}⟩ = ⟨E × E′, ϕ_{D_{s,f}} ⟩

defines a proper morphism μ_{s} : X_{0}(d) → A_{2} with image μ_{s}(X_{0}(N)) = H(q_{s}), where q_{s} is the quadratic form defined by (6).

Proof. Recall from Corollary 24 that D_{s,f} ∈ P(E × E′), so μ_{s}(f) ∈ A_{2}(K), i.e. μ_{s}(f) is a principally polarized abelian variety. Since this formation is compatible with isomorphisms, we thus see that this rule defines a map μ_{s} : X_{0}(d)(K) → A_{2}(K).

To show that μ_{s} comes from a morphism of varieties, we shall use the fact that both X_{0}(d) and A_{2} are the coarse moduli spaces of functors X_{0}(d) and A_{2} = A_{2,1,1} on \underline{Sch}_{/K}, respectively. It is thus enough to construct a morphism of functors μ̃_{s} = {μ̃_{s,S}}_{S} : X_{0}(d) → A_{2} which extends μ_{s} (i.e. μ̃_{s,S} = μ_{s} for S = Spec(K)).

To construct μ̃_{s}, we can use almost the same definition as for μ_{s}. Indeed, given a K-scheme S, then X_{0}(d)(S) consists of isomorphism classes ⟨f : E → E′⟩ in which f : E → E′ is an isogeny of elliptic curves /S which is cyclic in the sense of [22], p.

100. Moreover, A_{2}(S) consists of isomorphism classes ⟨A, λ⟩ of principally polarized abelian schemes A/S of dimension 2; cf. [32], p. 129. We now define

μ̃_{s,S}(⟨f : E → E′⟩) = ⟨E ×_{S} E′, λ_{s,f}⟩,

where λ_{s,f} : A := E ×_{S} E′ → Â is the principal polarization defined in Lemma 44 below.

It is clear that this definition is compatible with isomorphisms, and so we obtain a map μ̃_{s,S} : X_{0}(d) → A_{2}(S). Note that for S = Spec(K) we have λ_{s,f} = ϕ_{D_{s,f}} (cf. proof of Lemma 44 below) and so μ̃_{s,S} = μ_{s} agrees with the map μ_{s} as defined above. Moreover, since this construction is compatible with base change, the collection μ̃_{s} = {μ̃_{s,S}}_{S} defines a morphism of functors, which therefore induces a morphism μ_{s} : X_{0}(d) → A_{2} between the coarse moduli schemes.

By Proposition 28 we know that μ_{s}(X_{0}(d)) ⊂ H(q_{s}). On the other hand, the proof of Theorem 13 in §6 shows that if ⟨A, θ⟩ ∈ H(q_{s}), then (A, θ) ≃ (E × E′, D_{s,f}) for some cyclic isogeny f : E → E′ of degree d, and so ⟨A, θ⟩ = μ_{s}(⟨f⟩). Thus μ_{s}(X_{0}(d)) = H(q_{s}), as claimed.

It remains to show that μ_{s} is proper. Since X_{0}(d) and A_{2} are of finite type over K, it is enough to check that the functor μ̃_{s} satisfies the valuative criterion of properness. Thus, let S = Spec(R) be a discrete valuation ring with quotient field F ⊃ K and let y = ⟨A, λ⟩ ∈ A_{2}(S) be such that there exists x_{F} = ⟨E_{1}→^{h} E_{2}⟩ ∈ X_{0}(d)(F) with μ̃_{s,F} (x_{F}) = ⟨A_{F} , λ_{F} ⟩, where A_{F} = A ⊗ F and λ_{F} = λ ⊗ F. We want to show that x_{F} extends to x ∈ X_{0}(d)(S) and that μ̃_{s,S}(x) = y. For this we observe that since A_{F} ≃ E_{1} × E_{2}, and A_{F} has good reduction over R by hypothesis, it follows that the same is true for E_{i}, and so there exist elliptic curves Ẽ_{i}/R with Ẽ_{i} ⊗ F = E_{i}. By the Néron property we know that A ≃ Ẽ_{1} ×_{S} Ẽ_{2} and that h extends to h̃ : Ẽ_{1} → Ẽ_{2}. From [22], p. 162, it follows that h̃ is again cyclic, so x = ⟨h̃⟩ ∈ X_{0}(d)(S). We then have μ̃_{s}(x) = y because λ_{s,h̃} and λ agree on the generic fibre, and so μ̃_{s} is proper.

Lemma 44 Let f : E_{1} → E_{2} be an isogeny of degree d between two elliptic curves over a scheme S, and let s = (n_{1}, n_{2}, k) ∈ P (d). If λ_{i} : E_{i} →^{∼} Ê_{i} denotes the canonical polarization of E_{i}, and λ_{1} ⊗ λ_{2} the product polarization, then

λ_{s,f} = λ_{1} ⊗ λ_{2} ∘ ( \begin{matrix}[n_{1}]_{E_{1}} & kf^{t} \\ kf & [n_{2}]_{E_{2}} \end{matrix} )

is a principal polarization on E_{1} ×_{S} E_{2}.

Proof. First note that if S = Spec(K), then λ_{s,f} = ϕ_{D_{s,f}} by (19). Thus, since the formation of λ_{s,f} clearly commutes with base-change, it follows that λ_{s,f} is a principal polarization (in the sense of [32], p. 120) once we have shown that λ_{s,f} is an isomorphism. Now since f^{t}f = [d]_{E_{1}} and ff^{t} = [d]_{E_{2}} (cf. [22], p. 81), it follows from (5) that

( \begin{matrix}[n_{1}]_{E_{1}} & kf^{t} \\ kf & [n_{2}]_{E_{2}} \end{matrix} ) ( \begin{matrix}[n_{2}]_{E_{1}} & −kf^{t} \\ −kf & [n_{1}]_{E_{2}} \end{matrix} ) = ( \begin{matrix}1_{E_{1}} & 0 \\ 0 & 1_{E_{2}} \end{matrix} ) .

Thus, since the product polarization λ_{1} ⊗ λ_{2} (which is defined as in §11) is an isomorphism, we see that λ_{s,f} is an isomorphism.

Corollary 45 If q is a quadratic form of type d, then H(q) is a closed subvariety of A_{2} of dimension 1. Moreover, if char(K) ∤ d, then H(q) is an irreducible curve.

Proof. By Propositions 15 and 43 we have H(q) = μ_{s}(X_{0}(d)), for some s ∈ P (d), and so H(q) is a closed subset since μ_{s} is proper. Moreover, dim H(q) = dim X_{0}(d) = 1 because H(q) is infinite by Theorem 30. Finally, if char(K) ∤ d, then X_{0}(d) is irreducible (by Igusa), and hence so is its image H(q).

Proof of Theorem 3. By Corollary 45 and Theorem 13 we see that the H(q) for q ∈ Q̄_{d}^{∗} are the irreducible components of T (d). Since H(q_{1}) ≠ H(q_{2}) if q_{1} ≉ q_{2} (cf. Corollary 32), we see that the number of such components is precisely #Q̄_{d}^{∗}.

9 The action of Atkin-Lehner involutions

As is well-known, the curve X_{0}(d) comes equipped with a group of automorphisms called Atkin-Lehner involutions. In order to understand the birational structure of H(q), it is important to determine how these involutions act on the maps μ_{s} which were constructed in the previous section. Before stating the result, we first observe:

Proposition 46 Let s, s′ ∈ P (d). Then μ_{s} = μ_{s′} if and only if f_{d}(s) ≈ f_{d}(s′).

Proof. Suppose first that f_{d}(s) = f_{d}(s′)g with g ∈ GL_{2}(ℤ). Then by the proof of Lemma 20 we know that g ∈ Γ_{0}^{±}(d), and so f_{d}(s) = f_{d}(s′g) in the notation of (31). Thus, if x = ⟨f : E → E′⟩ ∈ X_{0}(d)(K), then α_{g,f} defines by Proposition 39 an isomorphism (E ×E′, D_{s′,f}) ≃ (E ×E′, D_{s,f}), and so μ_{s′}(x) = μ_{s}(x). This proves that μ_{s} = μ_{s′} provided that X_{0}(d) is reduced. In the general case (i.e. when char(K)|d), essentially the same argument (by replacing D_{s,f} by λ_{s,f} as in the proof of Proposition 43) shows that we actually have an equality μ̃_{s′} = μ̃_{s} of morphisms of functors, and so the induced morphisms μ_{s} and μ_{s′} on the coarse moduli spaces are equal.

Conversely, suppose μ_{s} = μ_{s′}. Then in particular μ_{s}(x) = μ_{s′}(x) for any point x = ⟨E →^{f} E′⟩ ∈ X_{0}(d)(K) which we can take to be a non-CM point, i.e. we have Hom(E, E′) = ℤf. Then the equality μ_{s}(x) = μ_{s′}(x) means that there is an α ∈ Aut(E × E′) such that α^{∗}D_{s,f} = D_{s′,f}. Now by Corollary 40 we know that α = α_{g,f} for some g ∈ Γ_{0}^{±}(d) and that f_{d}(s)g = f_{d}(s′). Thus, f_{d}(s) ≈ f_{d}(s′), as asserted.

We now come to the action on the Atkin-Lehner involutions on the maps μ_{s}. For this, recall that each Atkin-Lehner involution α on X_{0}(d) is uniquely defined by a divisor d_{1}||d of d, i.e. by a divisor d_{1}|d with the property that gcd(d_{1}, d/d_{1}) = 1. We can thus write α = α_{d_{1}}; this will be explained in more detail below.

Theorem 47 For each d_{1}||d, the Atkin-Lehner involution α_{d_{1}} permutes the μ_{s}’s. More precisely, if s ∈ P (d), then

μ_{s} ∘ α_{d_{1}} = μ_{s′}, where f_{d}(s′) ≈ f_{d}(s) ∘ a_{d_{1}}. (37)

Here a_{d_{1}} = [d_{1}, 0, d/d_{1}] if s ∈ P (d)^{odd} and a_{d_{1}} = [d_{1}, d_{1}, (d_{1}^{2}+d)/(4d_{1})], if s ∈ P (d)^{even}. Moreover, the orbits of the group of Atkin-Lehner automorphisms on {μ_{s}} are in oneto-one correspondence with the images H(q_{s}) = Im(μ_{s}); i.e. we have

(38) Im(μ_{s_{1}}) = Im(μ_{s_{2}}) ⇔ ∃d_{1}||d such that μ_{s_{1}} = μ_{s_{2}} ∘ α_{d_{1}}.

In order to prove this theorem, we need some auxiliary results concerning Atkin-Lehner involutions. We begin with their (functorial) definition, i.e. with their action on the functor X_{0}(d) which was discussed in the previous section.

Fix d_{1}||d and put d_{2} = d/d_{1}. Let h : E_{1} → E_{2} be a cyclic isogeny of degree d and for i = 1, 2, consider the quotient maps

h_{i1} = h_{i1}^{(h)} : E_{1} → E′_{i} := E_{1}/Ker(h)[d_{i}], where Ker(h)[d_{i}] = Ker(h) ∩ E_{1}[d_{i}].

Note that h_{i1} is a cyclic isogeny of degree deg(h_{i1}) = d_{i}, for i = 1, 2. By the universal property of quotients, there is a unique morphism h′_{i2} = (h′_{i2})^{(h)} : E′_{i} → E_{2} such that

h = h′_{i2} ∘ h_{i1}, for i = 1, 2. (39)

Note that h′_{i2} is cyclic of degree d/d_{i}, for i = 1, 2. Put h_{i2} = (h′_{i2})^{t} : E_{2} → E′_{i}; thus, h′_{i2} = h_{i2}^{t}. Finally, put

h′ = (h′)^{(h)} := h_{21} ∘ h_{11}^{t} = (h_{11} ∘ h_{12}^{t})^{t} : E′_{1} → E′_{2}.

Note that h′ is a cyclic isogeny of degree d = d_{1}d_{2} because h_{21} and h_{11}^{t} are cyclic of degree d_{2} and degree d_{1}, respectively, and because gcd(d_{1}, d_{2}) = 1. We observe that

h′ = h_{21} ∘ h_{11}^{t} = h_{22} ∘ h_{12}^{t}. (40)

(Indeed, the first equality is just the definition, whereas the second follows from the

fact that h_{21}h_{11}^{t}h_{11} = h_{21}[d_{1}] = [d_{1}]h_{21} = h_{22}h_{22}^{t}h_{12} =^{(39)}h_{22}h_{12}^{t}h_{11}, because is an isogeny.)

We now put

α_{d_{1}}(⟨E_{1} →^{h} E_{2}⟩) = ⟨E′_{1} →^{h′} E′_{2}⟩.

Note that the above construction works for elliptic curves over an arbitrary base scheme, and that it is compatible with base change. Thus, α_{d_{1}} defines a morphism of functors α_{d_{1}} : X_{0}(d) → X_{0}(d). In fact, α_{d_{1}} is an automorphism (and even an involution, i.e. α_{d_{1}} ∘ α_{d_{1}} = 1_{X_{0}(d)}) because with the above notation we have

α_{d_{1}}(⟨E′_{1} →^{h′} E′_{2}⟩) = ⟨E_{1} →^{h} E_{2}⟩.

(To see this, note that first that by (40) we have Ker(h′)[d_{i}] = Ker(h_{1i}^{t}), and so h_{i1}^{(h′)} = h_{1i}^{t} : E′_{i} → E_{i} and (h′_{i2})^{(h′)} = h_{2i}. Thus (h′)^{(h′)} = (h_{11}^{(h′)}(h′_{21})^{(h′)})^{t} = (h_{11}^{t}h_{21})^{t} = h_{21}^{t}h_{11} = h, and the assertion follows.)

Over ℂ, the Atkin-Lehner involutions on X_{0}(d)_{ℂ} = Γ_{0}(d)\ℌ can be defined by the Atkin-Lehner matrices of [1]. Although we don’t need this here, we do need these matrices in order to construct isomorphisms between E_{1} × E_{2} and E′_{1} × E′_{2}.

Notation. Put Γ_{0}^{±}(d_{2})_{d_{1}} = \{g ∈ Γ_{0}^{±}(d_{2}) : g ≡ \binom{0 ∗}{∗ ∗} (mod d_{1})\}. Thus, g ∈ Γ_{0}^{±}(d_{2})_{d_{1}} ⇔

(41) g = ( \begin{matrix}a_{11}d_{1} & a_{12} \\ a_{21}d_{2} & a_{22} \end{matrix} ) where a_{ij} ∈ ℤ and a_{11}a_{22}d_{1} − a_{12}a_{21}d_{2} = ±1.

If g ∈ Γ_{0}^{±}(d_{2})_{d_{1}}, then the associated Atkin-Lehner matrix is

g̃ := ( \begin{matrix}1 & 0 \\ 0 & d_{1} \end{matrix} ) g = ( \begin{matrix}a_{11}d_{1} & a_{12} \\ a_{21}d & a_{22}d_{1} \end{matrix} ) . (42)

Proposition 48 Let α_{d_{1}}(E_{1} →^{h} E_{2}) = (E′_{1} →^{h′} E′_{2}) and let g ∈ Γ_{0}^{±}(d_{2})_{d_{1}}. Put

α_{g} := ( \begin{matrix}a_{11}h_{11} & a_{12}h_{12} \\ a_{21}h_{21} & a_{22}h_{22} \end{matrix} ) where g = ( \begin{matrix}a_{11}d_{1} & a_{12} \\ a_{21}d_{2} & a_{22} \end{matrix} )

and where the h_{ij} = h_{ij}^{(h)} are as defined above. Then

(h_{12} × h_{22}) ∘ [g]_{E_{2}} = α_{g} ∘ (h^{t} × 1), (43)

and so α_{g} : E_{1} × E_{2} →^{∼} E′_{1} × E′_{2} is an isomorphism. Moreover,

((h′)^{t} × 1) ∘ [g̃]_{E′_{2}} = α_{g} ∘ (h^{t} × 1) ∘ (h_{22}^{t} × h_{22}^{t}). (44)

Proof. By (39) we have h_{i1}h^{t} = h_{i1}(h_{ii2}^{t}h_{11})^{t} = h_{i1}h_{i1}^{t}h_{i2} = d_{i}h_{i2}, and from this (43) follows immediately. Since det(g) = ±1, we see that deg([g]_{E_{2}}) = (±1)^{2} = 1; cf. Corollary 63. Thus, since deg(h_{12} × h_{22}) = d_{1}d_{2} = d = deg(h^{t} × 1), it follows from (43) that deg(α_{g}) = 1, i.e. that α_{g} is an isomorphism.

To prove (44), note first that (40) shows that (h_{12} × h_{22}) ∘ (h_{22}^{t} × h_{22}^{t}) = (h′)^{t} × [d_{1}] (because deg(h_{22}) = d/d_{2} = d_{1}), and so by (43) we obtain α_{g} ∘ (h^{t} × 1) ∘ (h_{22}^{t} × h_{22}^{t}) = (h_{12} ×h_{22})∘[g]_{E_{2}} ∘(h_{22}^{t} ×h_{22}^{t}) = (h_{12} ×h_{22})∘(h_{22}^{t} ×h_{22}^{t})∘[g]_{E′_{2}} = ((h′)^{t} ×[d_{1}])∘[g]_{E′_{2}} = ((h′)^{t} × 1) ∘ [g̃]_{E′_{2}}, which is (44).

In passing, we observe the following interesting fact concerning isomorphisms of product surfaces in the non-CM case; this will be used in the next section.

Proposition 49 Let (E_{1}, E_{2}) and (E′_{1}, E′_{2}) be two pairs of elliptic curves, and assume that Hom(E_{1}, E_{2}) = ℤh and Hom(E_{1}, E_{2}) = ℤh′. If d = deg(h), then

(45) E_{1} × E_{2} ≃ E′_{1} × E′_{2} ⇔ ∃ d_{1}||d such that ⟨E′_{1} →^{h′} E′_{2}⟩ = α_{d_{1}}(⟨E_{1} →^{h} E_{2}⟩).

Proof. The one direction follows from Proposition 48. Conversely, suppose that there exists an isomorphism f : E_{1} × E_{2} →^{∼} E′_{1} × E′_{2}. Then E′_{i} ∼ E_{1} ∼ E_{2}, and so Hom(E_{i}, E′_{j}) = ℤh_{ji}, for some (cyclic) h_{ji} ∈ Hom(E_{i}, E′_{j}), for all i, j = 1, 2. We can thus write f = (a_{ij}h_{ij}) with a_{ij} ∈ ℤ. Similarly, since Hom(E′_{j}, E_{i}) = ℤh_{ji}^{t}, we can write g := f^{−1} = (b_{ij}h_{ji}^{t}) with b_{ij} ∈ ℤ. Since 1_{E′_{1}×E′_{2}} = fg = \binom{[c_{11}] ∗}{∗ [c_{22}]}, we obtain the relations

c_{11} = a_{11}b_{11}d_{11} + a_{12}b_{21}d_{12} = 1 and c_{22} = a_{21}b_{12}d_{21} + a_{22}b_{22}d_{22} = 1,

where d_{ij} = deg(h_{ij}). From these we see that gcd(d_{11}, d_{12}) = 1 = gcd(d_{21}, d_{22}). Thus, h_{12}^{t}h_{11} ∈ Hom(E_{1}, E_{2}) is a composition of isogenies with cyclic kernels of relatively prime order, and hence also has cyclic kernel. This means that h_{12}^{t}h_{11} is a generator of Hom(E_{1}, E_{2}) and hence h_{12}^{t}h_{11} = ±h. By replacing h_{11} by −h_{11} if necessary, we thus have h = h_{12}^{t}h_{11}. Similarly, h_{22}^{t}h_{21} = h, (replacing h_{21} by −h_{21}, if necessary). Thus (39) holds with h′_{i2} = h_{i2}^{t}.

Next, using the fact that gf = 1_{E_{1}×E_{2}}, we obtain in a similar way the relations

a_{11}b_{11}d_{11} + a_{21}b_{12}d_{21} = 1 and a_{12}b_{21}d_{12} + a_{22}b_{22}d_{22} = 1,

and hence gcd(d_{11}, d_{21}) = 1 = gcd(d_{12}, d_{22}). Thus, since by (39) we have d_{12}d_{11} = d_{22}d_{21}, we see that d_{11}|d_{22} and d_{22}|d_{11}, and hence d_{11} = d_{22} and also d_{12} = d_{21}. Thus, if we put d_{i} = d_{i1}, then d = d_{1}d_{2} and (d_{1}, d_{2}) = 1, so d_{1}||d and Ker(h_{i1}) = Ker(h)[d_{i}], for i = 1, 2. Now h^{′(h)} = h_{21}∘h_{11}^{t} ∈ Hom(E′_{1}, E′_{2}) has cyclic kernel because h_{12} = (h′_{12})^{t} and h_{11}^{t} both have cyclic kernels of orders d_{12} = d_{2}, and d_{11} = d_{1}, respectively, and

(d_{1}, d_{2}) = 1. Thus, h^{′(h)} = ±h′, and so α_{d_{1}}(⟨E_{1} →^{h} E_{2}⟩) = ⟨E′_{1} →^{h′} E′_{2}⟩, as claimed.

Remark 50 In terms of the terminology of [20], p. 99, condition (39) means that (h, h_{11}, h′_{12}, h_{21}, h′_{22}) is an isogeny factor set representing the diamond configuration (h, Ker(h)[d_{1}], Ker(h)[d_{2}]). Thus, Proposition 49 gives a (partial) explanation of why such factor sets arise in the study of product surfaces.

We now want to compute the pullback of divisors with respect the isomorphism α_{g} defined in Proposition 48. For this, we shall use the embedding Ψ_{h} = Φ_{λ_{1}⊗λ_{2},h^{t}×1} which was defined in the proof of Proposition 39.

Proposition 51 In the situation of Proposition 48 we have

(h_{22} × h_{22})Ψ_{h}(α_{g}^{∗}D′)(h_{22}^{t} × h_{22}^{t}) = [g̃^{t}]_{E′_{2}}Ψ_{h′}(D′)[g̃]_{E′_{2}}, ∀D′ ∈ NS(E′_{1} × E′_{2}). (46)

In particular, if a′, b′, c′ ∈ ℤ, then

α_{g}^{∗}D(a′, b′, c′h′) = D(a, b, ch), (47)

where a, b, c ∈ ℤ are given by the matrix equation

( \begin{matrix}ad & cd \\ cd & b \end{matrix} ) = g^{t} ( \begin{matrix}a′d_{2} & c′d \\ c′d & b′d_{1} \end{matrix} ) g = \frac{1}{d_{1}} g̃^{t} ( \begin{matrix}a′d & c′d \\ c′d & b′ \end{matrix} ) g̃. (48)

Thus, if s′ ∈ P (d), then we have an isomorphism of principally polarized abelian surfaces

α_{g} : (E_{1} × E_{2}, D_{s′g̃,h}) →^{∼} (E′_{1} × E′_{2}, D_{s′,h′}), (49)

where s′g̃ ∈ P (d) is defined by the rule M(f_{d}(s′g̃)) = \frac{1}{d_{1}} g̃^{t}M(f_{d}(s′))g̃.

Proof. Since r_{λ′_{2}⊗λ′_{2},λ_{2}⊗λ_{2}}(h_{22}^{t} × h_{22}^{t}) = h_{22} × h_{22} (cf. (63)), it follows from the definitions and formula (55) of the appendix that the left hand side of (46) equals (h_{22}^{t} × h_{22}^{t})^{♭}(h^{t} × 1)^{♭}Φ_{λ_{1}⊗λ_{2}}(α_{g}^{∗}D′) = (h_{22}^{t} × h_{22}^{t})^{♭}(h^{t} × 1)^{♭}(α_{g})^{♭}Φ_{λ′_{1}⊗λ′_{2}}(D′) = (α_{g}(h^{t} × 1)(h_{22}^{t} × h_{22}^{t}))^{♭}Φ_{λ′_{1}⊗λ′_{2}}(D′) = (((h′)^{t} × 1)[g̃]_{E′_{2}})^{♭}Φ_{λ′_{1}⊗λ′_{2}}(D′) = ([g̃]_{E′_{2}})^{♭}Ψ_{h′}(D′), where we have used (57) and (44) in the last three equalities. Since r_{λ_{2}⊗λ′_{2}}([g̃]_{E′_{2}}) = [g̃^{t}]_{E′_{2}} by (63), we obtain ([g̃]_{E′_{2}})^{♭}Ψ_{h′}(D′) = [g̃^{t}]_{E′_{2}}Ψ_{h′}(D′)[g̃]_{E′_{2}}, which proves (46).

To prove (47), first note that the second equality of (48) follows immediately from the fact that g̃ = diag(1, d_{1})g. Furthermore, by multiplying out the right hand side of (48), we see that if g has the form (41), then a = a′d_{1}a_{11}^{2} + 2dc′a_{11}a_{21} + b′d_{2}a_{21}^{2}, b = a′d_{2}a_{12}^{2} + 2c′da_{12}a_{22} + b′d_{1}a_{22}^{2}, c = a′a_{11}a_{12} + c′(d_{2}a_{12}a_{21} + d_{1}a_{11}a_{22}) + b′a_{21}a_{22}, and so a, b, c ∈ ℤ. Now by (35) we have Ψ_{h}(D(a, b, ch)) = [g_{1}]_{E_{2}}, where g_{1} = \binom{ad cd}{cd b}, and similarly Ψ_{h′}(D(a′, b′, c′h′)) = [g′_{1}]_{E′_{2}} with g′_{1} = \binom{a′d c′d}{c′d b′}. Thus, if D′ = D(a′, b′, c′h′), then by (48) the right hand side of (46) equals [d_{1}g_{1}]_{E′_{2}} = (h_{22}×h_{22})(h_{22}^{t}×h_{22}^{t})[g_{1}]_{E′_{2}} = (h_{22} × h_{22})[g_{1}]_{E_{2}}(h_{22}^{t} × h_{22}^{t}) = (h_{22} × h_{22})Ψ_{h}(D)(h_{22}^{t} × h_{22}^{t}), where D = D(a, b, ch). Comparing this to the left hand side of (46) yields Ψ_{h}(α_{g}^{∗}(D′)) = Ψ_{h}(D) (because h_{22} × h_{22} and h_{22}^{t} × h_{22}^{t} are isogenies), and so (47) follows because Ψ_{h} is injective; cf. Corollary 58 of the appendix.

Finally, to prove (49), recall from Proposition 48 that α_{g} : E_{1} × E_{2} →^{∼} E′_{1} × E′_{2} is an isomorphism. Now by (47) we have α_{g}^{∗}D_{s′,h′} = D_{s′g̃,h}, and so (49) follows.

Proof of Theorem 47. Fix s = (n_{1}, n_{2}, k) ∈ P (d) and let g ∈ Γ_{0}^{±}(d_{2})_{d_{1}}. If g̃ is defined by (42), then a short computation shows that \frac{1}{d_{1}} g̃^{t}M(f_{d}(s))g̃ = M(f_{d}(s′)), for some s′ ∈ P (d) and that M(f_{d}(s′)) = g^{t}M(q)g, where q = [n_{1}d_{2}, 2k, n_{2}d_{1}]. Since g ∈ GL_{2}(ℤ), this implies that f_{d}(s′) ≈ q, and so Lemma 52 below shows that f_{d}(s′) ≈ q ∼ f_{d}(s) ∘ a_{d_{1}}. Thus, (37) follows once we have shown that μ_{s} ∘ α_{d_{1}} = μ_{s′}.

For this, let x = ⟨E_{1} →^{h} E_{2}⟩ ∈ X_{0}(d) and put x′ = α_{d_{1}}(x) = ⟨E′_{1} →^{h′} E′_{2}⟩. Then

μ_{s}(α_{d_{1}}(x)) = μ_{s}(x′) = ⟨E′_{1} × E′_{2}, D_{s,h′}⟩. Now by (49) we have α_{g} : (E_{1} × E_{2}, D_{s′,h}) →^{∼}

(E′_{1}×E′_{2}, D_{s,h}), and so μ_{s}(α_{d_{1}}(x)) = μ_{s′}(x). This proves that μ_{s}∘α_{d_{1}} = μ_{s′} when X_{0}(d) is reduced. In the general case a similar argument (generalized to elliptic curves over K-schemes) shows that we have an equality μ̃_{s} ∘ α_{d_{1}} = μ̃_{s′} of morphisms of functors, and so (37) holds in general.

It remains to prove (38). For this, let s_{1}, s_{2} ∈ P (d) be such that Im(μ_{s_{1}}) = Im(μ_{s_{2}}). Then Proposition 43 shows that H(q_{s_{1}}) = H(q_{s_{2}}) and so by Corollary 32 we have q_{s_{1}} ≈ q_{s_{2}}. We now distinguish two cases.

If s_{1} ∈ P (d)^{odd}, then q_{s_{1}} is primitive by Lemma 16 and hence so is q_{s_{2}}. Thus, also s_{2} ∈ P (d)^{odd}. By Corollary 18 (and Remark 12) we thus have that f_{d}(s_{1}) ∼ f_{d}(s_{1})∘a, where a ∈ Ker(π′_{d}). By Corollary 19 we have a ∼ a_{d_{1}}, for some d_{1}||d, and so (37) shows that μ_{s_{1}} ∘ α_{d_{1}} = μ_{s_{2}}, as desired.

Now suppose that s_{1} ∈ P (d)^{even}; then also s_{2} ∈ P (d)^{even}. Here f_{d}(s_{i}) = 2f′_{d}(s_{i}), where f′_{d}(s_{i}) ∈ Q_{−d}^{(1)}, and by Corollary 18 we thus have f′_{d}(s_{1}) ∼ f′_{d}(s_{2}) ∘ a with a ∈ Q̄_{−d}[2]. By genus theory, it follows that a ∼ a_{d_{1}} = [d_{1}, d_{1}, (d_{1} + d_{2})/4], for some d_{1}||d, and so (37) shows again that μ_{s_{1}} ∘ α_{d_{1}} = μ_{s_{2}}. This proves one direction of (38), and so (38) follows since the other direction is trivial.

Lemma 52 Let s = [n_{1}, n_{2}, k] ∈ P (d), and put q = [d_{2}n_{1}, 2k, d_{1}n_{2}], where d = d_{1}d_{2} with gcd(d_{1}, d_{2}) = 1. Then f_{d}(s) ∘ a_{d_{1}} ∼ q, where a_{d_{1}} = [d_{1}, 0, d_{2}] if s ∈ P (d)^{odd}, and a_{d_{1}} = [d_{1}, d_{1}, (d_{1} + d_{2})/4] if s ∈ P (d)^{even}.

Proof. If s ∈ P (d)^{odd}, then f_{d}(s) = [dn_{1}, 2kd, n_{2}] is primitive of discriminant −4d, and the composition algorithm of Shanks (cf. [2], p. 64) shows that a_{d_{1}} ∘f_{d}(s) ∼ q. Indeed, if 2 ∤ n_{1}, apply [2], Th. 4.12, to f_{1} = [d_{1}, 2d_{1}, d_{1} + d_{2}] ∼ [d_{1}, 0, d_{2}] and f_{2} = f_{d}(s). Then (with the notation there) m = n = d_{1}, and so we can take x = 1, y = 0 and z = dn_{1} − d_{1}, and so f_{1} ∘ f_{2} ∼ [d_{1}n_{1}d/d_{1}^{2}, 2d_{1} + 2z, ∗] = q. On the other hand, if 2|n_{1}, then f_{s}(s) ∼ [n_{2}d, −2kd, n_{1}], where 2 ∤ n_{2}, and then by the same argument [d_{2}, 0, d_{1}] ∘ [n_{2}d, −2kd, n_{1}] ∼ [n_{2}d_{1}, −2kd, n_{1}d_{2}] ∼ q. Thus a_{d_{1}} ∘ f_{d}(s) ∼ q because a_{d_{1}} ∼ [d_{2}, 0, d_{1}].

Now suppose s ∈ P (d)^{even}. Then f_{d}(s) = 2f′_{d}(s) where f′_{d}(s) = [n′_{1}d, kd, n′_{2}] is primitive of discriminant −d. Applying [2], Th. 4.12, to f_{1} = [d_{1}, d_{1}, (d_{1} + d_{2})/4] and f_{2} = f′_{d}(s) shows that f_{1} ∘ f_{2} ∼ [n′_{1}d_{2}, kd, n′_{2}d_{1}] because here again m = n = d_{1}, and so we can take x = 1, y = 0, and z = (kd − d_{1})/2. Thus f_{d}(s) ∘ a_{d_{1}} := 2(f′_{d}(s) ∘ a_{d_{1}}) ∼ 2[n′_{1}d_{2}, kd, n′_{2}d_{1}] = q.

10 The birational structure of H(q)

In order to determine the birational structure of H(q), we shall first calculate the automorphism group Aut(μ_{s}) of the morphism μ_{s} : X_{0}(d) → H(q_{s}). As we shall see, the Fricke involution w_{d} = α_{d} on X_{0}(d) always lies in Aut(μ_{s}). However, if q_{s} is an ambiguous form, then there is another Atkin-Lehner involution α_{s} in Aut(μ_{s}), as the following result shows.

Proposition 53 (a) If s ∈ P (d)^{odd}, then q_{s} ∈ Q̄_{−16d}^{2}[2] (i.e., q_{s} is ambiguous) if and only if f_{d}(s)^{2} ∈ Ker(π′_{d}). If this is the case, then there is a unique d_{1}||d with d_{1} ≤ d_{2} := d/d_{1} such that [d_{1}, 0, d_{2}] ∼ π_{−4d,2}(q_{s}) ∼ f_{d}^{2}(s).

(b) If s ∈ P (d)^{even}, then q′_{s} := \frac{1}{4}q_{s} ∈ Q̄_{−d}^{2}[2] (i.e., q_{s} is ambiguous) if and only if f′_{d}(s)^{2} ∈ Q̄_{−d}^{2}[2]. If this is the case, then there is a unique d_{1}||d with d_{1} ≤ d_{2} := d/d_{1} such that [d_{1}, d_{1}, (d_{1} + d_{2})/4] ∼ q′_{s} ∼ f′_{d}(s)^{2}.

(c) Let s ∈ P (d) and put α_{s} = α_{d_{1}}, where d_{1} is as above, if q_{s} is ambiguous, and d_{1} = 1 otherwise. Then G(q_{s}) := ⟨w_{d}, α_{s}⟩ ≤ Aut(μ_{s}), and hence μ_{s} factors over the quotient map π_{q_{s}} : X_{0}(d) → X_{0}(d)_{q_{s}}^{+} := X_{0}(d)/G(q_{s}).

(d) We have G(q_{s}) = ⟨w_{d}⟩ if and only if either q_{s} is not ambiguous or if \frac{1}{4}q_{s} ∼ 1_{−d} or if q_{s} ∼ q_{d}, where q_{d} is as in Theorem 36.

Proof. (a) By (8) we have f_{d}(s)^{2} ∼ π_{−4d,2}(q_{s}) and by (10) we have π′_{d}(π_{−4d,2}(q_{s})) ∼ q_{s}^{2}. Thus, f_{d}(s)^{2} ∈ Ker(π′_{d}) ⇔ q_{s}^{2} ∼ 1 ⇔ q_{s} ∈ Q̄_{−16d}^{2}[2]. This proves the first assertion, and the second follows from (13).

(b) By (9) we have f′_{d}(s)^{2} ∼ q′_{s}, so the first assertion is trivial. The second follows immediately from the fact that the forms [d_{1}, d_{1}, (d_{1} + d_{2})/2] represent all ambiguous classes in Q̄_{−d}; cf. proof of Theorem 47.

(c) It is enough to show that μ_{s} ∘ α′_{d} = μ_{s} for d′ = d and d′ = d_{1}, and this follows from Theorem 47 once we have shown that f_{d}(s) ≈ f_{d}(s) ∘ a_{d′}. This is clear if d′ = d (or d′ = d_{1} = 1) because then a_{d′} ∼ 1. (Indeed, if s ∈ P (d)^{odd}, then a_{d} = [d, 0, 1] ∼ 1_{−4d} = a_{1}, and if s ∈ P (d)^{even}, then a_{d} = [d, d, \frac{d+1}{4}] ∼ 1_{−d} = a_{1}.) On the other hand, if d′ = d_{1} and we are in the situation of (a), then f_{d}(s)^{2} ∼ a_{d_{1}} ∼ a_{d_{1}}^{−1} and then f_{d}(s) ≈ f_{d}(s)^{−1} ∼ f_{d}(s)∘a_{d_{1}}. Similarly, if we are in the situation of (b), then f′_{d}(s)^{2} ∼ a_{d_{1}} ∼ a_{d_{1}}^{−1} and then f′_{d}(s) ≈ f′_{d}(s)^{−1} ∼ f′_{d}(s)∘a_{d_{1}}, so again f_{d}(s) ≈ f_{d}(s)∘a_{d_{1}}.

(d) Since ⟨w_{d}⟩ = {α_{1}, α_{d}}, we see that G(s) = ⟨w_{d}⟩ ⇔ d_{1} = 1 (because d_{1} ≤ d/d_{1}). Thus, if q_{s} is not ambiguous, then the assertion is clear, so assume q_{s} is ambiguous. If q_{s} is not primitive, then by part (b) we see that d_{1} = 1 ⇔ a_{d_{1}} ∼ 1_{−d} ⇔ q′_{s} ∼ 1_{−d}, and if q_{s} is primitive, then by part (a) we have d_{1} = 1 ⇔ a_{d_{1}} ∼ 1_{−4d} ⇔ q_{s} ∈ Ker(π_{−4d,2}) ⇔ q_{s} ∼ q_{d}, the latter by (11).

We now show that Aut(μ_{s}) = G(q_{s}) by examining the fibres of μ_{s} at non-CM points.

Proposition 54 Let s ∈ P (d) and let x ∈ X_{0}(d)(K) be a non-CM point. Then

μ_{s}^{−1}(μ_{s}(x)) = G(q_{s})x = {x, w_{d}(x), α_{s}(x), w_{d}α_{s}(x)}, (50)

and so Aut(μ_{s}) = G(q_{s}), provided that char(K) ∤ d.

Proof. Write x = ⟨E_{1} →^{h} E_{2}⟩ and let y = ⟨E′_{1} →^{h′} E′_{2}⟩ ∈ X_{0}(d)(K). Then we have:

(51) μ_{s}(x) = μ_{s}(y) ⇔ y = α_{d_{1}}(x), for some d_{1}||d with f_{d}(s) ≈ f_{d}(s) ∘ a_{d_{1}}.

Indeed, if y = α_{d_{1}}(x) and f_{d}(s) ≈ f_{d}(s)∘a_{d_{1}}, then μ_{s}(y) = μ_{s}(α_{d_{1}}(x)) = μ_{s}(x) by (37). Conversely, if μ_{s}(x) = μ_{s}(y), then ∃α : E_{1} × E_{2} →^{∼} E′_{2} × E′_{2} such that α^{∗}D_{s,h′} = D_{s,h}. Then by Proposition 49 we know that ∃d_{1}||d = deg(h) such that y = α_{d_{1}}(x), and so by Theorem 47 we have μ_{s}(y) = μ_{s′}(x), where s′ ∈ P (d) is such that f_{d}(s′) ∼ f_{d}(s) ∘ a_{d_{1}}. Thus, μ_{s}(x) = μ_{s′}(x), which means that (E_{1} × E_{2}, D_{s,h}) ≃ (E_{1} × E_{2}, D_{s′,h}). From Corollary 40 it follows that f_{d}(s) ≈ f_{d}(s′) ∼ f_{d}(s) ∘ a_{d_{1}}, and so (51) holds.

We now analyze the condition that f_{d}(s) ≈ f_{d}(s) ∘ a_{d_{1}}. For this, assume first that f_{d}(s) is primitive, i.e. that s ∈ P (d)^{odd}. Then we have

f_{d}(s) ≈ f_{d}(s) ∘ a_{d_{1}} ⇔ a_{d_{1}} ∼ 1 or a_{d_{1}} ∼ f_{d}(s)^{2} ∼ π_{−4d,2}(q_{s}). (52)

Indeed, by Remark 12 we see that this condition holds if and only if either f_{d}(s) ∼ f_{d}(s) ∘ a_{d_{1}} or f_{d}(s)^{−1} ∼ f_{d}(s) ∘ a_{d_{1}}. In the first case this means that a_{d_{1}} is principal, and the second case we have a_{d_{1}} ∼ a_{d_{1}}^{−1} ∼ f_{d}(s)^{2} ∼ π_{−4d,2}(q_{s}), the latter by (8).

This proves (52). Note that the second condition implies by Proposition 53 that q_{s} ∈ Q̄_{−16d}^{2}[2] because a_{d_{1}} ∈ Ker(π′_{d}) by (13).

Thus, if q_{s} ∉ Q̄_{−16d}^{2}[2], or if q_{s} ∼ q_{d}, then the right hand side of (52) reduces to the condition a_{d_{1}} ∼ 1 (because Ker(π_{−4d,2}) = ⟨q_{d}⟩ by (11)), and so by reduction theory we see that this is the case if and only d_{1} = 1 or d_{1} = d. Thus, in this case it follows from (51) and (52) that μ_{s}(x) = μ_{s}(y) ⇔ y ∈ {x, w_{d}(x)} = G(q_{s})x.

Next, suppose that q_{s} ∈ Q̄_{−16d}^{2}[2] but q_{s} ≁ q_{d}. Then by Proposition 53(a) we have a = a_{d_{1}}, for some d_{1}||d with d_{1} ≤ d_{2} = d/d_{1}. Since a_{d_{2}} ∼ a_{d_{1}} and α_{d_{2}} = w_{d}α_{d_{1}}, it thus follows from (51)and (52) that (50) holds.

Now suppose that f_{d}(s) is not primitive, i.e. s ∈ P (d)^{even}. Then f_{d}(s) = 2f′_{d}(s) with f′_{d}(s) ∈ Q̄_{−d} and q_{s} = 4q′ with q′ ∼ f′_{d}(s)^{2}; cf. Lemma 16(b). In this case a similar argument to the one above shows that

f_{d}(s) ≈ f_{d}(s) ∘ a_{d_{1}} ⇔ a_{d_{1}} ∼ 1 or a_{d_{1}} ∼ f′_{d}(s)^{2} ∼ q′. (53)

Thus, if q′ ∉ Q̄_{−d}[2] or if q′ ∼ 1_{−d}, then the right hand side of (53) reduces to the condition a_{d_{1}} ∼ 1 and so as before we see that μ_{s}^{−1}(μ_{s}(x)) = {x, w_{d}(x)} = G(q_{s})x in this case. On the other hand, if q′ ∈ Q̄_{−d}[2] \ {1_{−d}}, then one concludes by a similar argument as above that (50) holds.

To verify the last assertion, assume char(K) ∤ d. Then μ_{s} : X_{0}(d) → H(q_{s}) is finite because it is a proper, surjective morphism between irreducible curves; cf. EGA (II, 7.4.4) and EGA (III, 4.4.2). Thus, from (50) we see that the separable degree deg_{s}(μ) of μ_{s} equals |G(q_{s})| because there are infinitely many non-CM points on X_{0}(d). We thus have |G(q_{s})| ≤ |Aut(μ_{s})| ≤ deg_{s}(μ_{s}) = |G(q_{s})|, and so we have equality throughout. In particular, G(q_{s}) = Aut(μ_{s}), as claimed.

Theorem 55 Let q ∈ Q̄_{d}^{∗}, and suppose that char(K) ∤ d. Then X_{0}(q)_{q}^{+} is the normalization of H(q). In particular, X_{0}(q)^{+} is the normalization of H(q) if and only if either q is not ambiguous or if \frac{1}{4}q ∼ 1_{−d} or if q_{s} ∼ q_{d}, where q_{d} is as in Theorem 36.

Proof. Since q ∼ q_{s}, for some s ∈ P (d) by Proposition 15, we see that the last assertion follows from the first assertion together with Proposition 53(d).

To prove the first assertion, recall that by Proposition 53(c) we have that μ_{s} = μ̄_{s} ∘ π_{q}, for some morphism μ̄_{s} : X_{0}(q)_{q}^{+} → H(q). Note that X_{0}(q)_{q}^{+} is affine and that hence μ̄_{s} is again finite (use EGA (II, 5.4.3)). Since X_{0}(q)_{q}^{+} is normal, we see that μ̄_{s} = ν ∘ μ̃_{s} factors over the normalization ν : H̃(q) → H(q). By the proof of Proposition 54 we know that deg_{s}(μ_{s}) = deg(π_{q}), and so we see that deg_{s}(μ̃_{s}) = 1, i.e. that μ_{s} is purely inseparable. Thus, the assertion follows once we have shown that μ̃_{s} or, equivalently, that μ_{s} is separable. Since this is automatic if char(K) = 0, it remains to verify this assertion if p = char(K) ≠ 0.

For this, we shall use a specialization argument. Let R = ℤ_{(p)} ⊂ ℚ denote the discrete valuation ring with residue field 𝔽_{p}, and let X_{0}(d)/R and A_{2}/R be the coarse moduli schemes of the functors X_{0}(d) and A_{2} on \underline{Sch}_{/R}, respectively. Since p ∤ d, we know that X_{0}(R)/R is smooth and that hence its fibres are the coarse moduli schemes of the corresponding fibre functors; cf. [22], p. 510. In addition, one has that the fibres of A_{2} are the coarse moduli schemes its fibre functors; cf. Igusa[17], for M_{2} in place of A_{2} (which suffices for our purposes). Now the method of proof of Proposition 43 extends to construct an R-morphism μ_{s} : X_{0}(d) → A_{2}, and the same proof shows that μ_{s} is again proper. Thus, by Fulton [9], Proposition 20.3(a), we have deg(μ_{s}^{∘}) = deg(μ_{s}^{s}), where μ_{s}^{∘} and μ_{s}^{s} are the restrictions of μ_{s} to the generic and special fibres of X_{0}(d), respectively. Since these can be identified with the previously constructed morphisms μ_{s} (over K = ℚ and over K = 𝔽_{p}, respectively), we have by (the proof of) Proposition 54 that deg_{s}(μ_{s}^{∘}) = |G(q_{s})| = deg_{s}(μ_{s}^{s}). But since deg_{s}(μ_{s}^{∘}) = deg(μ_{s}^{∘}), it follows that also deg_{s}(μ_{s}^{s}) = deg(μ_{s}^{s}), and so μ_{s}^{s} is separable.

Proof of Theorems 1 and 4. From the definition of G(q_{s}), it clear that Theorem 4 and the last part of Theorem 1 are special cases of Theorem 55. Moreover, the fact that T (d) is a closed subset (and that it is a finite union of curves) follows from Theorem 13 and Proposition 43.

11 Appendix: The Néron-Severi group

The purpose of this appendix is to present some basic facts about the Néron-Severi groups of abelian varieties which were used throughout the paper.

Let A be an abelian variety over an algebraically closed field K, and let NS(A) = Pic(A)/Pic^{0}(A) denote the Néron-Severi group of A. If A has a principal polarization λ = ϕ_{θ} : A →^{∼} Â (cf. Milne[30], p. 126), then NS(A) can be interpreted as a subgroup of End(A). More precisely, if r_{λ} denotes the Rosati involution on End(A) (which is defined by the rule r_{λ}(α) = λ^{−1}α̂λ), then by Mumford[33], p. 190, 209, the map D ↦ λ^{−1}ϕ_{D} defines an isomorphism

(54) Φ_{λ} : NS(A) →^{∼} End_{λ}(A) := {α ∈ End(A) : r_{λ}(α) = α}.

The isomorphism Φ_{λ} satisfies the following functorial property.

Proposition 56 If (A_{i}, λ_{i}), i = 1, 2, are two principally polarized abelian varieties, and h ∈ Hom(A_{1}, A_{2}),

Φ_{λ_{1}}(h^{∗}D) = r_{λ_{1},λ_{2}}(h)Φ_{λ_{2}}(D)h, ∀D ∈ NS(A_{2}), (55)

where r_{λ_{1},λ_{2}}(h) = λ_{1}^{−1}ĥλ_{2} ∈ Hom(A_{2}, A_{1}). In other words, Φ_{λ_{1}} ∘ h^{∗} = h^{♭} ∘ Φ_{λ_{2}}, where h^{♭} : End(A_{2}) → End(A_{1}) is defined by h^{♭}(α) = r_{λ_{1},λ_{2}}(h)αh. Moreover,

r_{λ_{1}} ∘ h^{♭} = h^{♭} ∘ r_{λ_{2}}, (56)

and hence Φ_{λ_{1}} ∘ h^{∗} defines a homomorphism Φ_{λ_{1},h} : NS(A_{2}) → End_{λ_{1}}(A_{1}).

Proof. The first formula follows immediately from the definitions and the fact that ϕ_{h^{∗}D} = ĥ∘ϕ_{D} ∘h, for D ∈ Pic(A). Similarly, (56) follows from the definitions together with the fact that r_{λ_{1},λ_{2}}(h) ∘ λ_{1} = λ_{2} ∘ h.

Remark 57 For later reference, let us observe here that the assignment h ↦ h^{♭} = h_{λ_{1},λ_{2}}^{♭} is functorial: if (A_{i}, λ_{i}), i = 1, 2, 3, are three principally polarized abelian varieties, and h_{i} ∈ Hom(A_{i}, A_{i+1}) for i = 1, 2, then

(h_{2} ∘ h_{1})_{λ_{1},λ_{2}}^{♭} = (h_{1})_{λ_{1},λ_{2}}^{♭} ∘ (h_{2})_{λ_{2},λ_{3}}^{♭}. (57)

This follows easily from the definitions and the fact that r_{λ_{1},λ_{3}}(h_{1} ∘ h_{2}) = r_{λ_{1},λ_{2}}(h_{1}) ∘ r_{λ_{2},λ_{3}}(h_{2}).

In the case that h is an isogeny, we can define h^{♭} in another way.

Corollary 58 If h : A_{1} → A_{2} is an isogeny, then the rule c_{h}(α) = h^{−1}αh defines a ring isomorphism c_{h} : End^{0}(A_{2}) →^{∼} End^{0}(A_{1}) which is related to h^{♭} by the formula

h^{♭}(α) = βc_{h}(α), where β = h^{♭}(1) = r_{λ_{1},λ_{2}}(h)h, (58)

and we have r_{λ_{1}}(c_{h}(α)) = βc_{h}(r_{λ_{2}}(α))β^{−1}, ∀α ∈ End^{0}(A_{2}). (59)

Thus Φ_{λ_{1},h} := Φ_{λ_{1}} ∘ h^{∗} = h^{♭} ∘ Φ_{λ_{2}} = β(c_{h} ∘ Φ_{λ_{2}}) : NS(A_{2}) → End_{λ_{1}}(A_{1}) is an injective group homomorphism which satisfies

Φ_{λ_{1},h}(α^{∗}D) = r_{λ_{1}}(c_{h}(α))Φ_{λ_{1},h}(D)c_{h}(α), ∀D ∈ NS(A_{2}), α ∈ End(A_{2}). (60)

Proof. It is clear that c_{h} is a ring isomorphism and that (58) holds. Thus, since r_{λ_{1}}(β) = r_{λ_{1}}(h^{♭}(1)) = h^{♭}(1) = β by (56), we see that r_{λ_{1}}(c_{h}(α))β = r_{λ_{1}}(c_{h}(α))r_{λ_{1}}(β) =

r_{λ_{1}}(βc_{h}(α)) =^{(58)} r_{λ_{1}}(h^{♭}(α)) =^{(56)} h^{♭}(r_{λ_{2}}(α)) =^{(58)} βc_{h}(r_{λ_{2}}(α), and so (59) follows.

Write Φ = Φ_{λ_{1},h}. Then Φ = h^{♭} ∘ Φ_{λ_{2}} by (55) and hence Φ = β(c_{h} ∘ Φ_{λ_{2}}) by (58). From the latter expression it is clear that Φ is an injective group homomor-

phism. Moreover, since c_{h} is multiplicative, we have Φ(α^{∗}D) = βc_{h}(Φ_{λ_{2}}(α^{∗}D)) =^{(55)}

βc_{h}(r_{λ_{2}}(α)Φ_{λ_{2}}(D)α) =^{(59)} r_{λ_{1}}(c_{h}(α))βc_{h}(Φ_{λ_{2}}(D))c_{h}(α), which proves (60).

Let (A_{i}, λ_{i}) be two principally polarized abelian varieties, and A = A_{1} × A_{2} be the product variety with projections p_{i} : A → A_{i} and inclusions e_{i} : A_{i} → A. Then

p := p̂_{1} + p̂_{2} : Â_{1} × Â_{2} →^{∼} Â is an isomorphism, and λ_{1} × λ_{2} := p ∘ λ_{1} × λ_{2} : A →^{∼} Â is

a principal polarization of A, called the product polarization. (Note that if λ_{i} = ϕ_{θ_{i}}, then λ_{1} ⊗ λ_{2} = ϕ_{θ}, where θ = p_{1}^{∗}θ_{1} + p_{2}^{∗}θ_{2}.)

If α ∈ End(A_{1} ×A_{2}), then we can identify α with the 2×2 matrix (α_{ij}) by putting α_{ij} = p_{i}αe_{j} ∈ Hom(A_{j}, A_{i}). Thus

End(A_{1} × A_{2}) = \{( \begin{matrix}α_{11} & α_{12} \\ α_{21} & α_{22} \end{matrix} ) : α_{ij} ∈ Hom(A_{j}, A_{i})\} .

Proposition 59 In the above situation we have

(61)End_{λ_{1}⊗λ_{2}}(A_{1} × A_{2}) = \{( \begin{matrix}α_{11} & α′_{21} \\ α_{21} & α_{22} \end{matrix} ) : α_{ii} ∈ End_{λ_{i}}(A_{i}), α_{21} ∈ Hom(A_{1}, A_{2})\} ,

where α′_{21} = r_{λ_{1},λ_{2}}(α_{21}). Thus, the rule (α_{1}, α_{2}, β) ↦ \binom{α_{1} β′}{β α_{2}} defines an isomorphism

μ = μ_{λ_{1},λ_{2}} : End_{λ_{1}}(A_{1}) ⊕ End_{λ_{2}}(A_{2}) ⊕ Hom(A_{1}, A_{2}) →^{∼} End_{λ_{1}⊗λ_{2}}(A_{1} × A_{2})

which induces an isomorphism

D = D_{λ_{1},λ_{2}} : NS(A_{1}) ⊕ NS(A_{2}) ⊕ Hom(A_{1}, A_{2}) →^{∼} NS(A_{1} × A_{2}).

Moreover, we have

D(D_{1}, D_{2}, 0) = p_{1}^{∗}D_{1} + p_{2}^{∗}D_{2}, ∀D_{i} ∈ NS(A_{i}). (62)

Proof. Since ê_{i}(λ_{1} ⊗ λ_{2}) = λ_{i}p_{i} and (λ_{1} ⊗ λ_{2})e_{j} = p̂_{j}λ_{j}, we see that p_{i}r_{λ_{1}⊗λ_{2}}(α)e_{j} = r_{λ_{i},λ_{j}}(p_{j}αe_{i}) = r_{λ_{i},λ_{j}}(α_{ji}). Thus

r_{λ_{1}⊗λ_{2}} ( \begin{matrix}α_{11} & α_{12} \\ α_{21} & α_{22} \end{matrix} ) = ( \begin{matrix}α′_{11} & α′_{21} \\ α′_{12} & α′_{22} \end{matrix} ) , (63)

where α′_{ji} = r_{λ_{i},λ_{j}}(α_{ji}) = λ_{i}^{−1}α̂_{ji}λ_{j}. From (63) we therefore see that α = (α_{ij}) ∈ End_{λ_{1}⊗λ_{2}}(A) ⇔ α_{ij} = α′_{ji}, ∀i, j = 1, 2 ⇔ α_{12} = α′_{21}, α_{ii} ∈ End_{λ_{i}}(A_{i}), i = 1, 2, the latter because the hypothesis α_{12} = α′_{21} implies that α′_{12} = (α′_{21})′ = α_{12}. This proves (61), and from this the assertion about μ follows immediately. Finally, if we put D_{λ_{1},λ_{2}} = Φ_{λ_{1}⊗λ_{2}}^{−1} ∘ μ_{λ_{1},λ_{2}} ∘ (Φ_{λ_{1}} ⊕ Φ_{λ_{2}} ⊕ id), then it is clear by (54) that D = D_{λ_{1},λ_{2}} yields the desired isomorphism.

To prove (62), we first note that since ê_{i}(λ_{1} ⊗ λ_{2}) = λ_{i}p_{i}, we have r_{λ_{1}⊗λ_{2},λ_{i}}(p_{i}) = e_{i} and hence Φ_{λ_{1}⊗λ_{2}}(p_{i}^{∗}D_{i}) = e_{i}Φ_{λ_{i}}(D_{i})p_{i} by (55). Thus Φ_{λ_{1}⊗λ_{2}}(p_{1}^{∗}D_{1} + p_{2}^{∗}D_{2}) = μ(Φ_{λ_{1}}(D_{1}), Φ_{λ_{2}}(D_{2}), 0), and so (62) follows.

Another useful formula is the following.

Proposition 60 Let (A, λ) be a principally polarized abelian variety. If m_{A} : A × A → A denotes the addition map and δ_{A} : A → A × A the diagonal map, then r_{λ⊗λ,λ}(m_{A}) = δ_{A} and hence

Φ_{λ⊗λ}(m_{A}^{∗}D) = δ_{A}Φ_{λ}(D)m_{A}, ∀D ∈ NS(A). (64)

Proof. Since ê_{i}(λ ⊗ λ) = λp_{i} and ê_{i}m̂_{A} = idÂ, we have p_{i}r_{λ⊗λ,λ}(m_{A}) = p_{i}(λ ⊗ λ)^{−1}m̂_{A}λ = λ^{−1}ê_{i}m̂_{A}λ = 1_{A}, and so r_{λ⊗λ,λ}(m_{A}) = δ_{A}. Thus (64) follows from (55).

We now specialize the above results to the case of products of two elliptic curves.

Proposition 61 Let A = E_{1} × E_{2} be a product of two elliptic curves, and let λ_{i} = ϕ_{0_{E_{i}}}. Then the isomorphism

D = D_{λ_{1},λ_{2}} : ℤ ⊕ ℤ ⊕ Hom(E_{1}, E_{2}) →^{∼} NS(A)

is given by the formula

D(a, b, f) = cl((a − deg(f))θ_{1} + (b − 1)θ_{2} + Γ_{−f}). (65)

Here θ_{i} = p_{i}^{∗}(0_{E_{i}}), Γ_{f} ∈ Div(A) is the graph of f, and cl(D) ∈ NS(A) denotes the class of a divisor D ∈ Div(A). Thus

(66)  (D(a, b, f).D(a, b, f)) = 2(ab − deg(f)),
(67)  (D(a, b, f).(xθ_{1} + yθ_{2})) = bx + ay.

Proof. First note that since NS(E_{i}) = ℤcl(0_{E_{i}}) ≃ ℤ, the map D yields the indicated isomorphism. To prove (65), it is in view of (62) enough to verify that

Φ_{λ_{1}⊗λ_{2}}(Γ_{−f}) = μ([deg(f)]_{E_{1}}, 1_{E_{2}}, f) (68)

and this follows from the identities Γ_{−f} = (f × 1)^{∗}m_{E_{2}}^{∗}(0_{E_{2}}), r_{λ_{1}⊗λ_{2},λ_{2}⊗λ_{2}}(f × 1_{E_{2}}) = f′ × 1_{E_{2}} and Φ_{λ_{2}}(0_{E_{2}}) = 1_{E_{2}} because by (55) and (64) we obtain Φ_{λ_{1}⊗λ_{2}}(Γ_{−f}) = (f′ × 1_{E_{2}})Φ_{λ_{2}⊗λ_{2}}(m_{E_{2}}^{∗}0_{E_{2}})(f × 1_{E_{2}}) = (f′ × 1_{E_{2}})δ_{E_{2}}Φ_{λ_{2}}(0_{E_{2}})m_{E_{2}}(f × 1_{E_{2}}) = (f′ × 1_{E_{2}})δ_{E_{2}}m_{E_{2}})(f × 1_{E_{2}}) = \binom{f′ 0}{0 1}\binom{1 1}{1 1}\binom{f 0}{0 1} = μ(f′f, 1, f) = μ([deg(f)], 1, f).

From (65), the formulae (66) and (67) follow immediately because (θ_{1}.θ_{2}) = (Γ_{−f}.θ_{1}) = 1, (Γ_{−f}.θ_{2}) = deg(−f) = deg(f) and θ_{1}^{2} = θ_{2}^{2} = Γ_{−f}^{2} = 0, the latter because θ_{1} = {0} × E_{2} ≃ E_{2} and θ_{2} ≃ Γ_{−f} ≃ E_{1} are elliptic curves.

Corollary 62 Let A′ = E′_{1} × E′_{2} be another product surface and let α = (α_{ij}) ∈ Hom(A′, A)), where α_{ij} ∈ Hom(E′_{j}, E_{i}). Then

deg(α) = |(d_{11} + d_{21})(d_{12} + d_{22}) − deg(f_{α})| , (69)

where d_{ij} = deg(α_{ij}) and f_{α} = α_{12}^{t}α_{11} + α_{22}^{t}α_{21}. Moreover, for f ∈ Hom(E_{1}, E_{2}) we have α^{∗}D(n_{1}, n_{2}, f) = D(n′_{1}, n′_{2}, f′) (70) where n′_{1}, n′_{2}, and f′ are determined by the matrix equation

( \begin{matrix}[n′_{1}]_{E′_{1}} & (f′)^{t} \\ f′ & [n′_{2}]_{E′_{2}} \end{matrix} ) = ( \begin{matrix}α_{11}^{t} & α_{21}^{t} \\ α_{12}^{t} & α_{22}^{t} \end{matrix} ) ( \begin{matrix}[n_{1}]_{E_{1}} & f^{t} \\ f & [n_{2}]_{E_{2}} \end{matrix} ) ( \begin{matrix}α_{11} & α_{12} \\ α_{21} & α_{22} \end{matrix} ) . (71)

In other words, we have explicitly

n′_{1} = n_{1}d_{11} + n_{2}d_{21} + tr(α_{21}^{t}fα_{11})
n′_{2} = n_{1}d_{12} + n_{2}d_{22} + tr(α_{12}^{t}fα_{22})
f′ = n_{1}α_{12}^{t}α_{11} + n_{2}α_{22}^{t}α_{21} + α_{12}^{t}f^{t}α_{21} + α_{22}^{t}fα_{11}

where tr(h) ∈ ℤ is defined by [tr(h)] = h + h^{t}, for h ∈ End(E′_{i}).

Proof. To prove (69), consider α̃ := r_{λ_{1}⊗λ_{2}}(α)α. Since deg(r_{λ_{1}⊗λ_{2}}(α)) = deg(α̂) = deg(α), we have deg(α)^{2} = deg(α̃). Now by (63) we have α̃ = \binom{α′_{11} α′_{21}}{α′_{12} α′_{22}}\binom{α_{11} α_{12}}{α_{21} α_{22}} = μ([d_{1}], [d_{2}], f_{α}), where d_{1} = d_{11} + d_{21} and d_{2} = d_{12} + d_{22}, and so 4 deg(α)^{2} = 4 deg(μ([d_{1}], [d_{2}], f_{α}) = (D(d_{1}, d_{2}, f_{α})^{2})^{2}, where the latter equality follows from the Riemann-Roch Theorem (cf. [33], p. 150) because μ([a], [b], f) = Φ_{λ_{1}⊗λ_{1}}(D(a, b, f)). From this (69) follows immediately by using (66).

To prove (70) and (71), note first that there exist unique n′_{1}, n′_{2} and f′ such that (70) holds. Then Φ_{λ′_{1}⊗λ′_{2}}(D(n′_{1}, n′_{2}, f′)) equals the left hand side of (71), where λ′_{i} denotes the canonical polarization of E′_{i}. On the other hand, by (a slight generalization of) formula (63), the right hand side of (71) equals r_{λ′_{1}⊗λ′_{2},λ_{1}⊗λ_{2}}(α)Φ_{λ_{1}⊗λ_{2}}(D(n_{1}, n_{2}, f))α. Since this equals Φ_{λ′_{1}⊗λ′_{2}}(α^{∗}D(n_{1}, n_{2}, f)) by (55), we see that (71) holds. The last assertion follows from this by multiplying out the right side of (71).

Corollary 63 Let g ∈ M_{2}(ℤ) be a 2 × 2 matrix and let [g]_{E} ∈ End(E × E) be the endomorphism induced by g. Then deg([g]_{E}) = det(g)^{2}.

Proof. Write g = (a_{ij}), and apply (69) to α = [g]_{E} = ([a_{ij}]_{E}). Here d_{ij} = deg([a_{ij}]_{E}) = a_{ij}^{2}, and deg(f_{α}) = deg([a_{12}a_{11} + a_{22}a_{21}]) = (a_{12}a_{11} + a_{22}a_{21})^{2}. Thus deg(α) = |(a_{11}^{2} + a_{21}^{2})(a_{12}^{2} + a_{22}^{2}) − (a_{12}a_{11} + a_{22}a_{21})^{2}| = |(a_{11}a_{22} − a_{12}a_{21})^{2}| = det(g)^{2}.

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