Elliptic Curves on Abelian Surfaces

Ernst Kani

The purpose of this paper is to present two theorems which give an overview of the set of elliptic curves lying on an abelian surface and to discuss several applications. One of these applications is a classical theorem of Biermann (1883) and Humbert (1893) on the characterization of abelian surfaces containing elliptic curves in terms of the “singular relations” of Humbert. As a by–product one obtains a purely algebraic description of such relations and hence also of Humbert surfaces.

1. Introduction

The principal aim of this note is to classify the set of elliptic curves lying on an abelian surface A defined over an algebraically closed field K. Since any elliptic curve on A may be translated to the origin, it is sufficient to classify the elliptic subgroups of A. The main result here is that these can be characterized numerically, that is, inside the Néron–Severi group NS(A) = Div(A)/ ≡ of numerical equivalence classes.

Theorem 1.1 The map E ↦ cl(E) ∈ NS(A) induces a bijective correspondence between the set of elliptic subgroups E ≤ A of A and the primitive classes cl(D) ∈ NS(A) with (D.D) = 0 and (D.Θ) > 0 for some (hence any) ample divisor Θ ∈ Div(A).

Here and below, we call an element x ∈ M of a finitely generated free ℤ-module M primitive if the quotient module M/ℤx is torsion-free.

As a first application of the above theorem, we note the following result which was first proved (for K = ℂ) by Bolza [Bo] and by Poincaré [Po] in 1886 (cf. Krazer[Kr], chapter XII, Satz IX, or Lange[La2]):

Corollary 1.2 If an abelian surface has more than two elliptic subgroups, then it has infinitely many.

On the other hand, the fact that NS(A) is a finitely generated group shows that there cannot be “too many” elliptic subgroups on an abelian surface. To make this precise, fix an ample divisor Θ ∈ Div(A) so that we can partially order the curves C on A according to their degrees:

deg(C) = deg_{Θ}(C) = (C.Θ).

We then have the following finiteness result:

Corollary 1.3 An abelian surface A has only finitely many elliptic subgroups E ≤ A of bounded degree.

Of course, there is no reason why a given abelian surface should contain any elliptic subgroups at all, and indeed, those that do are rather special. These were analyzed by Humbert as a special case of his notion of abelian surfaces “satisfying a singular relation”.

Although Humbert defined this notion analytically in terms of the period matrix of A = ℂ^{2}/Γ (and hence only in the case K = ℂ), it is possible to give an algebraic definition of this concept which therefore extends to an arbitrary ground field. Since this algebraic definition is perhaps of independent interest, we develop it here in some detail.

To define this notion, consider the quadratic form Δ = Δ_{Θ} on the Néron– Severi group NS(A) defined by the formula

Δ(D) = (D.Θ)^{2} − 2δ(D.D),

where δ = \frac{1}{2}(Θ.Θ) denotes the “polarization degree” of the polarized surface (A, Θ). From the Hodge Index Theorem it follows that if Θ is a primitive element of NS(A), then Δ defines a positive definite quadratic form on the “polarized Néron–Severi group”

NS(A, Θ) := NS(A)/ℤΘ.

As will be explained below in section 5, Δ(D) is essentially the invariant defined by Humbert [Hu1] in his study of Humbert surfaces and hence will be called the Humbert invariant of the divisor D ∈ Div(A). As a result, the notion of a “singular relation” can be defined purely algebraically in the following way.

Definition 1.4 A polarized abelian surface (A, Θ) is said to satisfy a singular relation with invariant N if there exists a primitive class [D] ∈ NS(A, Θ) with Humbert invariant Δ(D) = N.

By using the above Humbert invariant we obtain a second classification of the set of elliptic subgroups of A, at least if (A, Θ) is principally polarized (i.e. δ = 1):

Theorem 1.5 Let (A, Θ) be a principally polarized abelian surface, and let n ∈ ℕ be an integer. Then the map E ↦ [E] = class of E in NS(A, Θ) induces a one–to–one correspondence between

1) the set of all elliptic subgroups E ≤ A of A of degree n = deg(E), and 2) the set of primitive classes [D] ∈ NS(A, Θ) with invariant Δ(D) = n^{2}.

As a first application of this result, one obtains the following algebraic analogue of a classical theorem due to Biermann [Bi] and Humbert [Hu1] which (for K = ℂ) was recently reproved by Murabayashi [Mur]:

Corollary 1.6 (Biermann–Humbert) A principally polarized abelian surface (A, Θ) has an elliptic subgroup E ≤ A of degree n if and only if (A, Θ) satisfies a singular relation with invariant n^{2}.

Strictly speaking, the above formula relating the degree of the elliptic subgroup to the Humbert invariant is not stated explicitly in the papers of Biermann and Humbert, but a close look at the proof as presented in the book of Krazer [Kr], chapter XII, Satz V, shows that their arguments yield this formula as well.

The above result can also be interpreted in terms of the Humbert surfaces H_{N} introduced by Humbert [Hu1] (cf. also van der Geer[vdG]):

H_{N} = {(A, Θ) ∈ A_{2} : (A, Θ) satisfies a singular relation with invariant N}.

Here, as usual, A_{2} denotes the moduli space of principally polarized abelian varieties.

Corollary 1.7 A principally polarized abelian surface (A, Θ) has an elliptic subgroup E ≤ A of degree n if and only if (A, Θ) ∈ H_{n^{2}}.

Humbert proves in his papers that for K = ℂ the set H_{N} is a closed subset of the moduli space A_{2} and hence is an algebraic surface. In view of the above corollary, this is also true for any field K when N = n^{2} and char(K)|/n by the results of [Ka1], [Ka2], where the moduli space of principally polarized abelian varieties containing an elliptic curve is studied. (If n is a prime number, then this also follows from the work of Lange [La1]). Moreover, by combining the results of these papers with Corollary 1.7 we obtain the following interesting structure theorem of the Humbert surfaces with square invariant:

Corollary 1.8 If char(K)|/n, then H_{n^{2}} is a closed irreducible subvariety of A_{2} whose normalization is

H̃_{n^{2}} = (X(n) × X(n))/⟨τ, G(n)⟩,

where X(n) denotes the affine modular curve parametrizing elliptic curves with a level n-structure, τ is the involution of X(n) × X(n) which interchanges the two factors, and G(n) = Sl_{2}(ℤ/nℤ)/{±1} which acts in the usual way on the first factor and in a twisted fashion on the second factor.

By specializing to the case that A = J_{X} is the Jacobian of a curve X of genus 2, the above theorems can be translated to results about the (minimal) elliptic subcovers f : X → E of X, where the minimality means that f does not factor over any non-trivial isogeny of the elliptic curve E. Indeed, as is explained in section 4, it follows from duality theory that the elliptic subgroups E ≤ J_{X} correspond bijectively to isomorphism classes of minimal subcovers of X, and so we obtain from the above theorems:

Theorem 1.9 Let X be a curve of genus 2 with Jacobian J_{X} and canonical principal polarization Θ. Then there is a bijective correspondence between the following sets:

1) the set of isomorphism classes of (minimal) elliptic subcovers f : X → E of degree deg(f) = n;

2) the set of elliptic subgroups E ≤ J_{X} of J_{X} of degree deg_{Θ}(E) = n;

3) the set of primitive classes cl(D) ∈ NS(J_{X}) of degree deg_{Θ}(D) = n and with self–intersection (D.D) = 0;

4) the set of primitive classes [D] ∈ NS(J_{X}, Θ) with invariant Δ(D) = n^{2}.

In a similar manner the above corollaries can be translated to assertions about elliptic subcovers. For example, Corollary 1.3 becomes the genus two case of a theorem of Tamme [Ta] which is valid for arbitrary genus:

Corollary 1.10 (Tamme) There are only finitely many isomorphism classes of elliptic subcovers of X of bounded degree.

Finally, in section 5 we compare the algebraic theory developed so far with the analytic theory (for K = ℂ). In particular, we justify that the algebraic definitions of Humbert’s invariant and of Humbert surfaces as presented above agree with the classical definitions; cf. Remark 5.6. Moreover, by introducing the notion of an “elliptic submodule” of the singular cohomology group H^{1}(A, ℤ), we give an analytic reformulation of Theorem 1.1 together with a quick analytic proof.

This paper developed out of the joint work[FK] with G. Frey, whom I would like to thank very much for the many fruitful discussions we have had on this topic. In addition, I would like to thank P. Roquette for his interest in this paper.

This research was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

2. A numerical characterization of elliptic subgroups

Let A be an abelian surface defined over an algebraically closed field K of arbitrary characteristic. We first review some basic facts of such surfaces.

Since the canonical class ω_{A} of A is trivial, the Riemann–Roch theorem for a divisor D ∈ Div(A) assumes the form

χ(D) := h^{0}(D) − h^{1}(D) + h^{0}(−D) = \frac{1}{2}(D.D), (2.1)

where, as usual, h^{q}(D) = dim_{K} H^{q}(A, L(D)). Here we have also used Serre duality: h^{2}(D) = h^{0}(−D). Moreover, if D ≥ 0 is effective, then the adjunction formula becomes

p_{a}(D) − 1 = \frac{1}{2}(D.D). (2.2)

We shall also frequently make use of the following fact. For any x ∈ A, the translated divisor T_{x}(D) is numerically equivalent to D,

T_{x}(D) ≡ D, (2.3)

because T_{x}(D) is visibly algebraically equivalent to D.

We now want to classify the elliptic curves lying on A. The first step in this direction is accomplished by the following auxiliary result:

Proposition 2.1 Let C_{1} and C_{2} be two irreducible curves lying on A. Then (C_{1}.C_{2}) ≥ 0, and (C_{1}.C_{2}) = 0 if and only if C_{1} = T_{x}(C_{2}) = E, for some x ∈ A and some elliptic curve E on A.

Proof. This depends on the following well-known fact:

(2.4) There is no rational curve C lying on A.

Indeed, if i : C → A denotes the inclusion map and n : C̃ → C the normalization of C, then i ∘ n : C̃ → A factors over the Jacobian JC̃ of C̃, and so we must have g(C̃) = dim JC̃ > 0, and hence C is not rational.

If C_{1} ≠ C_{2} are distinct curves, then clearly (C_{1}.C_{2}) ≥ 0, so assume C_{1} = C_{2} = C. Since C is not rational, we have by the adjunction formula that (C.C) = 2p_{a}(C) − 2 ≥ 0, which proves the first assertion.

To prove the second, assume that (C_{1}.C_{2}) = 0 and choose x ∈ A such that T_{x}(C_{2}) ∩ C_{1} ≠ ϕ. Since (T_{x}(C_{2}).C_{1}) = (C_{2}.C_{1}) = 0, this forces T_{x}(C_{2}) = C_{1} because both are irreducible curves. Thus, by the adjunction formula we have 2p_{a}(C_{1}) − 2 = (C_{1}.C_{1}) = (C_{1}.T_{x}(C_{2})) = (C_{1}.C_{2}) = 0, so p_{a}(C_{1}) = 1. If C_{1} were not smooth, then C_{1} is rational, which contradicts (2.4), so C_{1} = E is an elliptic curve.

Corollary 2.2 a) If D_{1}, D_{2} ≥ 0 are effective divisors, then (D_{1}.D_{2}) ≥ 0, and

(D_{1}.D_{2}) = 0 ⟺ D_{i} = ∑ n_{ij}T_{x_{ij}}(E), i = 1, 2, for some elliptic curve E. (2.5)

b) If D ∈ Div(A) is any divisor on A, then (D.D) > 0 if and only if either D or −D is ample.

Proof. a) Write D_{i} = ∑ n_{ij}C_{ij}, where n_{ij} > 0 and each C_{ij} is an irreducible curve. Then (D_{1}.D_{2}) = ∑_{i,j} n_{1i}n_{2j}(C_{1i}.C_{2j}) ≥ 0 by Proposition 2.1. Moreover, (D_{1}.D_{2}) = 0 if and only if (C_{1i}.C_{2j}) = 0, for all i, j. By induction on i, j it follows from Proposition 2.1 again that C_{ij} = T_{x_{ij}}(C_{11}) and C_{11} = E, which proves the assertion.

b) The converse being trivial, assume that (D.D) > 0. From the Riemann– Roch formula we have h^{0}(D)+h^{0}(−D) ≥ \frac{1}{2}(D.D) > 0, so the linear equivalence class of either D or −D contains an effective divisor. Without loss of generality we may thus assume that D ≥ 0.

To prove that D is ample, by the Nakai–Moizeshon criterion it is enough to show that (D.C) > 0 for each irreducible curve C on A. Now by part a) this is automatic unless C = E is an elliptic curve and D = ∑_{x} n_{x}T_{x}(E). But then (D.D) = 0, contradiction. Thus (D.C) > 0 and hence D is ample.

The above proposition gives a characterization of elliptic curves E as divisors on A. The next and crucial step in their classification is to characterize their numerical equivalence classes cl(E) ∈ NS(A) within the Néron–Severi group NS(A) = Div(A)/ ≡ itself. While the above corollary gives a partial result in this direction in that it shows that the “elliptic classes” cl(mE) ∈ NS(A), where E is an elliptic curve and m ∈ ℕ a positive integer, are precisely the isotropic classes of the effective cone NS^{+}(A) = {cl(D) : D ≥ 0}, considerable work is still necessary to eliminate the effectivity condition “D ∈ NS^{+}(A)” in this characterization.

As we shall see below, the elliptic classes can be characterized numerically up to a sign. To be able to determine this sign, we shall fix from now on an ample divisor Θ ∈ Div(A) so that the the degree of a divisor D ∈ Div(A) with respect to Θ is defined:

(2.6) deg(D) = deg_{Θ}(D) = (D.Θ).

Clearly, all elliptic classes have positive degree (with respect to any ample divisor Θ), and hence their sign is determined in this way. are We then have the following result which should be viewed as the main cornerstone in the classification of the elliptic curves on A.

Proposition 2.3 Let D ∈ Div(A) be a divisor on A. Then D ≡ mE for some elliptic curve E and some m ∈ ℤ if and only if (D.D) = 0. Moreover, m > 0 if and only if deg_{Θ}(D) > 0.

Proof. The second assertion and one direction of the first assertion are clear, for if D ≡ mE, then clearly (D.D) = m^{2}(E.E) = 0 and deg(D) = m(E.Θ). It is therfore enough to prove the converse of the first assertion.

Thus, assume (D.D) = 0. If deg_{Θ}(D) = 0, then by the Hodge Index Theorem we have D ≡ 0, so we are done by taking m = 0. Thus, assume deg(D) ≠ 0; by replacing D by −D if necessary we may assume that deg(D) > 0.

Next, if D ≡ D′, for some D′ with h^{0}(D′) > 0, then by Corollary 2.2 we have that D ≡ mE, so we may henceforth assume that

h^{0}(D′) = 0, ∀D′ ≡ D; (2.7)

we will eventually see that this leads to a contradiction.

Step 1: There exists an elliptic curve E such that (D.E) = 0. Consider the closed subgroup K(D) = {x ∈ A : T_{x}^{∗}(D) ∼ D}; cf. Mumford [Mu], p. 60. If K(D) is finite, then (D.D) ≠ 0 by the Riemann–Roch Theorem of Mumford[Mu], p. 150, and if K(D) = A then D ∈ Pic^{0}(A), so D ≡ 0 (cf. Milne[Mi], Remark 9.3, p. 118), and hence deg(D) = 0, contradiction. We thus have that dim K(D) = 1, and so there is an elliptic subgroup E ≤ K(D).

Put L′ = L(D)_{|E}. Then T_{x}^{∗}L′ = (T_{x}^{∗}L(D))_{|E} ≃ (L(D))_{|E} = L′, for all x ∈ E, and so L′ ∈ Pic^{0}(A). Thus (D.E) = deg(L′) = 0, which completes the first step.

Step 2: There exists D̃ ≡ D such that L(D̃)_{|E} ≃ O_{E}. Since Θ′ = Θ_{|E} is ample on E, the map ϕ_{Θ′} : E → Pic^{0}(E) is surjective, and so there exists x ∈ E such that T_{x}^{∗}(L(Θ′)) ⊗ L(Θ′)^{−1} ≃ L(D)_{|E}. Thus, if we put D̃ = D + Θ − T_{x}^{∗}Θ, then D̃ ≡ D and L(D̃)_{|E} ≃ O_{E}.

Step 3: There is a surjective homomorphism f : A → E′ to an elliptic curve E′ such that Ker(f) = E. This follows either from the existence of quotients of smooth group schemes over a field (cf. SGA3, Exp. VI_{A}.5.4) or, more elementarily, from Poincaré’s complete reducibility theorem together with the existence of quotients for finite group schemes (cf. Mumford [Mu], §12).

Step 4: D̃ ∼ f^{∗}(D′), for some D′ ∈ Div(E′). By Hartshorne[Ha], Ex. III.12.4, it is enough to show that D̃_{t} ∼ 0, for all t ∈ E′. Moreover, since deg(D̃_{t}) = (D.E) = 0, it is enough to show that h^{0}(D̃_{t}) ≥ 1, for all t ∈ E′.

Suppose h^{0}(D̃_{t_{0}}) = 0 for some t_{0} ∈ E′. Then by semi-continuity we have h^{0}(D̃_{t}) = h^{1}(D̃_{t}) = 0 for almost all t ∈ E′, and so R^{1}f_{∗}L(D̃) is a skyscraper sheaf on E′. Using the Leray spectral sequence E_{2}^{pq} = H^{p}(E′, R^{q}f_{∗}L(D)) ⇒ H^{p+q}(A, L(D)), we obtain the exact sequence

0 → H^{1}(E′, f_{∗}L(D̃)) → H^{1}(A, L(D̃)) → H^{0}(E′, R^{1}f_{∗}L(D̃)) → H^{2}(E′, f_{∗}L(D̃))

of low-degree terms. But since H^{2}(E′, f_{∗}L(D)) = 0 because dim E′ = 1 and since h^{1}(D̃) = 0 by hypothesis (2.7), we must have R^{1}f_{∗}L(D̃) = 0. On the other hand, since H^{2}(A_{t}, D_{t}) = 0, ∀t ∈ E′, we have by base-change (cf. Mumford[Mu], p. 63) that

R^{1}f_{∗}L(D̃) ⊗ κ(t) →^{∼} H^{1}(A_{t}, D_{t})

is an isomorphism, which means that h^{1}(D_{t}) = 0, for all t ∈ E′. But this contradicts the choice of D̃, which was constructed in step 2 so that h^{0}(D_{t}) = h^{1}(D_{t}) = 1 for t = 0. Thus h^{0}(D̃_{t}) ≥ 1, ∀t ∈ E′, which concludes step 4.

To conclude the proof of the proposition, we first note that by steps 2 and 4 we have D ≡ D̃ ∼ f^{∗}(D′) ≡ mE, with m = deg(D′). Moreover, since m(E.Θ) = (D.Θ) > 0, we have that m > 0, whence the proposition.

Corollary 2.4 A divisor class cl(D) ∈ NS(A) lies in the effective cone (i.e. cl(D) ∈ NS^{+}(A)) if and only if (D.D) ≥ 0 and deg_{Θ}(D) > 0.

Proof. If D ≡ D′ ≥ 0, then (D.D) ≥ 0 by Corollary 2.2a) and deg(D) > 0 since D′ ≠ 0. Conversely, assume that (D.D) ≥ 0 and that (D.Θ) > 0. If (D.D) > 0, then by (the proof of) Corollary 2.2b) we have that D is ample and effective, and if (D.D) = 0, then D ≡ mE is an effective class by Proposition 2.3.

As another application of Proposition 2.3, we can now see that classes belonging to elliptic curves are primitive in the sense of the following definition.

Definition 2.5 An element 0 ≠ x ∈ Γ of a torsionfree abelian group Γ is called primitive if the quotient group Γ/ℤx is torsionfree (cf. eg. Barth/Peters/van der Ven [BPV], p. 13) or, equivalently, if x satisfies the following condition:

x = my, y ∈ Γ, m ∈ ℕ ⇒ m = 1. (2.8)

Corollary 2.6 The class cl(E) ∈ NS(A) of an elliptic curve E on A is primitive.

Proof. Suppose E ≡ mD for some D ∈ Div(A) and m > 0. Then (D.D) = 0 and deg(D) > 0, so by Proposition 2.3 D ≡ m′E′ for some m′ > 0 and some elliptic curve E′. But since (E.D) = 0, we must have E ≡ E′ by Proposition 2.1, and so we obtain E ≡ mD ≡ mm′E. Since E ≢ 0, it follows that m = m′ = 1, so E is primitive.

Remark 2.7 The last corollary can also be proved without recourse to Proposition 2.3 by using the following criterion of primitivity:

cl(D) ∈ NS(A) is primitive ⟺ A[N] ⊄ K(D), ∀N > 1,

where A[N] = Ker(N · id_{A}) denotes the subgroup scheme of N-torsion points of A; this follows from Mumford[Mu], Th. 23.3, p. 231, together with the fact that Pic^{0}(A) is divisible. Applying this to D = E, the primitivity follows easily once one has that

K(E) = E,

which can be proved by using the map f : A → A/E and functorial properties of K(D).

We are now ready to prove the numerical characterization of elliptic subgroups announced in Theorem 1.1 of the introduction:

Theorem 2.8 The map E ↦ cl(E) ∈ NS(A) induces a one–to–one correspondence between the set of elliptic subgroups E ≤ A of A and the primitive classes cl(D) ∈ NS(A) with (D.D) = 0 and deg(D) > 0.

Proof. First note that it follows from the above Corollary 2.6 that cl(E) has the asserted properties. Next we observe that the map E ↦ cl(E) is injective. Indeed, if E ≠ E′ are two distinct elliptic subgroups, then E ≠ T_{x}(E′), ∀x ∈ A, so (E.E′) > 0 by Proposition 2.2 and hence cl(E) ≠ cl(E′).

Finally, the map E ↦ cl(E) is surjective, for if cl(D) ∈ NS(A) is a primitive class with (D.D) = 0 and deg(D) > 0, then by Proposition 2.3 we have D = mE for some elliptic curve E and m > 0. But since cl(D) is primitive, we must have m = 1, and so the assertion is proved.

Corollary 2.9 (Bolza–Poincaré) If A has more than two elliptic subgroups, then it has infinitely many.

Proof. Let E_{1}, E_{2} and E_{3} be three distinct elliptic subgroups of A. We first observe that their classes in NS(A) are linearly independent. Indeed, if aE_{1} + bE_{2} + cE_{3} ≡ 0, and (say) a ≠ 0, then 0 = a^{2}(E_{1})^{2} = (abE_{2} + bcE_{3})^{2} = 2bc(E_{2}.E_{3}), so either b = 0 or c = 0. But if (say) b = 0, then aE_{1} ≡ −cE_{3}, which is impossible since (E_{2}.E_{3}) > 0 but (E_{2}.E_{2}) = 0. Thus, cl(E_{1}), cl(E_{2}) and cl(E_{3}) are linearly independent.

Now write a = (E_{1}.E_{2}) > 0, b = (E_{1}.E_{3}) > 0, c = (E_{2}.E_{3}) > 0 and D = cE_{1} +bE_{2} −aE_{3}. Then (D.E_{1}) = (D.E_{2}) = 0 and (D.D) = a(b+c) > 0. Thus, for every x ∈ ℚ, the rational divisor

D(x) = xE_{1} + \frac{(D.D)}{2x}E_{2} − D = (x − c)E_{1} + (\frac{(D.D)}{2x} − b) E_{2} + aE_{3}

satisfies (D(x).D(x)) = 0. Since E_{1}, E_{2}, and E_{3} are linearly independent, we see that by multiplying each D(x) by a suitable integer, we obtain infinitely many primitive classes D′(x) satisfying (D′(x).D′(x)) = 0 and (D′(x).Θ) > 0, and hence infinitely many elliptic curves.

Corollary 2.10 There are only finitely many elliptic subgroups E ≤ A of bounded degree deg_{Θ}(E) ≤ k.

Proof. Since NS(A) is finitely generated (cf. [Mu], p. 178), it follows from the Hodge Index Theorem that there are only finitely many effective classes of bounded degree (cf. [Ha], Ex. V.1.11a) for details), and so the assertion follows from Theorem 2.8.

3. Humbert’s invariant and Humbert surfaces

As before, let Θ be an ample divisor on an abelian surface A, so that the pair (A, cl(Θ)) defines a polarized abelian surface. Without loss of generality, we may and will assume that its class cl(Θ) ∈ NS(A) is primitive in the sense of Definition 2.5; we then say that (A, cl(Θ)) is a primitively polarized abelian surface of degree

δ = χ(Θ) = \frac{1}{2}(Θ.Θ).

Note that if Θ is a principal polarization (i.e. δ = 1), then Θ is automatically primitive.

If D ∈ Div(A) is any divisor, then define its Humbert invariant Δ(D) by

Δ(D) = Δ_{Θ}(D) = deg(D)^{2} − 2δ(D.D), (3.1)

where, as before, deg(D) = deg_{Θ}(D) = (D.Θ).

Since Δ(D + mΘ) = Δ(D), for all D ∈ Div(A) and m ∈ ℤ, it follows that

Δ(D) = −(D^{∗}.D^{∗}), (3.2)

where D^{∗} = 2δD − deg(D)Θ. Thus, since (D^{∗}.Θ) = 0, it follows from the Hodge index theorem that

(3.3) Δ(D) ≥ 0, and Δ(D) = 0 ⟺ D ≡ \frac{deg(D)}{2δ}Θ.

We therefore see that Δ defines a positive definite (non-degenerate) quadratic form on the “polarized Néron–Severi group”

NS(A, Θ) = NS(A)/ℤcl(Θ)

by the rule

Δ([D]) = Δ(D),

where [D] denotes the class of D ∈ Div(A) in NS(A, Θ).

Theorem 3.1 Suppose (A, Θ) is a principally polarized abelian variety. Then for any positive integer k ∈ ℕ, the map E ↦ [E] ∈ NS(A, Θ) induces a bijection between:

1) the set of elliptic subgroups E ≤ A of degree k = deg(E), and

2) the set of primitive classes [D] ∈ NS(A, Θ) with Humbert invariant Δ(D) = k^{2} .

Proof. We first show that [E] is primitive in NS(A, Θ). If not, there are m, n ∈ ℤ such that E + mΘ = nD. Since cl(E) is primitive (in NS(A)) by Corollary 2.6, we must have (m, n) = 1. Thus, since n^{2}D^{2} − 2mn(D.Θ) + m^{2}Θ^{2} = E^{2} = 0, it follows that n | Θ^{2} = 2. If 2 | n, then 4 | m^{2}Θ^{2} = 2m^{2}, which is impossible. Thus n = ±1, and so [E] is primitive.

Next we show that the map E ↦ [E] is injective. Thus, assume [E] = [E′], or E ≡ E′ + mΘ, where E, E′ are elliptic subgroups of A and m ∈ ℤ. We then obtain 0 ≥ −2(E.E′) = (E − E′)^{2} = m^{2}Θ^{2} ≥ 0, so equality holds throughout, which means that E = E′, as asserted.

It remains to show that the map is surjective. Thus, suppose D ∈ Div(A) is such that Δ(D) = k^{2} and [D] is primitive. Put d = deg(D) = (D.Θ). Then d^{2} = Δ(D) + 2(D.D) = k^{2} + 2(D.D), so d^{2} ≡ k^{2} (mod 2), and hence d ≡ k ( mod 2). Thus D′ = D− \frac{d−k}{2} Θ ∈ [D]; note that D′ is primitive in NS(A) since [D′] = [D] is primitive in NS(A, Θ). Now deg(D′) = d − \frac{d−k}{2} Θ^{2} = k and so 2(D′.D′) = deg(D′)^{2} − Δ(D′) = k^{2} − k^{2} = 0. Thus, by Proposition 2.8 we have D′ ≡ E, for some elliptic subgroup E, and so [E] = [D′] = [D], which proves the surjectivity.

Essentially the same proof shows that the above theorem can be generalized to arbitrary primitively polarized abelian surfaces as follows.

Theorem 3.2 Suppose (A, Θ) is a primitively polarized abelian surface of degree δ. Then for any positive integer k ∈ ℕ with (k, δ) = 1, the map E ↦ [E] ∈ NS(A, Θ) induces a bijection between:

1) the set of elliptic subgroups E ≤ A of degree k = deg(E), and

2) the set of primitive classes [D] ∈ NS(A, Θ) with Δ(D) = k^{2} and deg(D) ≡ k (mod 2δ).

As a first application of Theorem 3.1, we can prove an algebraic analogue of a theorem of Biermann [Bi] and Humbert [Hu1] on the characterization of (principally polarized) abelian surfaces containing an elliptic curve in terms of the “singular relations” of Humbert, which we define algebraically as follows.

Definition 3.3 A primitively polarized abelian surface (A, Θ) is said to satisfy a singular relation with invariant N if there exists a primitive class [D] ∈ NS(A, Θ) with Humbert invariant Δ([D]) = Δ_{Θ}([D]) = N.

Remark 3.4 Note that (A, Θ) satisfies a singular relation (for some N) if and only if rk(NS(A)) ≥ 2. Moreover, N is uniquely determined if and only if rk(NS(A)) = 2; in fact, in the case that rk(NS(A)) ≥ 3 there are infinitely many singular relations (and hence N’s).

Using this terminology, the above–mentioned result of Biermann and Humbert may be formulated as follows.

Corollary 3.5 (Biermann–Humbert) A principally polarized abelian surface (A, Θ) has an elliptic subgroup E ≤ A of degree n if and only if (A, Θ) satisfies a singular relation with invariant n^{2}.

Proof. This is immediate by Theorem 3.1.

Remark 3.6 As was mentioned above, this corollary is just an algebraic analogue of the actual result proved by Biermann [Bi], pp. 981–983 and Humbert [Hu1], Theorem 15 (p. 308). To obtain the original result, one still has to relate the present algebraic terminology to the analytic terminology of Humbert; this is done in section 5; cf. Remark 5.6.

The above result can also be interpreted in terms of the Humbert surfaces introduced by Humbert[Hu1]. Although we require below only the case of a principal polarization (i.e. δ = 1), it is just as easy to give the definition in the general case, as is done in van der Geer[vdG], ch. IX.2.

Definition 3.7 Let A_{2}(δ) denote the moduli space consisting of all isomorphism classes of primitively polarized abelian varieties (A, Θ) of degree δ. Then for each integer N ≥ 1 the subset

H_{N} = {(A, Θ) ∈ A_{2}(δ) : (A, Θ) satisfies a singular relation with invariant N}

is called the Humbert surface of invariant N in A_{2}(δ).

Remark 3.8 Once we have shown that the above algebraic definition of a singular relation coincides with that of Humbert, it follows also that this algebraic definition of a Humbert surface coincides with the analytic definition of Humbert (cf. van der Geer[vdG], p. 211). As a result, it follows from the work of Humbert and others that for K = ℂ the set H_{N} is a closed subset of the moduli space A_{2}(δ) and hence is an algebraic surface. While the corresponding result for ground fields K of positive characteristic has not yet been proved in general, there is little doubt that it is correct, provided one restricts to the case that char(K)|/N. For a partial result in this direction, see Corollary 3.10 below.

With the above definition, Corollary 3.5 can be rephrased as follows.

Corollary 3.9 A principally polarized abelian surface (A, Θ) has an elliptic subgroup E ≤ A of degree n if and only if (A, Θ) ∈ H_{n^{2}}.

The above corollary may be viewed as the umbilical cord connecting the Humbert surfaces with square invariant to the moduli spaces A_{g,1}(N) studied in [Ka1] and [Ka2]. The latter is the moduli space parametrizing isomorphism classes of principally polarized abelian varieties (A, Θ) of dimension g containing an elliptic subgroup E ≤ A of degree N:

A_{g,1}(N) = {⟨A, Θ, E⟩ : ⟨A, Θ⟩ ∈ A_{g}, E ≤ A, deg_{Θ}(E) = N}.

For g = 2 and char(K)|/N this moduli space can be explicitly determined; in fact, we have by [Ka1], [Ka2] that

A_{2,1}(N) = (X(N) × X(N))/G(N),

where X(N) denotes the (affine) modular curve parametrizing elliptic curves with a level N structure (of fixed determinant), and G(N) = Sl_{2}(ℤ/Nℤ)/{±1} which acts in the usual manner on the first factor and in a certain twisted manner on the second factor. To be precise, this twisted action is given by (g, x) ↦ g^{ε}.x, where ε ∈ Gl_{2}(ℤ/Nℤ)/{±1} is a fixed element with det(ε) = −1 , g^{ε} = ε^{−1}gε, and (g, x) ↦ g.x denotes the usual action of G(N) on X(N).

Thus, since the image of the “forget map”

f_{N} : A_{2,1}(N) → A_{2}

is precisely H_{N^{2}} by Corollary 3.9, we obtain from [Ka1] and [Ka2]:

Corollary 3.10 If char(K)|/N, then H_{N^{2}} is a closed irreducible subvariety of A_{2} whose normalization is A_{2,1}(N)/⟨τ⟩, where τ is the involution on A_{2,1}(N) which is induced by the involution of X(N) × X(N) which interchanges the factors.

4. Application to elliptic subcovers of curves

Let C be a (connected, smooth, projective) curve defined over K, and let J_{C} denote its Jacobian variety. As in Milne[Mi], we view J_{C} as a Picard variety; i.e. J_{C} represents the Picard functor Pic^{0}, so in particular we have a canonical identification Pic^{0}(C) = J_{C}(K). Thus, if f : C → C′ is any covering of curves, then by functoriality we have an induced homomorphism

f^{∗} : J_{C′} → J_{C}

between their Jacobian varieties. By autoduality of the Jacobian (i.e. by the universal property of J_{C} as an Albanese variety) we also have an induced homomorphism

f_{∗} : J_{C} → J_{C′}

which is characterized by the relation

j_{f(P)} ∘ f = f_{∗} ∘ j_{P} , for any P ∈ C, (4.1)

where j_{P} : C → J_{C} denotes the canonical embedding of C in its Jacobian defined by the point P, i.e. j_{P} (Q) = cl(Q − P) ∈ Pic^{0}(C) = J_{C}(K). These two homomorphisms are actually dual to each other in the sense that we have

(4.2)  (f^{∗})^{∧} ∘ λ_{C} = λ_{C′} ∘ f_{∗},
(4.3)  (f_{∗})^{∧} ∘ λ_{C′} = λ_{C} ∘ f^{∗},

where λ_{C} = ϕ_{Θ_{C}} : J_{C} → Ĵ_{C} denotes the canonical principal polarization of J_{C} induced by the theta–divisor Θ_{C} and, for any homomorphism h : A → B of abelian varieties, ĥ : B̂ → Â denotes the dual map.

Here we shall be particularly interested in the case that target curve C′ = E has genus 1 (so E is an elliptic curve but we do not fix the identity point) and f is in a certain sense “minimal”. More precisely, we shall study the following class of subcovers:

Definition 4.1 An elliptic subcover of a curve C is a finite morphism f : C → E to some a curve E of genus 1 which does not factor over any non–trivial isogeny of E, i.e. the only factorizations f = f′′ ∘ f′ : C → E′ → E, where E′ is another curve of genus 1, are those for which f′′ is an isomorphism.

If f′ : C → E′ is another elliptic subcover, then f′ is said to be equivalent to f if there is an isomorphism h : E′ → E such that f = h ∘ f′.

Proposition 4.2 For a morphism f : C → E to an elliptic curve E, the following conditions are equivalent:

(i) f is an elliptic subcover;

(ii) f_{∗} : J_{C} → J_{E} does not factor over a non–trivial isogeny;

(iii) f^{∗} : J_{E} → J_{C} is injective (i.e. a closed immersion). Moreover, if these conditions are satisfied, then we have

deg(f) = (f^{∗}J_{E}.Θ_{C}). (4.4)

Proof. The equivalence of (i) and (ii) follows from the universal (Albanese) property of (J_{C}, j_{P}), and the equivalence of (ii) and (iii) follows by duality since f^{∗} is essentially the dual of f_{∗} by (4.1).

It remains to prove (4.4), which is equivalent to the assertion

(f^{∗})^{∗}(Θ_{C}) ≡ deg(f)Θ_{E} (4.5)

because (f^{∗}J_{E}.Θ_{C}) = deg_{J_{E}}((f^{∗})^{∗}(Θ_{C})), and two divisors on J_{E} are numerically equivalent if and only if they have the same degree.

To prove (4.5), we first note that for any homomorphism h : A → B of abelian varieties and divisor D ∈ Div(B) we have

ϕ_{h^{∗}D} = ĥ ∘ ϕ_{D} ∘ h;

cf. the argument in Mumford[Mu], p. 143. (The assumption that h be an isogeny is not necessary for this formula.) Applying this to h = f^{∗} : J_{C′} → J_{C} and D = Θ_{C} we obtain by (4.1):

ϕ_{h^{∗}(D)} = λ_{C′} ∘ f_{∗} ∘ f^{∗} = deg(f)λ_{C′},

since f_{∗} ∘ f^{∗} = deg(f)id_{J_{C′}}. By the theorem of the square ([Mu], p. 59/60) we have dλ_{C′} = dϕ_{Θ_{C′}} = ϕ_{dΘ_{C′}}, where d = deg(f). Thus ϕ_{h^{∗}D} = ϕ_{dΘ_{C′}}, or equivalently, (f^{∗})^{∗}Θ_{C} ≡ deg(f)Θ_{C′}. (4.6)

This proves (4.5) and hence (4.4). Note, however, that the last equation is true for any covering f : C → C′ of curves.

Corollary 4.3 For a curve C, the map f ↦ f^{∗}J_{E} induces a one–to–one correspondence between the set of isomorphism classes of elliptic subcovers f : C → E of C of degree n = deg(f) and the set of elliptic subgroups E ≤ J_{C} of its Jacobian J_{C} of degree n = (E.Θ_{C}).

Proof. It is clear that the assignment f ↦ f^{∗}J_{E} is constant on equivalence classes of elliptic subcovers, and hence defines a map Φ from the first set to the second by Proposition 4.2. We shall now construct an inverse.

For this, fix a point P ∈ C and let E ≤ J_{C} be an elliptic subgroup. Let ι : J_{E} → J_{C} denote the inclusion map (here we identify E with J_{E} via j = j_{0} :

E →^{∼} J_{E}), and define f = f_{E,P} : C → J_{E} by

f = λ_{E}^{−1} ∘ ι̂∘ λ_{C} ∘ j_{P} .

Then f_{∗} = λ_{E}^{−1} ∘ ι̂∘ λ_{C}, so

(f_{∗})^{∧} = λ̂_{C} ∘ ˆ ι̂∘ λ̂_{E}^{−1} = λ_{C} ∘ ι ∘ λ_{E}^{−1},

using the canonical double–dual identifications ˆ Ĵ_{C} = J_{C} etc. From equation (4.2) it thus follows that f^{∗} = ι, so f^{∗} is a closed immersion and hence f : C → E is an elliptic subcover by Proposition 4.2. It is immediate that the map E ↦ (f_{E,P} : C → J_{E}) is an inverse of Φ, and hence Φ is a bijection.

Remark 4.4 As indicated, the above corollary is true for a curve C of arbitrary genus. For genus 2 curves, however, it is possible to give a somewhat different proof; cf. [FK], pp. 154–156.

If we combine Corollary 4.3 with Theorems 2.8 and 3.1 we obtain the following result:

Theorem 4.5 Let C be a curve of genus 2 with Jacobian J_{C} and canonical principal polarization Θ. Then there is a bijective correspondence between the following sets:

1) the set of isomorphism classes of elliptic subcovers f : C → E of degree deg(f) = n;

2) the set of elliptic subgroups E ≤ J_{X} of J_{X} of degree deg_{Θ}(E) = n;

3) the set of primitive classes cl(D) ∈ NS(J_{X}) of degree deg_{Θ}(D) = n and with self-intersection (D.D) = 0;

4) the set of primitive classes [D] ∈ NS(J_{X}, Θ) with invariant Δ(D) = n^{2}.

5. Analytic theory

We now want to re-examine at the results developed so far from an analytic viewpoint; in particular, we want to connect the algebraic definition of Humbert’s invariant with the classical analytic definition and also give an analytic version of the main theorem.

Thus, suppose K = ℂ is the field of complex numbers, so each abelian variety A has an analytic uniformization A = V/U, where U is a lattice in a ℂ-vector space V.

Summary 5.1 We recall the following basic facts about the cohomology of a complex abelian variety A = V/U of dimension g; cf. Mumford[Mu], ch. 1, or Griffiths/Harris[GH], ch. 2.6:

1. Since V is simply–connected, we have a canonical identification

U = π_{1}(A, 0) = H_{1}(A, ℤ), (5.1)

which induces an isomorphism

H^{1}(A, ℤ) = Hom(π_{1}(A, 0), ℤ) ≃ Hom(U, ℤ).

Via the cup–product, this isomorphism yields a canonical identification

H^{n}(A, ℤ) ≃ ∧^{n}H^{1}(A, ℤ) = Alt^{n}(U, ℤ) (5.2)

of the (singular) integral cohomology group H^{n}(A, ℤ) with the group of alternating n-forms a : U^{n} → ℤ; cf. [Mu], p. 3.

2. We can identify V with the tangent space T (A) of A at 0. Thus, if T = Hom_{ℂ}(V, ℂ), and \overline{T} = Hom_{ℂ-antilinear}(V, ℂ), then the canonical decomposition H^{1}(A, ℂ) = Hom_{ℝ}(V, ℂ) = T ⊕ \overline{T} (5.3)

is just the Hodge decomposition of H^{1}(A, ℂ), which gives rise to canonical isomorphisms

H^{p,q}(A) = H^{q}(A, Ω^{p}) ≃ ∧^{p}T ⊗ ∧^{q}\overline{T} , (5.4)

H^{n}(A, ℂ) = ∧^{n}Hom_{ℝ}(V, ℂ) = ∧^{n}(T ⊕ \overline{T}) = ⨁_{p+q=n} ∧^{p}T ⊗ ∧^{q}\overline{T}; (5.5)

cf. [Mu], p. 4/5). In particular, if g = 2 and z = (z_{1}, z_{2}) are Euclidean coordinates of V, then the associated global 1-forms {dz_{1}, dz_{2}}, respectively \{d\overline{z}_{1}, d\overline{z}_{2}\}, form a ℂ-basis of T, respectively of \overline{T} , and

(5.6) H^{1,1}(A) = ⟨dz_{1} ∧ d\overline{z}_{1}, dz_{1} ∧ d\overline{z}_{2}, dz_{2} ∧ d\overline{z}_{1}, dz_{2} ∧ d\overline{z}_{2}⟩_{ℂ}
= \{ω ∈ H^{2}(A, ℂ) : ω ∧ dz_{1} ∧ dz_{2} = \overline{ω} ∧ dz_{1} ∧ dz_{2} = 0\}.

3. The Chern class map c_{1} : Pic(A) = H^{1}(A, O^{∗}) → H^{2}(A, ℤ) induces an isomorphism NS(A) ≃ H^{2}(A, ℤ) ∩ H^{1,1}(A) (5.7)

which we will henceforth view as an identification (cf. [BPV], p. 119). Thus, by (5.6) we have (when g = 2)

NS(A) = {ω ∈ H^{2}(A, ℤ) : ω ∧ dz_{1} ∧ dz_{2} = 0}. (5.8)

4. If A is an abelian surface, let μ ∈ H^{4}(A, ℤ) = H_{0}(A, ℤ) ≃ ℤ be the cohomology class defined by the canonical orientation of A. For ω_{1}, ω_{2} ∈ H^{2}(A, ℤ) = ∧^{2}H^{1}(A, ℤ) define their “intersection number” ⟨ω_{1}, ω_{2}⟩ ∈ ℤ by the rule ω_{1} ∧ ω_{2} = ⟨ω_{1}, ω_{2}⟩μ. (5.9)

If D_{1}, D_{2} ∈ Div(A) are divisors on A, then their usual intersection number is related to the one just defined by the expected formula:

(D_{1}.D_{2}) = ⟨c_{1}(L(D_{1})), c_{1}(L(D_{2}))⟩; (5.10)

cf. [BPV], p. 66. Thus, if θ = c_{1}(L(Θ)) denotes the Chern class of an ample divisor Θ of degree δ = \frac{1}{2}(Θ.Θ), then we can define the degree and Humbert invariant of ω ∈ H^{2}(A, ℤ) by

deg(ω) = ⟨ω, θ⟩ and Δ(ω) = deg(ω)^{2} − 2δ⟨ω, ω⟩.

We now want write down explicit formulae for these invariants in terms of the coordinates of a period lattice of an abelian surface A. To this end we fix the following notation.

Notation 5.2 Fix a primitive ample divisor Θ on A of degree δ with associated cohomology class θ = c_{1}(L(Θ)) ∈ H^{2}(A, ℤ) = Alt^{2}(U, ℤ). Viewing θ as a (non– degenerate) alternating bilinear form θ(x, y) on U, we can choose a symplectic basis {u_{1}, u_{2}, u_{3}, u_{4}} of U with respect to θ, i.e. a basis such that

θ (∑ n_{1j}u_{j}, ∑ n_{2j}u_{j}) = ∑_{j=1}^{2} δ_{j}(n_{1j}n_{2j+2} − n_{2j}n_{1j+2}), (5.11)

where 1 ≤ δ_{1} | δ_{2}; cf. [GH], p. 304. Thus, in terms of the dual basis x_{1}, …, x_{4} of Hom_{ℂ}(U, ℂ) with associated 1-forms dx_{1}, …, dx_{4} ∈ H^{1}(A, ℂ) = H^{1}(A, ℤ) ⊗ ℂ we can express θ in the form

θ = δ_{1}dx_{1} ∧ dx_{3} + δ_{2}dx_{2} ∧ dx_{4}. (5.12)

It thus follows from (5.9) and (5.10) that the orientation class μ is given by

μ = \frac{1}{2δ} θ ∧ θ = −\frac{δ_{1}δ_{2}}{2δ} dx_{1} ∧ dx_{2} ∧ dx_{3} ∧ dx_{4} = −dx_{1} ∧ dx_{2} ∧ dx_{3} ∧ dx_{4}, (5.13)

where the latter equality follows because μ and −dx_{1} ∧ dx_{2} ∧ dx_{3} ∧ dx_{4} are both generators of H^{4}(A, ℤ) = ℤ. We note that since Θ was assumed to be primitive in NS(A), we must have δ_{1} = 1, as is evident from (5.7), and hence δ_{2} = δ.

Moreover, we can choose a ℂ-basis {e_{1}, e_{2}} of V such that the period matrix Ω = (ω_{jk}), defined by u_{k} = ∑_{j} ω_{jk}e_{j}, has the form

Ω = ( \begin{matrix}1 & 0 & τ_{1} & τ_{2} \\ 0 & δ & τ_{2} & τ_{3} \end{matrix} ) ,

where the matrix \binom{τ_{1} τ_{2}}{τ_{2} τ_{3}} has positive definite imaginary part; cf. [GH], pp. 304– 306. The associated 1-forms dz_{1}, dz_{2}, d\overline{z}_{1}, d\overline{z}_{2} ∈ H^{1}(A, ℂ) are then related to the basis dx_{1}, …, dx_{4} by the formulae (cf. [GH], p. 304):

\begin{aligned}dz_{1} & = & dx_{1} + τ_{1}dx_{3} + τ_{2}dx_{4}, & d\overline{z}_{1} & = & dx_{1} + \overline{τ}_{1}dx_{3} + \overline{τ}_{2}dx_{4} \\ dz_{2} & = & δdx_{2} + τ_{2}dx_{3} + τ_{3}dx_{4}, & d\overline{z}_{2} & = & δdx_{2} + \overline{τ}_{2}dx_{3} + \overline{τ}_{3}dx_{4}. \end{aligned}

In terms of the above bases, the Humbert invariant Δ may be expressed as follows.

Lemma 5.3 Let ω ∈ H^{2}(A, ℤ), and write

ω = ∑_{i<j} a_{ij}dx_{i} ∧ dx_{j} = \frac{1}{2} ∑_{i,j} a_{ij}dx_{i} ∧ dx_{j}. (5.14)

Then the following formulae are valid:

(5.15) deg(ω) = δa_{13} + a_{24},
(5.16)  ⟨ω, ω⟩ = −2pf((a_{ij})) = −2(a_{12}a_{34} − a_{13}a_{24} + a_{14}a_{23}),
(5.17)  Δ(ω) = (δa_{13} − a_{24})^{2} + 4δ(a_{12}a_{34} + a_{14}a_{23}).

Moreover, ω ∈ NS(A) if and only if

δa_{34} − δa_{14}τ_{1} + (δa_{13} − a_{24})τ_{2} + a_{23}τ_{3} + a_{12}(τ_{1}τ_{3} − τ_{2}^{2}) = 0. (5.18)

Proof. In view of (5.9), (5.12) and (5.13) we have

ω ∧ θ = −(δa_{13} + a_{24})dx_{1} ∧ dx_{2} ∧ dx_{3} ∧ dx_{4} = (δa_{13} + a_{24})μ,

which proves (5.15). Moreover, a short calculation yields

ω ∧ ω = 2(a_{12}a_{34} − a_{13}a_{24} + a_{14}a_{23})dx_{1} ∧ dx_{2} ∧ dx_{3} ∧ dx_{4},

and so (5.16) follows since, as is well–known, the Pfaffian of a 4 × 4 skewsymmetric matrix (a_{ij}) is given by the indicated formula. Thus, from (5.15) and (5.16) we obtain

Δ(ω) = (δa_{13} + a_{24})^{2} − 2δ(−2(a_{12}a_{34} − a_{13}a_{24} + a_{14}a_{23}))
= (δa_{13} − a_{24})^{2} + 4δ(a_{12}a_{34} + a_{14}a_{23}),

which shows that (5.17) is valid.

To prove the last assertion, we first note that

(5.19)dz_{1} ∧ dz_{2} = (dx_{1} + τ_{1}dx_{3} + τ_{2}dx_{4}) ∧ (δdx_{2} + τ_{2}dx_{3} + τ_{3}dx_{4})
= δdx_{1} ∧ dx_{2} + τ_{2}dx_{1} ∧ dx_{3} + τ_{3}dx_{1} ∧ dx_{4} − δτ_{1}dx_{2} ∧ dx_{3}
−δτ_{2}dx_{2} ∧ dx_{4} + (τ_{1}τ_{3} − τ_{2}^{2})dx_{3} ∧ dx_{4},

and so

ω ∧ dz_{1} ∧ dz_{2} = (δa_{34} − δa_{14}τ_{1} + (δa_{13} − a_{24})τ_{2} + a_{23}τ_{3} + a_{12}(τ_{1}τ_{3} − τ_{2}^{2}))
·dx_{1} ∧ dx_{2} ∧ dx_{3} ∧ dx_{4}.

In view (5.8), we thus see that ω ∈ NS(A) if and only if (5.18) holds.

The above lemma can be rephrased in the following more convenient way.

Proposition 5.4 There exists a basis v_{1}, …, v_{5} of H(A, θ) := H^{2}(A, ℤ)/ℤθ such that, under the resulting identification H(A, θ) ≃ ℤ^{5}, the induced quadratic form Δ on H(A, θ) assumes the form

Δ(a, b, c, d, e) = b^{2} − 4δ(ac + de) (5.20)

and such that the group NS(A, Θ) = NS(A)/ℤΘ, viewed as a subgroup of H(A, θ) via (5.7), is given by

NS(A, Θ) = {(a, b, c, d, e) : aδτ_{1} + bτ_{2} + cτ_{3} + d(τ_{2}^{2} − τ_{1}τ_{3}) + eδ = 0}. (5.21)

Proof. Put v_{1} := −dx_{1}∧dx_{4}, v_{2} := −dx_{2}∧dx_{4}, v_{3} := dx_{2}∧dx_{3}, v_{4} := −dx_{1}∧dx_{2}, and v_{5} := dx_{3} ∧ dx_{4}. Then v_{1}, …, v_{5}, θ is a basis of H^{2}(A, ℤ); more precisely, each ω ∈ H^{2}(A, ℤ) can be written in the form

ω = av_{1} + bv_{2} + cv_{3} + dv_{4} + ev_{4} + fθ,

with a = −a_{14}, b = δa_{13} − a_{24}, c = a_{23}, d = −a_{12}, e = a_{34}, and f = a_{13} in the notation of (5.14). Substituting these expressions in (5.17) and (5.18) and passing to the respective quotients yields the assertions.

We can now relate the algebraic definition of a singular relation (cf. Definition 3.3) to the analytic definition of Humbert[Hu1], p. 297, 301.

Corollary 5.5 The primitively polarized abelian surface (A, Θ) of degree δ satisfies a singular relation with invariant N if and only if there are integers a, b, c, d, e ∈ ℤ with (a, b, c, d, e) = 1 such that

b^{2} − 4δ(ac + de) = N, (5.22)

and aδτ_{1} + bτ_{2} + cτ_{3} + d(τ_{1}τ_{3} − τ_{2}^{2}) + eδ = 0, (5.23)

where \binom{τ_{1} τ_{2}}{τ_{2} τ_{3}} is a period matrix of (A, Θ).

Proof. Put ω = av_{1} + … + ev_{5} ∈ H^{2}(A, ℤ). Then by Proposition 5.4 we have that ω ∈ NS(A) if and only if (5.23) holds. If this is the case, then ω is primitive in NS(A) if and only if ω is primitive in H^{2}(A, ℤ) if and only if (a, b, c, d, e) = 1. Moreover, by (5.20) we have Δ(ω) = N if and only if (5.22) holds.

Remark 5.6 The above Proposition 5.4 and its Corollary 5.5 show that the Definition of a “singular relation with invariant Δ” as defined by Humbert[Hu1], pp. 297 and 301, coincides with the algebraic definition presented in Definition 3.3 above (for δ = 1). Moreover, since the latter definition does not depend on the choice of the period lattice, it is clear that Δ(a, b, c, d, e) is an invariant of the polarized abelian variety (A, Θ), and so the extensive calculations of Humbert[Hu1], pp. 298–301 are superfluous. Finally, the above formulae also show that the definition of a Humbert surface as defined by van der Geer[vdG], p. 211 is essentially identical to the algebraic definition given above (cf. Definition 3.7), except that van der Geer uses a different normalization of the period matrix so that his formulae are slightly different.

We now want to present an analytic proof of Theorem 2.8. This depends on the following interesting “dictionary” presented in Proposition 5.9 below which relates the elliptic subgroups E ≤ A of a complex abelian surface A to certain submodules M of H^{1}(A, ℤ) which we shall call elliptic submodules. These submodules are also closely connected to certain subspaces W_{M} of H^{1,0}(A) = H^{0}(A, Ω_{A}) and H_{M} of H^{1}(A, ℂ) which we shall call elliptic differentials and elliptic subspaces respectively; they are defined as follows.

Definition 5.7 Let X be a compact algebraic variety. Then an elliptic submodule of X is a rank 2 submodule M ⊂ H^{1}(X, ℤ) which is primitive (i.e. H^{1}(X, ℤ)/M is torsionfree) and which satisfies

W_{M} := (ℂ ⊗ M) ∩ H^{1,0}(X) ≠ 0. (5.24)

An elliptic differential on an algebraic variety X is a one–dimensional ℂsubspace W = ℂω ⊂ H^{1,0}(X) of the space H^{1,0}(A) = H^{0}(X, Ω_{X}^{1}) of holomorphic differentials such that

M_{W} = (W + \overline{W}) ∩ H^{1}(X, ℤ) (5.25)

is a submodule of rank 2 of H^{1}(X, ℤ).

An elliptic subspace of X is a two-dimensional ℂ-subspace H ⊂ H^{1}(X, ℂ) which has a Hodge structure of weight 1 (cf. [BPV], p. 36) compatible with that of H^{1}(X, ℂ), i.e. we have

H = (H ∩ H^{1}(X, ℤ)) ⊗ ℂ = (H ∩ H^{1,0}(X)) ⊕ (H ∩ H^{0,1}(X)). (5.26)

The set of all elliptic submodules (respectively, of all elliptic differentials, respectively, of all elliptic subspaces) of X will be denoted by Ell(X) (respectively, by Ell_{d}(X), respectively by Ell_{sp}(X)).

Remark 5.8 a) The above three concepts “elliptic submodule”, “elliptic differential” and “elliptic subspace” are essentially equivalent notions. Indeed, if W is an elliptic differential, then H_{W} := W ⊕ \overline{W} = ℂ ⊗ W_{M} is an elliptic subspace and hence M_{W} is an elliptic submodule. Similarly, if M is an elliptic submodule, then H_{M} := ℂ ⊗ M = W_{M} ⊗ \overline{W}_{M} is an elliptic subspace, so W_{M} is an elliptic differential. Finally, if H is an elliptic subspace, then W_{H} = H ∩ H^{1,0}(A) and M_{H} = H ∩ H^{1}(A, ℤ) are an elliptic differential and an elliptic submodule, respectively. Thus, the maps M ↦ W_{M} and W ↦ H_{W} define bijections between the sets Ell(X), Ell_{d}(X) and Ell_{sp}(X) of elliptic submodules, elliptic differentials and elliptic subspaces of A.

b) If X is a complex curve (i.e. X is a compact Riemann surface), then the quotient Jac(X) = H^{1}(X, Ω_{X}^{1})^{∗}/H_{1}(X, ℤ) is an abelian variety. (Note that Jac(X) = Ĵ_{X} is the dual of the Jacobian defined earlier.) Moreover, for each P ∈ X, the morphism j_{P} : X → Jac(X) defined by j_{P} (Q)(ω) = ∫_{P}^{Q} ω induces an isomorphism

j^{∗} = j_{P}^{∗} : H^{1}(Jac(X), ℂ) →^{∼} H^{1}(X, ℂ)

which does not depend on P and which is compatible with the integral structures and with the Hodge decompositions. Thus, j^{∗} induces a natural identification

Ell(X) = Ell(Jac(X))

between the set of elliptic submodules of X and those of Jac(X), and similarly, the sets of elliptic differentials (respectively, elliptic subspaces) of X and Jac(X) correspond.

We now relate the elliptic submodules of a complex abelian surface A its elliptic subgroups E ≤ A. For the sake of clarity and since it is it involves no extra work, we shall do this more generally for an abelian variety of arbitrary dimension. In this general case, however, one has work with the co-elliptic subgroups, i.e. with abelian subvarieties B ≤ A of codimension 1 of A (so E = A/B is an elliptic curve), in place of the elliptic subgroups of A.

Proposition 5.9 a) Let M ∈ Ell(A) be an elliptic submodule of a complex abelian variety A = V/U. Using the identification H^{1}(A, ℂ) = Hom_{ℝ}(V, ℂ), put V_{M} = M^{⊥} = {v ∈ V : m(v) = 0, ∀m ∈ M}. (5.27)

Then V_{M} is a ℂ-subspace of V of codimension 1 and hence B_{M} = (V_{M}+U)/U ≤ V/U = A is a co-elliptic subgroup of A. Moreover, the Chern class of B_{M}, viewed as a divisor on A, is given by

c_{1}(L(B_{M})) = ω_{M} := ω_{1} ∧ ω_{2}, (5.28)

where {ω_{1}, ω_{2}} is any suitably ordered basis of M.

b) The above map M ↦ B_{M} is a bijective correspondence between the set Ell(A) of elliptic submodules of A and the set of co-elliptic subgroups of A. Moreover, the inverse map is given by B ↦ M_{B} = p^{∗}H^{1}(A/B, ℤ), where p : A → A/B = E denotes the quotient map, and similarly, the map B ↦ W_{B} = p^{∗}H^{1}(A/B, Ω_{A/E}^{1}) is the inverse of the map W ↦ B_{M_{W}}.

Proof. a) By construction, U ∩ V_{M} is a (full) lattice in V_{M}, which is a real subspace of V of (real) codimension 2, so the first assertion follows once we have shown that V_{M} is a complex subspace of V. For this, it is enough to show that V_{M} is stable under the map I : V → V defined by multiplication by i = \sqrt{−1}. Now since T and \overline{T} are the ±1-eigenspaces of the dual map I^{∗} ∈ End_{ℝ}(H^{1}(A, ℂ)), it follows that H_{M} ∩ T and H_{M} ∩ \overline{T} are stable under I^{∗}, and hence so is H′ := (H_{M} ∩ T) + (H_{M} ∩ \overline{T}). Since M is an elliptic submodule we have H′ = H_{M}, and so V_{M} = M^{⊥} = (H_{M})^{⊥} is stable under I. This proves the first statement; the second one (i.e. (5.28)) will be proved at the end.

b) It is enough to show that the map B ↦ M_{B} is the inverse of M ↦ B_{M}. For this, let B ≤ A be a co-elliptic subgroup. Then B = (Ṽ + U)/U, where Ṽ is a ℂ-subspace of V of codimension 1 such that Ũ = U ∩ Ṽ is a lattice in Ṽ. Put V′ = V/Ṽ and U′ = (Ṽ + U)/Ṽ. Then E′ = V′/U′ is an elliptic curve, and we have an exact sequence

0 → B →^{i} A →^{p} E′ → 0.

Using the identification (5.1) we obtain an induced exact sequence

0 → H_{1}(B, ℤ) = Ũ →^{i_{∗}} H_{1}(A, ℤ) = U →^{p_{∗}} H_{1}(E′, ℤ) = U′ → 0,

which by dualizing yields the exact sequence

0 → H^{1}(E′, ℤ) →^{p^{∗}} H^{1}(A, ℤ) →^{i^{∗}} H^{1}(B, ℤ) → 0.

We thus see that M_{B} = p^{∗}H^{1}(E′, ℤ) is a direct factor of H^{1}(A, ℤ) and hence is a primitive submodule. Moreover, since p^{∗} is injective, M_{B} has rank 2. Finally, since p is a holomorphic map, the induced map p^{∗} respects the Hodge decompositions, and so M_{B} is an elliptic submodule. It is immediate from the above construction that B ↦ M_{B} is inverse to M ↦ B_{M}, and so b) follows.

Finally, to prove (5.28) we note first that since B = p^{−1}(D) is the pullback of the divisor D = 0_{E′} on E′ consisting of the point 0_{E′}, it follows that c_{1}(L(B)) = p^{∗}c_{1}(L(D)). Since its deg_{E′}(D) = 1, we have c_{1}(D) = ω′_{1} ∧ ω′_{2}, for any oriented basis {ω_{1}, ω_{2}} of H^{1}(E′, ℤ), and so (5.28) follows.

We can supplement this dictionary by defining the degree of an elliptic submodule.

Definition 5.10 Let Θ be an ample divisor on A with associated Chern class θ = c_{1}(L(Θ)) ∈ H^{2}(A, ℤ) = ∧^{2}H^{1}(A, ℤ). If M ⊂ H^{1}(A, ℤ) is elliptic submodule with an oriented basis {ω_{1}, ω_{2}}, then its degree deg_{θ}(M) is defined by

deg_{θ}(M) = \frac{1}{(g − 1)!} ∫_{A} θ^{g−1} ∧ ω_{1} ∧ ω_{2}.

Note that this is a positive integer because if D = Θ_{|B_{M}} then by (5.28)

deg_{θ}(M) = \frac{1}{(g − 1)!}(Θ^{g−1}.B_{M}) = \frac{1}{(g − 1)!}(D)^{g−1} = χ(L(D)) (5.29)

by the Riemann–Roch Theorem on B_{M}. In particular, if A is an abelian surface, then deg_{θ}(M) = (Θ.B_{M}) = deg_{Θ}(B_{M}) as defined above.

Applying this to the case of a Jacobian A = Jac(X), we obtain the following result, which may be viewed as a sharpening of a classical result due to Weierstraß; cf. Krazer[Kr], ch. XII, Satz I, p. 271:

Corollary 5.11 If X is a complex curve, then the assignment (f : X → E) ↦ M_{f} := f^{∗}H^{1}(E, ℤ) induces a natural bijection between the isomorphism classes of elliptic subcovers of X of degree n and the elliptic submodules M ⊂ H^{1}(X, ℤ) ≃ H^{1}(Jac(X), ℤ) of degree n.

Proof. If M is an elliptic submodule of A = Jac(X), then the projection map p = p_{M} : A → E_{M} = A/B_{M} does not factor over any isogeny, so f_{M} = p_{M} ∘ j_{P} is an elliptic subcover. Clearly the assignment M ↦ f_{M} is inverse to f ↦ M_{f}, so both are bijections. To see that they are degree–preserving, note first that

deg(f_{M}) = deg f_{M}^{∗}(0) = deg(j_{P}^{∗} (B_{M})) = (X.B_{M}).

By Poincaré’s formula (cf. [GH], p. 350), X ≡ \frac{1}{(g−1)!}Θ_{C}^{g−1}, so we see by (5.29) that (X.B_{M}) = deg_{Θ}(B_{M}) = deg_{θ}(M), and hence deg(f_{M}) = deg_{θ}(M), as desired.

We now want to formulate an analytic version of Theorem 2.8. This depends on the notion of a “primitively decomposable” p-form, which is defined as follows.

Definition 5.12 Let M be a free ℤ-module of finite rank. A p-vector ω ∈ ∧^{p}M is called primitively decomposable if there exist m_{1}, …, m_{p} ∈ M with M/(∑ ℤm_{i}) torsionfree such that ω = m_{1} ∧ … ∧ m_{p}.

We note the following two elementary facts, of which the second is essentially a restatement of Grassmann’s relation for n = 4 and p = 2; cf. Bourbaki[Bou], III.11.13 (p. 611).

Lemma 5.13 a) Each primitively decomposable ω ∈ ∧^{p}M is primitive in ∧^{p}M.

b) Suppose rank(M) = 4 and p = 2. Then ω ∈ ∧^{2}M is primitively decomposable if and only if ω is primitive and ω ∧ ω = 0.

We are now ready to give a quick proof of the following analytic version of Theorem 2.8.

Theorem 5.14 Let A be a complex abelian surface with ample divisor Θ. Then there are natural degree–preserving bijections between the following sets:

1) the set of elliptic subgroups E ≤ A;

2) the set Ell(A) of elliptic submodules;

3) the set of primitively decomposable 2-forms ω ∈ H^{2}(A, ℤ) = ∧^{2}H^{1}(A, ℤ) with ω ∧ dz_{1} ∧ dz_{2} = 0 and ⟨ω, θ⟩ > 0;

4) the set of primitive divisor classes cl(D) ∈ NS(A) with (D.D) = 0 and deg_{Θ}(D) > 0.

Proof. The bijection between 1) and 2) was already established in Proposition 5.9.

Next, consider the map M ∈ Ell(A) ↦ ω_{M}. Clearly, ω_{M} is a primitively decomposable 2-form. Moreover, ω_{M} ∈ H^{1,1}(A) by (5.28), so ω_{M} ∧dz_{1}∧dz_{2} = 0 by (5.6). Thus, ω_{M} belongs to the set described in 3). Conversely, suppose ω = ω_{1} ∧ ω_{2} is as in 3) and put M_{ω} = ℤω_{1} + ℤω_{2}. Since ω_{1} ∧ ω_{2} ∧ dz_{1} ∧ dz_{2} = 0, we must have a relation a_{1}ω_{1} + a_{2}ω_{2} + a_{3}dz_{1} + a_{4}dz_{2} = 0, with a_{1}, …, a_{4} ∈ ℂ not all zero. This means that (ℂ ⊗ M) ∩ H^{1,0}(A) ≠ 0, so M_{ω} is an elliptic submodule. It is easily checked that the map ω ↦ M_{ω} is inverse to M ↦ ω_{M}, so both define bijections between the sets 2) and 3) which are degree–preserving by (5.29).

To construct a bijection between the sets 4) and 3), let D be as in set 4), and put ω_{D} = c_{1}(L(D)) ∈ NS(A) = H^{2}(A, ℤ) ∩ H^{1,1}(A). Since ω_{D} is primitive in NS(A), it is also primitive in H^{2}(A, ℤ). Thus, since ω_{D} ∧ ω_{D} = 0, it follows from Lemma 5.11b) that ω_{D} is primitively decomposable and hence lies in the set 3). Conversely, each ω in the set 3) lies in NS(A) = H^{2}(A, ℤ) ∩ H^{1,1}(A), and hence is of the form ω = c_{1}(L(D)) for some divisor D. By Lemma 5.11a), ω is primitive in H^{2}(A, ℤ) and hence also in NS(A). Moreover, since clearly ω ∧ ω = 0, we have (D.D) = 0. It is thus immediate that the map cl(D) ↦ ω_{D} is the desired (degree–preserving) bijection between the sets 4) and 3).

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Ernst Kani Department of Mathematics and Statistics Queen’s University Kingston, Ontario K7L 3N6 Canada