Hurwitz spaces of genus 2 covers of an elliptic curve

Ernst Kani

1 Introduction

Let E be an elliptic curve over a field K of characteristic ≠ 2 and let N > 1 be an integer prime to char(K). The purpose of this paper is to study the family of genus 2 covers of E of fixed degree N, i.e. those covers f : C → E for which C/K is a curve of genus 2 and deg(f) = N. Since we can (without loss of generality) restrict our attention those covers that are normalized in the sense of section 2, this investigation is essentially equivalent to the study of the set

Cov_{E/K,N}(K) := {f : C → E : g_{C} = 2, deg(f) = N and f is normalized}/≃

of isomorphism classes of normalized genus 2 K-covers of E of degree N, where, as usual, two covers f_{i} : C_{i} → E are called isomorphic (f_{1} ≃ f_{2}) if there is an isomorphism φ : C_{1} → C_{2} such that f_{1} = f_{2} ∘ φ.

The main result here is that this set can be naturally identified with the set H_{E/K,N}(K) of K-rational points of an explicit affine curve H_{E/K,N}/K, and that this identification also holds for families of such covers (over an arbitrary base). More precisely, we first show in section 3 that the assignment L ↦ Cov_{E/K,N}(L) extends in a natural way to a Hurwitz functor H_{E/K,N} : \underline{Sch}_{/K} → \underline{Sets} which is analogous to the Hurwitz functors considered by Fulton[Fu] (where the base is ℙ^{1}). The main result of this paper can then be stated as follows.

Theorem 1.1 If N ≥ 3, then the functor H_{E/K,N} is finely represented by a smooth, affine and geometrically connected curve H_{E/K,N}/K which is an open subset of a certain twist X_{E/K,N,−1} of the modular curve X(N) of level N; in particular, H_{E,N} ⊗ \overline{K} is isomorphic to an open subset of X(N)_{/\overline{K}}, where K denotes the algebraic closure of K.

Remarks. 1) As the proof shows, exactly the same assertion holds for elliptic curves E/K over an arbitrary ring (or scheme) K; cf. Theorem 5.18.

2) The “twisted modular curve” X_{E/K,N,−1}/K is defined and constructed in section 4 as the moduli space of the functor X_{E/K,N,−1} which assigns to any extension field L/K the set X_{E/K,N,−1}(L) of isomorphism classes of pairs (E′, ψ) where E′/L is an elliptic curve and ψ : E[N] →^{∼} E′[N] is an L-rational anti-isometry of the N-torsion subgroups of E and E′.

3) The above representability results are obtained by purely algebraic techniques and hence do not use (not even implicitly) the Riemann Existence Theorem.

The above theorem constitutes a refinement and extension of the “basic construction” of genus 2 curves (with elliptic differentials) which was presented in [FK], [Ka2]. Indeed, this construction shows that each genus 2 cover f : C → E of degree N determines a unique pair (E′_{f}, ψ) ∈ X_{E/K,N,−1}(K) and that conversely for each such pair (satisfying a suitable additional hypothesis) one can reconstruct the cover f : C → E. Thus, the “basic construction” defines for each extension field L/K an injection

Ψ_{L} : H_{E/K,N}(L) ↪ X_{E/K,N,−1}(L) = X_{E/K,N,−1}(L)

and identifies the image. Here we shall see in Theorem 5.18 that this idea can be refined to obtain an open embedding of functors

Ψ : H_{E/K,N} ↪ X_{E/K,N,−1}.

However, instead of proving this directly, it is more convenient to divide the above construction into two steps, which amounts to a factorization of Ψ as

Ψ = Ψ′ ∘ τ : H_{E/K,N} ↪ A_{E/K,N} →^{∼} X_{E/K,N,−1},

in which τ is (essentially) the Torelli map which associates to a curve its (polarized) Jacobian and A_{E/K,N} is the functor that classifies principally polarized abelian surfaces with an embedding “of degree N” of E (cf. subsection 5.1). The fact that Ψ′ is an isomorphism is proved in Theorem 5.10 and that τ is a monomorphism is shown in Proposition 5.12. Finally, the image of τ is analyzed in Proposition 5.17: it is the subfunctor J_{E/K,N} of A_{E/K,N} consisting of the “theta-smooth” elements (cf. subsection 5.3).

The fact that the Hurwitz functor H_{E/K,N} is representable by an open subset of the modular curve X_{E/K,N,−1} has a number of interesting consequences. For example, it shows that over any number field K there are only finitely many genus 2 covers f : C → E of fixed degree N ≥ 7 because the genus of (the compactification of) X_{E/K,N,−1} is ≥ 2 for N ≥ 7 and so #X_{E/K,N,−1}(K) < ∞ by Faltings’ Theorem. This means in particular that one cannot write down parametric families of such covers when N ≥ 7, which is in sharp contrast to the fact that for N ≤ 5 (in which case X_{E/K,N,−1} is a rational curve) such families are known to exist and can be written down explicitly; cf. Krazer[Kr], p. 477ff, Kuhn[Ku], p. 48, and Frey[Fr2], p. 96ff. Another application of the explicit description of the Hurwitz space H_{E/K,N} is given in [Ka4], where it is used to compute the number of genus 2 covers of E/K with a given discriminant divisor.

In order to complete the description of the Hurwitz space H_{E/K,N}, one should also describe the complement

D_{E/K,N} = X_{E/K,N,−1} \ H_{E/K,N},

which can be viewed as the “degeneracy locus” of the basic construction. (Alternately, one can view D_{E/K,N} as the boundary of the Hurwitz space H_{E/K,N}; cf. [FKV].) However, since a description of D_{E/K,N} was already given in essence in [Ka2], we merely need to cite and/or translate the relevant result; cf. Theorem 6.1 below. Similarly, the number of points in D_{E/K,N} was computed in [Ka2] and [Ka3] (cf. Theorem 6.2); a slightly weaker version of this result may be stated as follows:

Theorem 1.2 If K is algebraically closed, then the number of points in the degeneracy locus satisfies the inequality

#D_{E/K,N} ≤ \frac{1}{24N} (5N − 6)#SL_{2}(ℤ/Nℤ). (1)

Furthermore, equality holds in (1) if and only if char(K)|̸ N!, i.e. if and only if either char(K) = 0 or char(K) > N.

Acknowledgments. This paper developed out the joint article [FK] with G. Frey, whom I would like to thank very much for the many stimulating and fruitful discussions we have had over the last decade as well as for his continued interest in this research. In addition, I would like to thank him for his kind hospitality at the Institut für Experimentelle Mathematik, where parts of this paper were written.

Finally, I would like to gratefully acknowledge receipt of funding from the Natural Sciences and Engineering Research Council of Canada (NSERC) and also from the Deutsche Forschungsgemeinschaft (DFG – Forschergruppe und Graduiertenkolleg Essen) which made this research possible.

Contents

1 Introduction  1
2 Normalized genus 2 covers  5
3 Families of genus 2 covers and the Hurwitz functor H_{E/K,N}  14
4 The modular curves X_{E/K,N} and X_{E/K,N,ε}  21
5 The basic construction  24
5.1 The functor A_{E/K,N}  24
5.2 The isomorphism Ψ′ : A_{E/K,N} → X_{E/K,N,−1}  25
5.3 The Torelli map τ : H_{E/K,N} → A_{E/K,N}  35
6 The degeneracy locus D_{E/K,N}  42
7 Appendix: Jacobians of relative curves  44
References  53

2 Normalized genus 2 covers

As in the introduction, let E/K be an elliptic curve over a field K. It will be convenient (but not absolutely necessary) to assume in the sequel that char(K) ≠ 2 so as to avoid case distinctions.

A covering f : C → E is called minimal (or optimal ([Ku]) or maximal ([Se])) if the induced map f^{∗} : J_{E} → J_{C} on the Jacobians is a closed immersion. Note that if C is a curve of genus 2, then this is equivalent to the condition that f does not factor over an isogeny of E of degree ≥ 2 (use Kuhn[Ku], Corollary on p. 45). Thus, by replacing E/K by an isogenous curve if necessary, we may always assume “without loss of generality” that f is minimal.

Minimal genus 2 covers of elliptic curves are partially analogous to the simple covers of ℙ^{1} studied by Fulton[Fu]; for example, they do not have any internal automorphisms, as the following result shows.

Proposition 2.1 Let f : C → E be a minimal genus 2 cover of E/K of degree N ≥ 3. If α ∈ Aut_{K}(C) is an automorphism such that f ∘α = f, then α = id_{C}.

Proof. Without loss of generality, we may assume that K is algebraically closed. Let G = ⟨α⟩ be the (finite) group generated by α. If α ≠ id_{C}, then the quotient map π : C → \overline{C} = G\C has degree deg(π) ≥ 2 and f factors over π, i.e. f = f′ ∘ π. By Riemann-Hurwitz we then have 2 > g\overline{C} ≥ g_{E} = 1, so g\overline{C} = 1. Since f is minimal, this forces f′ to be an isomorphism, i.e. f is a (ramified) Galois cover with group G. Furthermore, if we replace α by a power α^{n}, then the same argument shows that G cannot have any proper subgroup, and hence N = |G| is prime.

Since f is Galois, the degree of its different satisfies deg(Diff(f)) ≥ N ∑_{P ∈R_{f}} (1 − \frac{1}{e_{P} (f)}) where R_{f} = \{P ∈ E(K) : e_{P} (f) > 1\}. (Note that equality holds if char(K)|̸ N.) Thus, by the Riemann-Hurwitz relation we obtain 2 = deg(Diff(f)) ≥ \frac{N}{2} #R_{f} ≥ \frac{N}{2} , and hence N ≤ 4.

Since N is prime, this leaves only the case N = 3. In this case there are three fibres of f which have isolated Weierstrass points in them (see the proof of Prop. 2.2 below). Thus, each of these points is fixed by α, and so we obtain deg(Diff(f)) ≥ 3, which contradicts the Riemann-Hurwitz relation.

It is immediate that if f : C → E is a minimal genus 2 cover of E/K, then so is T_{x}∘f, for any x ∈ E(K), where T_{x} : E → E denotes the translation map.

Since T_{x} ∘ f is essentially the same cover as f, it is useful to “normalize” the genus 2 covers of E/K in a certain way so as to avoid redundant translates. This will be done by means of the Weierstrass (or hyperelliptic) divisor of C.

Notation. If C/K is a curve of genus 2, then σ_{C} denotes its hyperelliptic involution and W_{C} its hyperelliptic divisor. Recall that W_{C} is the divisor of fixed points of σ_{C} and is thus an effective divisor of degree 6. Furthermore, W_{C} is reduced if and only if char(K) ≠ 2; cf. Lønsted-Kleiman[LK].

Similarly, if E/K is an elliptic curve with zero 0_{E}, then the minus map [−1]_{E} has the divisor E[2] of 2-torsion points as its fixed point divisor. Note that the divisor [0_{E}] defined by the point 0_{E} is contained in E[2], and so E[2]^{#} := E[2] − [0_{E}] is an effective divisor of degree 3 on E.

Definition. A morphism f : C → E is called normalized if it is minimal and if the norm (or direct image) f_{∗}W_{C} of the hyperelliptic divisor has the form

f_{∗}W_{C} = 3ε[0_{E}] + (ε + 1)E[2]^{#} (2)

where ε = 0 if deg(f) is even and ε = 1 if deg(f) is odd. Thus, deg(f) ≡ ε(2), and we have more explicitly f_{∗}W_{C} = 3[0_{E}] + E[2]^{#}, if deg(f) is odd, and f_{∗}W_{C} = 2E[2]^{#}, if deg(f) is even.

This terminology is justified by the following result:

Proposition 2.2 If f : C → E is a minimal genus 2 cover, then there is a unique point x ∈ E(K) such that f_{norm} := T_{x} ∘f is normalized. Furthermore, f_{norm} is “pseudo-normalized” in the sense that it satisfies the relation

f_{norm} ∘ σ_{C} = [−1]_{E} ∘ f_{norm}. (3)

Proof. It is well-known (cf. e.g. [Ku], p. 42) that there is a (unique) involution σ_{E} ∈ Aut(E) such that f ∘ σ_{C} = σ_{E} ∘ f. If \overline{K} denotes the algebraic closure of K, then the unique extension σ_{C} ⊗ \overline{K} of σ_{C} to C ⊗ \overline{K} has four distinct fixed points P_{0}, P_{1}, P_{2}, P_{3} ∈ E(\overline{K}), which can be numbered in such a way that

f\overline{K}_{∗}W_{C\overline{K}} = 3εP_{0} + (ε + 1)(P_{1} + P_{2} + P_{3}), (4)

where f\overline{K} : C\overline{K} = C ⊗ \overline{K} → E\overline{K} = E ⊗ \overline{K} denotes the induced cover over \overline{K} (and where, as above, deg(f) ≡ ε(2)). Indeed, if ε = 1 then this follows from [FK], Lemma 2.1, or from Kuhn[Ku], p. 44, and if ε = 0, then this follows from the discussion of [Ku] on the top of p. 48.

From the above description is clear that P_{0} is Gal(\overline{K}/K)-stable and hence is rational over K. Thus, the first assertion holds with x = P_{0} (and only for this point). Furthermore, in [FK] and/or [Ku] it was already noted the second assertion follows, for we have f_{norm}∘σ_{C} = σ′_{E}∘f_{norm}, where σ′_{E} = T_{x}∘σ_{E}∘T_{x}^{−1}, and σ′_{E} = [−1]_{E} because the fixed point set of σ′_{E} is E[2].

Corollary 2.3 Let f : C → E be a minimal genus 2 cover of E/K. Then f is normalized if and only if f satisfies (3) and

#(f^{−1}(0_{E}) ∩ W_{C}) = 3ε. (5)

Proof. If f is normalized, then clearly (5) holds by definition (cf. equation (2). Furthermore, Proposition 2.2 shows that f satisfies (3).

Conversely, suppose f satisfies (3). Then the proof of Prop. 2.2 shows that the points P_{0}, …, P_{3} of (4) are fixed under [−1]_{E} and hence are 2-torsion points. Furthermore, equation (5) guarantees that P_{0} = 0_{E}, and so (2) holds.

As we shall see below in Theorem 2.6, normalized covers f : C → E are intimately connected to the induced homomorphisms f_{∗} : J_{C} → J_{E} and/or f^{∗} : J_{E} → J_{C} on the Jacobians. For this we first show:

Proposition 2.4 Let f : C → E be a normalized genus 2 cover of degree N, and let λ_{E} : E → Ê = J_{E} be the canonical polarization on E which is defined by λ_{E}(P) = cl(O_{E}([P] − [0_{E}])).

(a) If N is odd, then there is a unique effective divisor W′_{0} ≤ W_{C} of degree 3 such that f_{∗}W′_{0} = E[2]^{#}.

(b) If N is even, and the Weierstrass points of C are K-rational, then there is a divisor W′_{0} ≤ W_{C} with f_{∗}W′_{0} = E[2]^{#}. Furthermore, if W̃′_{0} is another divisor with this property, then cl(O_{C}(W′_{0} − W̃′_{0})) ∈ Ker(f_{∗})[2].

(c) Let W′_{0} ≤ W_{C} be a divisor such that f_{∗}W′_{0} = E[2]^{#}, and put L = O_{C}(W′_{0}) ⊗ ω_{C}^{−1}, where ω_{C} denotes the canonical sheaf of C/K. Then the map x ↦ cl(O_{C}(x − W′_{0}) ⊗ ω_{C}) = cl(O_{C}(x) ⊗ L^{−1}) defines an embedding j = j_{L} : C ↪ J_{C} which satisfies

0_{J} ∉ j(C) and j ∘ σ_{C} = [−1]_{J} ∘ j and λ_{E} ∘ f = f_{∗} ∘ j. (6)

Proof. (a) Let W_{0} = f^{∗}(0_{E}) ∩ W_{C} and W′_{0} = W_{C} − W_{0}. Then W_{0} and W′_{0} are both effective divisors and f_{∗}W_{0} = 3(0_{E}) by construction (and (2)) and hence deg(W_{0}) = deg(W′_{0}) = 3 and also f_{∗}W′_{0} = E[2]^{#}, as claimed.

(b) The existence of W′_{0} is clear. If W̃′_{0} is another divisor, then we have cl(2(W′_{0} − W̃′_{0})) = ω_{C}^{⊗3} ⊗ (ω_{C}^{⊗3})^{−1} ≃ O_{C} (cf. part(c)), so cl(W′_{0} − W̃′_{0}) ∈ J_{C}[2]. Moreover, f_{∗}(cl(O_{C}(W′_{0}−W̃′_{0}))) = cl(O(f_{∗}W′_{0}−f_{∗}W̃′_{0})) = 0, so cl(W′_{0}−W̃′_{0}) ∈ Ker(f_{∗})[2].

(c) Since deg(O_{C}(W′_{0}) ⊗ ω_{C}^{−1}) = 1, the rule x ↦ cl(O_{C}(x − W′_{0}) ⊗ ω_{C}) is represented by a closed immersion j : C ↪ J_{C}.

We now verify that j satisfies (6). For this we may assume that K is algebraically closed, so W′_{0} = W_{1} + W_{2} + W_{3} with W_{i} ∈ C(K).

To prove the first equation of (6) we shall use the fact that for each P ∈ C(K), the divisor P + σ_{C}(P) ∼ ω_{C} is a canonical divisor on C. In particular, for each Weierstrass point W_{i} ∈ C(K) we have 2W_{i} ∼ ω_{C}, and so 2W′_{0} ∼ ω_{C}^{⊗3}. Thus j(P) + j(σ_{C}(P)) = cl(O_{C}(P + σ_{C}(P) − 2W′_{0}) ⊗ ω_{C}^{⊗2}) = 0, which proves the first equality of (6).

To prove the second equality, we first note that the points P_{i} := f(W_{i}), for i = 1, 2, 3 are (by construction) precisely the non-trivial 2-torsion points of E, and so cl(O_{C}(P_{2} − P_{3})) = cl(O_{C}(P_{1} − 0_{E})) = λ_{E}(P_{1}). Moreover, by the Albanese property of J_{C} we know that f_{∗}∘j = T_{x}∘λ_{E}∘f, for some x ∈ J_{E}(K). Thus, since 2W_{2} ∼ ω_{C}, we see that f_{∗}(j(W_{1})) = f_{∗}(cl(O_{C}(W_{1} − W_{1} + W_{2} − W_{3}))) = cl(O_{C}(P_{2} − P_{3})) = λ_{E}(P_{1}) = λ_{E}(f(W_{1})), and so T_{x}(λ_{E}f(W_{1})) = f_{∗}(j(W_{1})) = λ_{E}(f(W_{1})). This forces x = 0 and so f_{∗} ∘ j = λ_{E} ∘ f.

Next we observe that 0_{J} ∉ j(C). Indeed, if 0_{J} ∈ j(C), then there is a point P ∈ C(K) such that P ∼ O_{C}(W′_{0}) ⊗ ω_{C}^{−1} ∼ W_{1} − W_{2} + W_{3}, which is impossible as h^{0}(W_{1} − W_{2} + W_{3}) = 0 (because h^{0}(W_{1} + W_{2}) = 1).

Corollary 2.5 The curve θ := j(C) ⊂ J_{C} of Proposition 2.4(c) is a symmetric theta-divisor of J_{C}, i.e. θ is an effective divisor on J_{C} such that [−1]_{J_{C}}θ = θ and such that λ_{O(θ)} = λ_{C} : J_{C} →^{∼} Ĵ_{C} is the canonical principal polarization of J_{C}. Furthermore:

(a) If N is odd, then θ is the unique symmetric theta divisor such that

(7) θ ∩ Ker(f_{∗})[2] = Ker(f_{∗})[2]^{#} := Ker(f_{∗})[2] − [0].

(b) If N is even, then θ is a symmetric theta divisor satisfying

θ ∩ Ker(f_{∗})[2] = ∅. (8)

Moreover, if θ′ is any symmetric theta-divisor on J_{C} satisfying (8), then θ′ = T_{x}(θ), for some x ∈ Ker(f_{∗})[2].

Proof. Since j is of the form P ↦ cl(O(P) ⊗ L^{−1}), where L is a suitable invertible sheaf of degree 1, it is clear that θ is an effective theta-divisor. Furthermore, θ is symmetric by the first equation of (6).

(a) To prove that θ satisfies (7) we may assume without loss of generality that K is algebraically closed. We first observe the first equation of (6) shows that j(P) ∈ J_{C}[2] if and only if P is a Weierstrass point.

We now claim that #Ker(f_{∗})[2] = 4. This follows either by observing that Ker(f_{∗}) is an elliptic curve (cf. Prop. 2.7 below) or by noting that (6) shows that f_{∗}(J_{C}[2]) = J_{E}[2] and hence it follows that #Ker(f_{∗})[2] = #(Ker(f_{∗}) ∩ J_{C}[2]) = #J_{C}[2]/#J_{E}[2] = 4.

Thus, if W_{01}, W_{02}, W_{03} ∈ C(\overline{K}) are the three Weierstrass points of C such that f(W_{0i}) = 0_{E}, then by (6) we have Ker(f_{∗})[2] = {0_{J}, j(W_{01}), …, j(W_{03})}, and so (7) follows.

Now let θ′ be another theta-divisor of J_{C} satisfying (7). Then θ′ = T_{x}θ, for some x, and so θ′ is also irreducible. Moreover, by (7) we have θ′ ∩ θ ⊃ Ker(f_{∗})[2]^{#}, which has 3 distinct points (over \overline{K}), and so (θ′.θ) ≥ 3, if θ′ ≠ θ. But (θ′.θ) = (θ)^{2} = 2 (by Riemann-Roch), contradiction. Thus θ′ = θ.

(b) A slight modification of the proof of part (a) shows that θ satisfies (8) and that (f_{∗})(θ ∩ J[2]) = E[2]^{#} (as sets). Now suppose θ′ is another symmetric theta-divisor satisfying (8). Then θ′ = T_{x}θ, for some x ∈ J_{C}[2] (because θ and θ′ are both symmetric). If x ∉ Ker(f_{∗}), then f_{∗}(x) ∈ E[2]^{#}, and so there exists y ∈ θ ∩ J[2] such that f_{∗}(y) = f_{∗}(x). But then x + y ∈ T_{x}θ ∩ Ker(f_{∗})[2] = θ′ ∩ Ker(f_{∗})[2], contradiction. Thus x ∈ Ker(f_{∗}), as claimed.

The above proposition and its corollary lead to the fundamental fact that normalized covers can be characterized by their induced homomorphisms on the Jacobians. To state this more precisely, it is useful to introduce some equivalence relations on covers and on homomorphisms.

Definition. Two (normalized) K-covers f_{i} : C_{i} → E of genus 2 are called equivalent if there exists an a K-isomorphism α : C_{1} →^{∼} C_{2} such that f_{1} = f_{2} ∘ α. If such an α exists, then we write f_{1} ≃ f_{2}.

Two injective homomorphims h_{i} : J_{E} → J_{i} := J_{C_{i}} are called equivalent if there exists a K-isomorphism α : J_{1} →^{∼} J_{2} such that h_{2} = α ∘ h_{1} and α̂ ∘ λ_{2} ∘ α = λ_{1}, where λ_{i} = λ_{C_{i}} : J_{i} →^{∼} Ĵ_{i} is the canonical polarization of the Jacobian J_{i}, for i = 1, 2. We write h_{1} ≃ h_{2} if such an isomorphism α exists.

Finally, we say that an injective homomorphism h : E → J_{C} has degree N if ĥ ∘ λ_{C} ∘ h = λ_{J_{E}} ∘ [N]_{J_{E}}; we then write N = deg_{λ_{C}}(h).

Theorem 2.6 Let C/K be a curve of genus 2 and let N ≥ 2 be an integer. Then the assignment f ↦ f^{∗} induces a surjection

ρ = ρ_{C,E,N} : Cov_{K}(C, E, N) → InjHom_{K}(J_{E}, J_{C}, N)

from the set Cov_{K}(C, E, N) of equivalence classes of normalized K-covers f : C → E of degree N to the set InjHom_{K}(J_{E}, J_{C}, N) of equivalence classes of injective K-homomorphisms h : J_{E} ↪ J_{C} of degree N. Furthermore, ρ is a bijection if N > 2 or if the Weierstrass points of C are K-rational.

Proof. First note that if f : C → E is a normalized cover, then f^{∗} defines an equivalence class cl(f^{∗}) in the set InjHom_{K}(J_{E}, J_{C}, N). Indeed, f^{∗} is injective because f is minimal, and f^{∗} has degree N because f_{∗} = λ_{J_{E}}^{−1} ∘ (f^{∗})^{}ˆ ∘ λ_{C}, and so (f^{∗})^{}ˆ ∘ λ_{C} ∘ f^{∗} = λ_{J_{E}} ∘ (f_{∗} ∘ f^{∗}) = λ_{J_{E}} ∘ [N]_{J_{E}}. Furthermore, the rule f ↦ cl(f^{∗}) is compatible with the equivalence relation on Cov_{K}(C, E, N) because if f_{1} ≃ f_{2}, then f_{1} = f_{2} ∘ α for some α ∈ Aut_{K}(C), and so f_{1}^{∗} = (f_{2} ∘ α)^{∗} = α^{∗} ∘ f_{2}^{∗}. Clearly, α^{∗} ∈ Aut_{K}(J_{C}) and we have (α^{∗})^{}ˆ ∘ λ_{C} ∘ α^{∗} = λ_{C} ∘ (α_{∗} ∘ α^{∗}) = λ_{C} ∘ [deg(α)] = λ_{C}. This means that f_{1}^{∗} ≃ f_{2}^{∗} and so the rule f ↦ cl(f^{∗}) defines a map ρ : Cov_{K}(C, E, N) → InjHom_{K}(J_{E}, J_{C}, N).

We first show that ρ is surjective. Thus, let h : E ↪ J_{C} be an injective homomorphism of degree N. Assume temporarily that C/K satisfies:

(†) W_{C} ⊂ C(K) and J_{C}[2] ⊂ J_{C}(K).

Since the Weierstrass points of C are now assumed to be rational, there exists an embedding j = j_{P} : C ↪ J_{C} such that θ := j(C) is a symmetric thetadivisor on J_{C}; cf. Prop. 2.4 and Cor. 2.5. Then [−1]_{J}θ = θ, and so [−1]_{J} induces a unique automorphism σ on C such that j ∘ σ = [−1]_{J} ∘ j. It is then immediate that σ has the Weierstrass points as fixed points and hence σ = σ_{C} is the hyperelliptic involution on C.

Define f := f_{h} : C → E by f = h^{∗} ∘ j where h^{∗} = ĥ ∘ λ_{C} : J_{C} → Ĵ_{E} = E. Thus, since h^{∗} is a homomorphism we have [−1]_{E} ∘ f = [−1]_{E} ∘ h^{∗} ∘ j = h^{∗} ∘ [−1]_{J} ∘ j = h^{∗} ∘ j ∘ σ_{C} = f ∘ σ_{C}, so f is pseudo-normalized, i.e. satisfies condition (3). By Prop. 2.2 we know that there exists a (unique) point x ∈ E(K) such that f_{norm} := T_{x} ∘f is normalized. Since f and f_{norm} both satisfy (3), it follows that [−1]_{E}x = x, i.e. x ∈ E[2]. Now since h^{∗} : J[2] → E[2] is surjective (because h^{∗} is surjective and the fibres of h^{∗} are connected), there is a point x′ ∈ J[2] such that h^{∗}(x′) = x, and then f_{norm} = h^{∗} ∘ j′, where j′ = T_{x′} ∘ j. Using the autoduality property of the Jacobian, i.e., the fact that (j′)^{∗} = j^{∗} = −λ_{C}^{−1} : Ĵ_{C} → J_{C}, we obtain by dualizing the above relation that f_{norm}^{∗} = (j′)^{∗} ∘ (h^{∗})^{∗} = −h. Thus, f_{norm} : C → E is a normalized cover such that f_{norm}^{∗} = −h ≃ h, i.e. τ is surjective (provided that condition (†) holds).

To finish the proof of the surjectivity of ρ, we now remove the hypothesis (†) and hence consider an arbitrary curve C/K. Then there exists a finite Galois cover K′/K such that C_{K′} = C ⊗ K′ satisfies (†) and so, by what was just shown, there exists a normalized map f = f_{h} : C_{K′} → E_{K′} such that f^{∗} = −h. Furthermore, f = λ_{E}^{−1} ∘ h^{∗} ∘ j, where j = j_{L} is the embedding defined by a suitable L ∈ Pic(C′_{K}). Consider the Galois twist f^{g} of f by g ∈ Gal(K′/K). Since j^{g} = T_{x(g)} ∘ j with x(g) = cl(L^{g} ⊗ L^{−1}) ∈ Pic^{0}(C′_{K}) = J(K′), we obtain f^{g} = λ_{E}^{−1} ∘ h^{∗} ∘ T_{x(g)} ∘ j. Now θ′ := T_{x(g)}(j(C)) = j(C)^{g} is again a symmetric theta-divisor of λ_{C} which satisfies θ′ ⊃ Ker(h^{∗})[2]^{#} (resp., θ∩Ker(h^{∗})[2] = ∅) if N is odd (resp. if N is even) because Ker(h^{∗})^{g} = Ker(h^{∗}). Thus, θ′ = j(C) (resp. θ′ = T_{x′}j(C) with x′ ∈ Ker(h^{∗})[2]) and so x(g) = 0 (resp. x(g) = x′ ∈ Ker(h^{∗})[2]). Thus, in both cases h^{∗} ∘ T_{x(g)} = h^{∗}, and so f^{g} = f, for all g ∈ Gal(K′/K). This means that f is defined over K, and so τ is surjective in general.

It remains to show that ρ is injective (under the stated hypotheses). For this, suppose that f_{i} : C → E are two normalized covers such that f_{1}^{∗} ≃ f_{2}^{∗}; we then want to show that also f_{1} ≃ f_{2}. The hypothesis f_{1}^{∗} ≃ f_{2}^{∗} means that there exists an α ∈ Aut_{K}(J_{C}) with property that f_{1}^{∗} = α ∘ f_{2}^{∗} and that α̂ ∘ λ_{C} ∘ α = λ_{C}. We then also have (f_{1})_{∗} ∘ α = (f_{2})_{∗}, for by dualizing the first relation we obtain (f_{1})_{∗} = λ_{J_{E}}^{−1} ∘ (f_{1}^{∗})^{}ˆ ∘ λ_{C} = λ_{J_{E}}^{−1} ∘ (f_{2}^{∗})^{}ˆ ∘ α̂ ∘ λ_{C} = λ_{J_{E}}^{−1} ∘ (f_{2}^{∗})^{}ˆ ∘ λ_{C} ∘ α^{−1} = (f_{2})_{∗} ∘ α^{−1}, as claimed.

To show that f_{1} ≃ f_{2}, suppose first that N is odd. Then by Prop. 2.4 there exists a K-embedding j_{i} : C ↪ J_{C} such that λ_{E} ∘ f_{i} = (f_{i})_{∗} ∘ j_{i} and such that θ_{i} = j_{i}(C) is theta divisor associated to λ_{C}, for i = 1, 2. Since f_{i} is normalized, we have by Cor. 2.5(a) that θ_{i} ⊃ Ker((f_{i})_{∗})[2]^{#}. Now α^{−1}(θ_{1}) is again a symmetric theta-divisor (associated to λ_{C}) and α^{−1}(θ_{1}) ⊃ α^{−1}(Ker((f_{1})_{∗})[2]^{#}) = Ker((f_{1})_{∗} ∘ α)[2]^{#} = Ker((f_{2})_{∗})[2]^{#}, so θ_{2} = α^{−1}(θ_{1}) because θ_{2} is uniquely characterized by this property. Thus φ := ((j_{2})^{−1} ∘ α^{−1} ∘ j_{1} ∈ Aut_{K}(C) satisfies j_{2} ∘ φ = α^{−1} ∘ j_{1}, and so by the above identity we have λ_{E} ∘ f_{1} = (f_{1})_{∗} ∘ j_{1} = (f_{2})_{∗} ∘ α^{−1} ∘ j_{1} = (f_{2})_{∗} ∘ j_{2} ∘ φ = λ_{E} ∘ f_{2} ∘ φ, or f_{1} = f_{2} ∘ φ with φ ∈ Aut_{K}(C), i.e. f_{1} ≃ f_{2}.

Next assume that N is even. Then by Prop. 2.4 there exists a finite Galois extension K′/K such that over K′ we have embeddings j_{i} : C′ = C ⊗ K′ ↪ J′ = J_{C′} such that λ_{E} ∘ f_{i} = (f_{i})_{∗} ∘ j_{i} and such that θ_{i} = j_{i}(C) are theta divisors associated to λ_{C} (defined over K′). Then by Cor. 2.5(b) we have θ_{i} ∩ Ker((f_{i})_{∗})[2] = ∅, and so also α^{−1}(θ_{1}) ∩ Ker((f_{2})_{∗})[2] = α^{−1}(θ_{1}) ∩ Ker((f_{1})_{∗} ∘ α)[2] = α^{−1}(θ_{1} ∩ Ker((f_{1})_{∗})[2]) = ∅. Thus, by the uniqueness property of θ_{2} it follows that θ_{2} = T_{x}(α^{−1}(θ_{1})) for some x ∈ Ker((f_{2})_{∗})[2], and so φ := (j_{2})^{−1} ∘ T_{x} ∘ α^{−1} ∘ j_{1} ∈ Aut_{K′}(C′) satisfies j_{2} ∘ φ = T_{x} ∘ α^{−1} ∘ j_{1}. Now since x ∈ Ker((f_{2})_{∗}), we have (f_{2})_{∗} ∘T_{x} = (f_{2})_{∗}, and so (f_{2})_{∗} ∘α^{−1} ∘j_{1} = (f_{2})_{∗}∘j_{2}∘φ = λ_{E}∘f_{2}∘φ. Thus λ_{E}∘f_{1} = (f_{1})_{∗}∘j_{1} = (f_{2})_{∗}∘α^{−1}∘j_{1} = λ_{E}∘f_{2}∘φ, which means that f_{1} = f_{2} ∘ φ, with φ ∈ Aut_{K′}(C′).

To conclude that f_{1} ≃ f_{2}, we still have to show that φ is defined over K. This is automatic if the Weierstrass points are defined over K, for then we can choose K′ = K. Thus, assume that this is not the case but that N > 2. Let g ∈ Gal(K′/K), and consider the Galois twist φ^{g} ∈ Aut_{K′}(C′) of φ. Since the f_{i}’s are defined over K we have f_{1} = f_{2} ∘ φ^{g}, and so, if φ′ = φ^{g} ∘ φ^{−1}, then we have f_{2} ∘ φ′ = f_{1} ∘ φ^{−1} = f_{2}. Now since N ≥ 3, we have by Prop. 2.1 that φ′ = id_{C′}, and so φ^{g} = φ, for all g ∈ Gal(K′/K), which means that φ ∈ Aut_{K}(C). Thus f_{1} ≃ f_{2}, as desired, and so ρ is injective.

Remark. It is easy to see that the sets of Theorem 2.6 are finite; in fact, the cardinality of Cov_{K}(C, E, N) can bounded by the number of (primitive) representations of N^{2} by a suitable positive definite quadratic form associated to C (multiplied by \frac{1}{2}#Aut(E)); cf. [Ka1], Theorem 4.5.

As was mentioned in [FK] or [Ku], each minimal genus 2 cover f : C → E induces a splitting of the Jacobian J_{C} up to isogeny, i.e. J_{C} ∼ E × E′, and the complementary elliptic curve E′ = E′_{f} can be chosen in a canonical way by using the (canonical) principal polarization λ_{C} : J_{C} →^{∼} Ĵ_{C} of C. This curve E′_{f} plays an important role in the “basic construction” of [FK], which will be reviewed (and extended) in section 5.

Proposition 2.7 If f : C → E is a minimal genus 2 cover of degree N, then E′ := Ker(f_{∗}) is an elliptic curve such that

E′ ∩ f^{∗}J_{E} = f^{∗}J_{E}[N] = E′[N]. (9)

Thus, if π : J_{E} × E′ → J_{C} denotes the unique map such that π ∘ i_{J_{E}} = f^{∗} and π ∘ i_{E′} = −h′, where i_{J_{E}} : J_{E} ↪ J_{E} × E′, i_{E′} : E′ ↪ J_{E} × E′, and h′ : E′ = Ker(f_{∗}) ↪ J_{C} denote the canonical inclusions, then Ker(π) = f^{∗}J_{E} ∩ E′ and hence π is an isogeny of degree N^{2}; in particular, J_{C} ∼ J_{E} × E′ ≃ E × E′.

Proof. Since f_{∗} = λ_{E}^{−1} ∘ (f^{∗})^{}ˆ ∘ λ_{C} is the “dual” of f^{∗} and since f^{∗} is a closed immersion, it follows that f_{∗} is surjective and has (geometrically) connected fibres of dimension (dim J_{C} − dim J_{E}) = 1 (cf. section 7 below). Thus, E′ is an elliptic curve.

Moreover, since f^{∗} is an injection and since E′ = Ker(f_{∗}), we have E′ ∩ f^{∗}J_{E} = f^{∗}Ker(f_{∗} ∘ f^{∗}) = f^{∗}([N]_{J_{E}}) = f^{∗}(J_{E}[N]). Thus, E′ ∩ f^{∗}J_{E} is a finite group (scheme) of order N^{2} and of exponent N, and so E′ ∩ f^{∗}J_{E} ≤ E′[N]. But E′[N] also has order N^{2}, and so E′ ∩ f^{∗}J_{E} = E′[N], which proves the first statement. The second statement follows immediately from the first.

Corollary 2.8 The above embedding h′ : E′ ↪ J_{C}, has degree N, and hence there exists a “complementary” minimal K-cover f′ : C → E′ of degree N such that (f′)^{∗} = h′ ∘ λ_{E′}^{−1}.

Proof. By Theorem 2.6 there is a normalized K-cover f′ : C → E′ of degree N′ = deg_{λ_{C}}(h′) such that (f′)^{∗} = h′ ∘ λ_{E′}^{−1}, and then (f′)_{∗} = λ_{E′} ∘ (h′)^{∗}, where (h′)^{∗} = λ_{E′}^{−1} ∘ (h′)^{}ˆ ∘ λ_{C} : J_{C} → E′.

Recall that the sequence

0 → E′ →^{h′} J_{C} →^{f_{∗}} J_{E} → 0

is exact by the definition of h′. Thus, dualizing this sequence and applying λ_{C}, λ_{E′} and λ_{J_{E}}(= λ_{E}^{−1}) yields the sequence

0 → J_{E} →^{f^{∗}} J_{C} →^{(h′)^{∗}} E′ → 0

which is again exact (cf. [La], p. 216); here we have used the identification f^{∗} = λ_{C}^{−1} ∘ (f_{∗})^{}ˆ ∘ λ_{J_{E}}. This means in particular that Ker(f′_{∗}) = Ker((h′)^{∗}) = f^{∗}J_{E}, and thus, if we now apply Prop. 2.7 to f′, then we obtain

E′ ∩ f^{∗}J_{E} = (f′)^{∗}J_{E′} ∩ Ker(f′_{∗}) = E′[deg(f′)].

On the other hand, since f^{∗}J_{E} ∩ E′ = E′[N] by (9), we conclude that N = deg(f′) = deg_{λ_{C}}(h′), as desired.

3 Families of genus 2 covers and the Hurwitz functor H_{E/K,N}

We now want to study families of normalized genus 2 covers over an arbitrary base scheme S, i.e. (normalized) covers f : C → E where E/S is a (relative) elliptic curve and C/S is a relative curve of genus 2 in the sense of the appendix (cf. section 7). For this, we first observe that every genus 2 curve C/S is hyperelliptic in the sense of Lønsted/Kleiman[LK], p. 101:

Lemma 3.1 If p : C → S is a relative curve of genus 2, then there exists a unique S-automorphism σ_{C/S} ∈ Aut_{S}(C) which induces the hyperelliptic involution on each fibre C_{s} of p.

Proof. If ω = ω_{C/S} denotes the relative canonical sheaf of C/S, then p_{∗}(ω_{C/S}) is locally free of rank g = 2, and the canonical S-morphism φ_{ω} : C → ℙ(ω_{C/S}) is surjective (because this is true fibre-by-fibre), and so condition (i) of Th. 5.5 of [LK] holds, which means that C/S is hyperelliptic.

As in section 2, let E/K be an elliptic curve over a field K with char(K) ≠ 2 (or, more generally, over any ring (or any scheme) K in which 2 is invertible). For any K-scheme S let E_{S} := E ×_{K} S be the elliptic curve over S obtained from E/K by base-change.

Definition. A genus 2 cover f : C → E_{S} of E_{S}/S of degree N is called normalized if it is minimal (cf. §7) and if the direct image f_{∗}W_{C/S} of the hyperelliptic divisor W_{C/S} (cf. [LK]) has the form

f_{∗}(W_{C/S}) = 3ε[0_{E_{S}/S}] + (ε + 1)E_{S}[2]^{#} (10)

where (as before) ε = 0 if N is even and ε = 1 if N is odd, and [0_{E_{S}/S}] denotes the Cartier divisor associated the zero-section 0_{E_{S}/S} and E_{S}[2]^{#} := E_{S}[2] − [0_{E/S}] (viewed as effective relative Cartier divisors on E_{S}).

The basic properties of normalized genus 2 covers of E_{S}/S are summarized in the following theorem.

Theorem 3.2 (a) If f : C → E_{S} is a normalized genus 2 cover of degree N, then so is any base-change f_{(T)} : C_{T} = C ×_{S} T → E_{T} = E_{S} ×_{S} T.

(b) If C/S is flat and locally of finite presentation, and if S is reduced, then an S-morphism f : C → E_{S} is a normalized genus 2 cover of degree N if and only if f_{s} : C_{s} → E_{s} = E ⊗ κ(s) is a normalized genus 2 cover of degree N, for all s ∈ S.

(c) If f : C → E_{S} is a normalized genus 2 cover then we have

f ∘ σ_{C/S} = [−1]_{E_{S}} ∘ f. (11)

Conversely, if f : C → E_{S} is a minimal cover satisfying (11) and if for at least one s ∈ S the induced map f_{s} : C_{s} → E_{s} is normalized, then f is normalized.

(d) Let f : C → E_{S} be a normalized genus 2 cover of degree N. If N is odd, or if N is even and C/S has 6 distinct Weierstrass sections, then there is a closed S-immersion j : C ↪ J = J_{C/S} of C into its Jacobian (cf. §7) such that

j ∘ σ_{C} = [−1]_{J} ∘ j, λ_{E} ∘ f = f_{∗} ∘ j, and 0_{J}(S) ∩ j(C) = ∅. (12)

In particular, there is always a suitable etale faithfully flat base change S′/S such that there exists an immersion j : C_{(S′)} ↪ J_{(S′)} satisfying (12).

(e) In the situation of (d), the image θ = j(C) ⊂ J is a symmetric theta-divisor on J. If N is odd, then θ is the unique symmetric theta-divisor satisfying

(13) θ ∩ Ker(f_{∗})[2] = Ker(f_{∗})[2]^{#} := Ker(f_{∗})[2] \ [0].

If N is even, then θ is a symmetric theta divisor satisfying

θ ∩ Ker(f_{∗})[2] = ∅. (14)

Moreover, if θ′ is any symmetric theta-divisor on J satisfying (14), then θ′ = T_{x}(θ), for some x ∈ Ker(f_{∗})[2](S′).

(f) Let C/S be a curve of genus 2. If f : C → E_{S} is a minimal cover of degree N, then f^{∗} : J_{E_{S}/S} → J_{C/S} is an injective homomorphism of degree N, i.e. we have λ_{E_{S}/S}^{−1} ∘(f^{∗})^{}ˆ ∘λ_{C/S} ∘f^{∗} = [N]_{J_{E_{S} /S}}. Conversely, if h : J_{E_{S}} ↪ J_{C/S} is an injective homomorphism of degree N, then there exists a normalized cover f : C → E_{S} of degree N such that f^{∗} = h.

Proof. (a) First note that f_{(T)} is minimal because f is (cf. §7). Moreover, since the formation of W_{C/S} commutes with base-change (cf. [LK], Prop. 6.3), and since the same is true for the direct image of relative Cartier divisors by Lemma 7.2 of the appendix, we see that the analogue of (10) also holds for f_{(T)}, and so f_{(T)} is also normalized.

(b) If f is a normalized genus 2 cover, then so is f_{s}, ∀s ∈ S, by part (a). Conversely, if f_{s} : C_{s} → E_{s} is a genus 2 cover of degree N for all s ∈ S, then C/S is a smooth curve of genus 2 (use [BLR], 2.4/8), and so f is a genus 2 cover of degree N which is minimal (cf. §2.2). Moreover, since by hypothesis (and base-change) we have (f_{∗}W_{C/S})_{s} = (f_{s})_{∗}W_{C_{s}/κ(s)} = 3ε[0_{E_{s}/κ(s)}] + (ε + 1)E_{s}[2]^{#} = (3ε[0_{E_{S}/S}] + (ε + 1)E_{S}[2]^{#})_{s}, for all s ∈ S, we can conclude from (46) that (10) holds, and so f is normalized.

(c) Since it is enough to prove (11) after a faithfully flat base-change, we may assume that there exists a Weierstrass section w of C/S. Then P := f(w) ∈ E_{S}[2](S) because [P] = f_{∗}[w] ≤ f_{∗}W_{C/S} ≤ 3E_{S}[2].

Write f_{1} = f ∘ σ_{C/S} and f_{2} = [−1]_{E_{S}} ∘ f. Then f_{i} : C → E_{S} are two Smorphisms which by Prop. 2.2 satisfy (f_{1})_{s} = (f_{2})_{s}, ∀s ∈ S, and so by rigidity we there is a section η ∈ E_{S}(S) such that f_{1} = (η ∘p_{C})+f_{2}; cf. [Mu1], p. 116. But since σ_{C/S} (resp. [−1]) fixes w (resp. P), we have f_{1}(w) = P = f_{2}(w) and so η = 0_{E_{S}}, which means that f_{1} = f_{2}, as desired.

We now prove the converse. Again, it is enough to prove this after a faithfully flat base change (because distinct relative Cartier divisors stay distinct after a faithfully flat base change). Thus, we may assume that

(†) C/S has six Weierstrass sections and J_{C/S}/S has sixteen 2-torsion sections.

Then E_{S}/S has four 2-torsion sections P_{0} = 0, P_{1}, P_{2}, P_{3} (because (†) implies that f_{∗} maps J_{C/S}[2](S) surjectively to J_{E_{S}/S}[2](S) as f_{∗} is surjective with integral fibres). Thus, if w_{1}, …, w_{6} denote the Weierstrass sections, then we have by (11) that f(w_{i}) ∈ E_{S}[2](S) and so f_{∗}W_{C/S} = ∑ f_{∗}[w_{j}] = ∑_{i=0}^{3} n_{i}[P_{i}]. Specializing this equation at the given s ∈ S yields (f_{s})_{∗}W_{C_{s}} = (f_{∗}W_{C/S})_{s} = ∑ n_{i}[(P_{i})_{s}]. But since the 2-torsion sections of E_{S}/S are mapped injectively to those of E_{s}/κ(s), it follows that (P_{0})_{s} = 0, (P_{1})_{s}, …, (P_{3})_{s} are precisely the 2-torsion points of E_{s}. Thus, since f_{s} is normalized we obtain n_{0} = 3ε and n_{i} = (ε + 1), for i > 0, and so f is also normalized.

(d) Suppose first that N is odd, and put W_{0} := f^{∗}([0_{E_{S}}]) ×_{C} W_{C/S}. We claim that W_{0} is an effective relative Cartier divisor on C/S of degree 3. For this we shall use Lemma 7.3 of the appendix. Since the formation of W_{0} commutes with base-change, we see by Prop. 2.4(a) that deg((W_{0})_{s}) = 3 for all s ∈ S. Choose a finite faithfully flat base-change S′/S such that we have 6 distinct Weierstrass sections w_{i} ∈ W_{C_{S′}/S′}. By the proof of (c) we know that P_{i} = f(w_{i}) ∈ E_{S}[2](S′) and so by specializing to any fibre we see that (after renumbering) 0_{E_{S}}, P_{1}, P_{2}, P_{3} are distinct and that P_{4} = P_{5} = P_{6} = 0_{E_{S}}.

Thus, w_{4}, w_{5}, w_{6} are 3 sections of W_{0}, and so by Lemma 7.3 we conclude that W_{0} is a relative Cartier divisor of C/S.

Thus, W′_{0} = W_{C/S} − W_{0} is a also relative Cartier divisor of degree 3 and we have f_{∗}W′_{0} = E_{S}[2]^{#} (since this is obviously true after a suitable faithfully flat base change). Put L := O_{C}(W′_{0}) ⊗ ω_{C/S}^{−1} ∈ Pic(C). Then L has relative degree 1, and therefore gives rise to a closed embedding j_{L} : C → J (cf. §7). By Prop. 2.4, properties (12) hold fibre-by-fibre, and so by the same argument as in (c) we conclude that (12) holds over S.

If N is even, and we have 6 Weierstrass sections of C/S, then we choose 3 of these to obtain a relative Cartier divisor W′_{0} with the property that (f_{∗}W′_{0})_{s} = E_{s}[2]^{#}, for one fixed s ∈ S. It then follows that f_{∗}W′_{0} = E_{S}[2]^{#} because both sides are sums of 2-torsion sections, and the 2-torsion sections of E_{S}/S are mapped bijectively to those of the fibre E_{s}/κ(s). Then the invertible sheaf L = O_{C}(W′_{0}) ⊗ ω_{C/S}^{−1} defines an embedding j = j_{L} : C → J which (by the same argument as in the case that N is odd) satisfies (12).

The last assertion clearly follows from the first because we can always choose a suitable (étale) faithfully flat base-change so as to obtain 6 distinct Weierstrass sections (since W_{C/S} is etale).

(e) Since j = j_{L} is the embedding defined by an invertible sheaf L ∈ Pic(C) of relative degree 1 = g −1, it is clear that θ = j(C) is a theta-divisor (cf. §7). Furthermore, [−1]θ = j(σ_{C}(C)) = θ by the first equation of (12), and so θ is symmetric.

Next we show that if N is odd, then θ satisfies (13). For this we first note that Ker(f_{∗})[2] is a finite etale group scheme of rank 4 (because Ker(f_{∗})/S is an elliptic curve by (42); cf. also Prop. 5.2 below). Furthermore, since it is enough to verify (13) after a faithfully flat base-change, we may assume without loss of generality that W_{C/S} has six Weierstrass sections w_{1}, …, w_{6}. Then by the discussion of (e) we see that Ker(f_{∗})[2](S) = {0, j(w_{4}), j(w_{5}), j(w_{6})} and so (13) follows.

To see that θ is uniquely characterized by this property, we argue as follows. If θ′ is another symmetric theta-divisor, then by Lemma 7.1 of the appendix we know that θ′ = T_{x}(θ), for some x ∈ J_{C/S}(S). Moreover, x ∈ J_{C/S}[2](S) since θ and θ′ are both symmetric. By specializing to a fibre J_{s} we conclude by the uniqueness assertion of Cor. 2.5 that θ′_{s} = T_{x(s)}(θ_{s}) and that hence x(s) = 0 = 0(s). But then x = 0 since J_{C/S}[2] is a finite etale group scheme, and so θ′ = θ. This proves the assertion in case that N is odd, and the case that N is even is proved similarly.

(f) Since the first assertion is just a restatement of (44), we only need to prove the second statement.

Suppose first that C/S satisfies (†), and let w be a Weierstrass section. Then the image θ = j_{w}(C) of the embedding j_{w} : C ↪ J_{C/S} is a symmetric theta-divisor of λ_{C/S}, and so there is a unique automorphism σ ∈ Aut_{S}(C) such that j_{w} ∘ σ = [−1] ∘ j_{w}. Clearly, σ fixes the 6 Weierstrass sections and hence σ = σ_{C} is the hyperelliptic involution. Put f′ = h^{∗} ∘ j_{w}, where h^{∗} = ĥ ∘ λ_{C/S} : J_{C/S} → Ĵ_{E_{S}/S} = E_{S}. Then f′ ∘ σ_{C} = (h^{∗}) ∘ [−1]_{J} ∘ j_{w} = [−1]_{E_{S}} ∘ (h^{∗}) ∘ j_{w} = [−1]_{E_{S}} ∘ f′, so f′ satisfies (11). Furthermore, (f′)^{∗} = j_{w}^{∗} ∘ (h^{∗}) = −λ_{C/S}^{−1} ∘ (λ_{C/S} ∘ h) = −h, so f′ is minimal.

We now claim that there exists an x ∈ E_{S}[2](S) such that f = T_{x} ∘ f′ is normalized. To see this, fix an s ∈ S. Then by Theorem 2.6 there is a normalized cover f_{s} : C_{s} → E_{s} such that (f_{s})^{∗} = −h_{s}, and so by the Albanese property (cf. §7) there exists x_{s} ∈ E_{s}(k) (where k = κ(s)) such that f′_{s} = T_{x_{s}} ∘ f_{s}. Since f_{s} and f′_{s} both satisfy (11), it follows that x_{s} ∈ E_{s}[2](k). Since (†) implies that E_{S}[2](S) → E_{s}[2](k) is surjective, there exists an x ∈ E_{S}[2](S) such that f := T_{x} ∘ f′ specializes at s to the given f_{s}. Then f also satisfies (11), and so f is normalized by part (c) above. In addition, f^{∗} = (f′)^{∗} = −h. Thus, replacing f by f ∘ σ_{C} yields the desired f (since σ^{∗} = [−1]_{J}).

We now remove the hypothesis that C/S satisfies (†) and hence consider an arbitrary genus 2 curve C/S. Then there exists a finite, faithfully flat base change β : S′ → S such that C_{(S′)}/S′ satisfies (†) and so, by what was just shown, there exists a normalized morphism f′ : C_{(S′)} → E_{(S′)} such that (f′)^{∗} = h_{(S′)}.

It is clearly enough to show that f′ = f_{(S′)} for some morphism f : C → E_{S}, for then f is automatically normalized (since f′ is) and satisfies f^{∗} = h (because we have f_{(S′)}^{∗} = h_{(S′)}). Furthermore, the existence of f will follow by faithfully-flat descent (cf. [BLR], Th. 6.1/6(a)) once we have shown that p_{1}^{∗}f′ = p_{2}^{∗}f′, where p_{i} = pr_{i} : S′′ := S′ ×_{S} S′ → S′, i = 1, 2, denote the two projections and p_{i}^{∗}f′ : C_{(S′′)} → E_{(S′′)} denotes the base-change of f′ via p_{i}.

Since f′ is normalized, we have by part (d) (or by construction) that there exists L ∈ Pic(C_{(S′)}) such that the embedding j = j_{L} : C_{(S′)} → J_{C_{(S′)}/S′} satisfies [−1] ∘ j = j ∘ σ_{C_{(S′)}} and f′ = h_{(S′)}^{∗} ∘ j. Here, as above, h^{∗} = ĥ ∘ λ_{C/S} and so h_{(S′)}^{∗} = λ_{E_{S′}/S′}^{−1} ∘ (f′)_{∗}. Put L_{i} = p_{i}^{∗}L ∈ Pic(C_{(S′′)}). Since λ_{O(L_{i})} = (λ_{C/S})_{(S′′)} and λ_{C/S} is a principal polarization, it follows that x := cl(L_{1}⊗L_{2}^{−1}) ∈ Pic_{C/S}^{0}(S′′) = J_{C/S}(S′′), and so p_{1}^{∗}j = j_{L_{1}} = T_{x}∘j_{L_{2}} = T_{x}∘p_{2}^{∗}j.

In addition we note that x ∈ J_{C/S}[2](S′′) because [−1]∘p_{i}^{∗}j = p_{i}^{∗}j ∘σ_{C_{(S′′)}}, for i = 1, 2. Now since h^{∗} is defined over S, p_{i}^{∗}h^{∗} = h_{(S′′)}^{∗} does not depend on i and so we obtain p_{1}^{∗}f′ = h_{(S′′)}^{∗} ∘ p_{1}^{∗}j = h_{(S′′)}^{∗} ∘ T_{x} ∘ p_{2}^{∗}j = T_{x′} ∘ h_{(S′′)}^{∗} ∘ p_{2}^{∗}j = T_{x′} ∘ p_{2}^{∗}f′ with x′ = h_{(S′′)}^{∗}(x) ∈ E_{S}[2](S′′).

To prove that p_{1}^{∗}f′ = p_{2}^{∗}f′, it is thus enough to show that x′ = 0 or, equivalently, that x ∈ Ker(h^{∗})(S′′). For this, consider θ_{i} := p_{i}^{∗}j(C_{(S′′)}), for i = 1, 2; note that θ_{1} = T_{x}(θ_{2}). Since p_{i}^{∗}f′ is normalized, we have by part (e) that θ_{i} is a symmetric theta-divisor of λ_{C_{(S′′)}/S′′} which satisfies θ_{i} ∩Ker(h_{(S′′)}^{∗})[2] = Ker(h_{(S′′)}^{∗})[2]^{#} (resp., θ_{i} ∩ Ker(h_{(S′′)}^{∗})[2] = ∅) if N is odd (resp. if N is even) because Ker(p_{i}^{∗}(f′)_{∗}) = Ker(h^{∗}). Thus, by the uniqueness assertion of part (e) we obtain θ_{1} = θ_{2} (resp. θ_{1} = T_{y}θ_{2} with y ∈ Ker(h^{∗})[2](S′′)) and so x = 0 (resp. x = y ∈ Ker(h^{∗})[2](S′′)). (Recall that if O(θ_{1}) ≃ T_{z}^{∗}(O(θ_{1})), then z = 0 because λ_{C/S} is a principal polarization.) Thus, in both cases x ∈ Ker(h^{∗})(S′′) and so p_{1}^{∗}f′ = p_{2}^{∗}(f′), as desired.

For later use let us also observe that the analogue of Proposition 2.1 carries over to genus 2 families and thus has the following important consequence.

Proposition 3.3 (a) If f : C → E_{S} is a minimal genus 2 cover of degree N ≥ 3 and α ∈ Aut_{S}(C) is an automorphism such that f ∘ α = f, then α = id_{C}.

(b) Let f_{i} : C_{i} → E_{S}, i = 1, 2 be two normalized genus 2 covers of E_{S}/S of degree N ≥ 3, and let p : S′ → S be a faithfully flat, quasi-compact cover of S. If there exists an S′-isomorphism α′ : (C_{1})_{(S′)} →^{∼} (C_{2})_{(S′)} such that (f_{2})_{(S′)} ∘ α′ = (f_{1})_{(S′)}, then there is a unique S-isomorphism α : C_{1} →^{∼} C_{2} such that α_{(S′)} = α′, and we have f_{2} ∘ α = f_{1}.

Proof. (a) First note that it follows from Prop. 2.1 (and Th. 3.2(a)) that α_{s} = id_{C_{s}}, for all s ∈ S. Thus, if S is reduced, then so is C and thus it follows that α = id_{C} because C/S is separated. (Indeed, let γ : Ker(α, β) → X be the subscheme of coincidences of α and β = id_{C}. Then Ker(α, β) is a closed subscheme of X by [EGA], (I, 5.2.5). Furthermore, γ is surjective (since this is true fibre-by-fibre), hence schematically dominant since C is reduced (use [EGA], (I, 5.4.3)). Thus, γ is an epimorphism (by [EGA], (I, 5.4.6)), and so, since α ∘ γ = β ∘ γ by definition, it follows that α = β.)

Now suppose that S is an arbitrary scheme; without loss of generality, however, we may assume that S is locally noetherian. Then the map S_{red} → S is defined by a locally nilpotent ideal of O_{S} (cf. [EGA], (I, 4.5.8)). Now since the scheme of automorphisms Aut_{S}(C) is finite and unramified (this is a special case of [DM], Th. (1.11)), it follows that the induced map Aut_{S}(C) → Aut_{S_{red}}(C_{red}) is injective (cf. [EGA], (IV, 17.1.2)(iv)). Thus, α = id_{C}, as desired.

(b) It is clearly enough to construct α such that α_{(S′)} = α′, for then the second property follows since p is faithfully flat. Furthermore, the existence of α will follow by faithfully-flat descent (cf. [BLR], Th. 6.1/6(a)) once we have shown that p_{1}^{∗}α′ = p_{2}^{∗}α′, where p_{i} = pr_{i} : S′′ := S′ ×_{S} S′ → S′ denote the two projections.

Now since p_{1}^{∗}(f_{i})_{(S′)} = p_{2}^{∗}(f_{i})_{(S′)} = (f_{i})_{(S′′)}, for i = 1, 2, we have (f_{2})_{(S′′)} ∘ p_{j}^{∗}α′ = p_{j}^{∗}((f_{2})_{(S′)} ∘ α′) = p_{j}^{∗}((f_{1})_{(S′)}) = (f_{1})_{(S′′)}, for j = 1, 2. Thus, if we put β = (p_{1}^{∗}α′)^{−1} ∘ p_{2}^{∗}α′ ∈ Aut_{S′′}((C_{1})_{(S′′)}), then we obtain (f_{1})_{(S′′)} ∘ β = (f_{1})_{(S′′)}. Now since (f_{1})_{(S′′)} : (C_{1})_{(S′′)} → E_{(S′′)} is a normalized genus 2 cover by Th. 3.2(a), we have by part (a) that β = id_{(C_{1})_{S′′}}, and so p_{1}^{∗}α′ = p_{2}^{∗}α′, as desired.

Notation. Let us now put, for any K-scheme S,

H_{E/K,N}(S) = {C →^{f} E_{S} : f is a normalized genus 2 cover of degree N}/≃,

where two covers f_{i} : C_{i} → E_{S} are called isomorphic (notation: f_{1} ≃ f_{2}) if there is an isomorphism φ : C_{1} → C_{2} such that f_{1} = f_{2} ∘ φ.

By Theorem 3.2(a) we see that if β : S′ → S is a K-morphism, then the rule f ↦ f_{(S′)} induces a map H_{E/K,N}(β) : H_{E/K,N}(S) → H_{E/K,N}(S′), and so we obtain a contravariant functor

H_{E/K,N} : \underline{Sch}_{/K} → \underline{Sets},

called the Hurwitz functor of genus 2 covers of E/K.

Remark. Part (b) of the above Proposition 3.3 shows that if N ≥ 3, then the functor H_{E/K,N} is a separated presheaf in the fpqc-topology (cf. [Mi1], p. 49 and/or [BLR], p. 199), which is a necessary condition for the functor to be (finely) representable; cf. [BLR], Proposition 8.1/1.

In fact, if N is invertible in K, then we shall see later (cf. Theorem 5.18) that H_{E/K,N} is representable by a smooth affine curve H_{E/K,N} which is an open subscheme of the (affine) modular curve X_{E,N,−1} which will be defined and studied in the next section.

4 The modular curves X_{E/K,N} and X_{E/K,N,ε}

For a given elliptic curve E/K, let

X_{E/K,N} : \underline{Sch}_{/K} → \underline{Sets}

denote the functor which “classifies E[N]-structures of elliptic curves”. Thus, if S is any K-scheme, then X_{E/K,N}(S) = {(E′, ψ)}/ ≃ is the set of isomorphism classes of pairs (E′, ψ) consisting of an elliptic curve E′/S and an Sisomorphism ψ : E_{S}[N] →^{∼} E′[N] of the S-group schemes of N-torsion points of E_{S} and E′; here two such pairs are equivalent, i.e. (E′_{1}, ψ_{1}) ≃ (E′_{2}, ψ_{2}), if there is an S-isomorphism φ : E′_{1} →^{∼} E′_{2} such that ψ_{2} = φ ∘ ψ_{1}. Furthermore, if β : S′ → S is any K-morphism, then the map X_{E/K,N}(β) : X_{E/K,N}(S) → X_{E/K,N}(S′) is given by base-change, i.e. by the assignment (E′, ψ) ↦ (E′_{S′}, ψ_{(S′)}).

Theorem 4.1 If N ≥ 3 is invertible in K, then the functor X_{E/K,N} is representable by a smooth affine curve X_{E/K,N}/K which is a twist of the usual (affine) modular curve X′_{N}/K which classifies elliptic curves with level-N-structures. More precisely, if K′/K is any extension field such that the Ntorsion points of E are K-rational over K′, then X_{E/K,N} ⊗ K′ ≃ X′_{N} ⊗ K′.

Proof. This is analogous to Corollary 4.7.2 of [KM] and is proved by exactly the same techniques. To explain this more precisely, fix a finite étale group scheme G/K of rank N^{2} and consider the moduli problem P_{G/K} “which classifies G-structures of elliptic curves”. By this we mean the contravariant functor

P_{G/K} : \underline{Ell}_{/K} → \underline{Sets}

from the category \underline{Ell}_{/K} of elliptic curves E′/S over variable K-schemes S (cf. [KM], (4.1), (4.13)) defined by P_{G/K}(E/S) = Isom_{S−gp}(G_{S}, E′[N]), the set of all S-group isomorphisms α : G_{S} = G ×_{K} S →^{∼} E′[N]. (Each such α will be called a G-structure of E′/S.)

Note that in the case that G = G_{N} := ℤ/Nℤ × ℤ/Nℤ is the constant group scheme of rank N^{2}, a G-structure is the same as a (naive) level-N-structure (or Γ(N)-structure) in the sense of [KM], (3.1). Furthermore, since the above functor X_{E/K,N} depends only on the K-group scheme E[N] (which has rank N^{2}), we see that this functor is closely related to the functor P_{E[N]/K}; in fact, we have

X_{E/K,N} = P̃_{E[N]/K} : \underline{Sch}_{/K} → \underline{Sets},

where P̃ is the functor on \underline{Sch}_{/K} associated to P in the sense of [KM], p. 108 and p. 125; i.e. P̃ classifies isomorphism classes of elliptic curves with P-structures.

Now the same argument as that of [KM], Corollary 4.7.2 shows that P_{G} is representable by a smooth affine curve. Indeed, P_{G/K} is relatively representable and finite étale (since for a given E′/S, the functor (of Sschemes) T ↦ Isom_{T −gp}(G_{T} , E′_{T} [N]) is represented by a finite étale S-scheme; cf. [KM], 1.6.7). Moreover, P_{G/K} is rigid for N ≥ 3 (cf. [KM], Cor. 2.7.2), and so P_{G/K} is representable by [KM], (4.7.0); in particular, P̃_{G/K} is representable by [KM], A.4.2 (p. 126).

Applying this to the case that G = E[N] shows that X_{E/K,N} = P̃_{E[N]/K} is representable by a K-scheme X_{E/K,N}. Furthermore, if K′/K is as given, then E′_{K}[N] ≃ (G_{N})_{/K′}, and so P_{E_{K′}[N]} ≃ P_{(G_{N})_{/K′}/K′}, which implies that X_{E/K,N} ⊗ K′ = X_{E_{K′}/K′,N} = (X_{N})_{/K′}. Since the latter is a smooth affine curve over K′ (cf. [KM], Cor. 4.7.2), it follows (by faithfully flat descent) that X_{E/K,N} is a smooth affine curve over K.

It is well-known that the curves X_{N}/K and hence X_{E/K,N} are not geometrically connected; indeed, each is a sum of ϕ(N) irreducible components over \overline{K}. Now for the curves X_{E/K,N}, this decomposition already takes place over K (even though (X_{N})_{/K} may still be irreducible as a K-scheme). The reason for this is that it is possible to define a determinant of an isomorphism ψ : E_{S}[N] → E′[N], as the following lemma shows.

Lemma 4.2 Let E/S and E′/S be elliptic curves over a scheme S, and suppose ψ : E[N] →^{∼} E′[N] is an S-isomorphism, where N is invertible in S. Then there is a unique automorphism det(ψ) ∈ Aut_{S}(μ_{N}) ≃ ((ℤ/Nℤ)^{×})_{/S} such that e′_{N} ∘ (ψ × ψ) = det(ψ) ∘ e_{N}, (15)

where e_{N} : E[N] × E[N] → μ_{N} and e′_{N} : E′[N] × E′[N] → μ_{N} denote the e_{N}-pairings of E and E′.

Proof. Viewing E[N] as a (locally constant) étale sheaf of ℤ/Nℤ-modules on S_{et}, let ∧_{A}^{2}E[N] be its second exterior power with respect to A = ℤ/Nℤ (which is formed analogous to the tensor product of étale sheaves; cf. [Mi1], p. 79); thus, ∧_{A}^{2}E[N] is a (locally constant) étale sheaf of A-modules. Since e_{N} is an alternating pairing (cf. [KM], p. 90, 505), we have an induced Ahomomorphism of étale sheaves \overline{e}_{N} : ∧^{2}E[N] → μ_{N} because for any étale sheaves M,N of A-modules we have the canonical isomorphism (analogous to [Mi1], Prop. II.3.19)

Hom_{A}(∧_{A}^{2}M, N) ≃ Alt_{A}(M^{2}, N).

Now since e_{N} is non-degenerate (cf. [KM], p. 90) it follows that \overline{e}_{N} is an isomorphism, and hence det(ψ) := \overline{e}′_{N} ∘ ∧^{2}(ψ) ∘ \overline{e}_{N}^{−1} is the unique automorphism of μ_{N} such that (15) holds.

By the above lemma we thus see that for any ε ∈ (ℤ/NZ)^{×}, the rule

X_{E/K,N,ε}(S) = {(E′, ψ) ∈ X_{E/K,N}(S) : det(ψ) = [ε]_{μ_{N}} }

defines an open and closed subfunctor X_{E/K,N,ε} of X_{E/K,N} which is therefore represented by an open and closed K-subscheme X_{E/K,N,ε} of X_{E/K,N}, and we have the decomposition

X_{E/K,N} = ∐_{ε∈(ℤ/Nℤ)^{×}} X_{E/K,N,ε}.

In fact, each X_{E/K,N,ε} is geometrically irreducible, as we shall now show:

Corollary 4.3 If N ≥ 3 is invertible in K, then for each ε ∈ (ℤ/Nℤ)^{×}, the functor X_{E/K,N,ε} is representable by a smooth, affine, geometrically irreducible curve X_{E/K,N,ε}/K which is a twist of the usual (affine) modular curve X′(N)/K(ζ_{N}) of level N which classifies elliptic curves with level-N-structures (of fixed determinant ζ_{N}). More precisely, if K′/K is any extension field such that the N-torsion points of E are rational over K′, then X_{E/K,N,ε} ⊗ K′ ≃ X′(N)_{/K′}.

Proof. By Igusa and/or Deligne/Rapoport[DM], Cor. IV.5.6, the modular curve X(N)/K(ζ_{N}) is smooth and geometrically irreducible and hence so is X′(N) = X(N) \ {cusps}. Since the functor X_{E_{K′}/K′,N,ε} is isomorphic (over K′) to the functor X (N) defining X′(N), it follows that X_{E/K,N,ε} ⊗ K′ ≃ X′(N)_{/K′}, and so X_{E/K,N,ε} is geometrically irreducible.

Remarks. (a) The modular curves X_{E/K,N} and X_{E/K,N,1} were already studied by Frey[Fr1] and Kraus-Oesterlé[KO], respectively.

(b) Although we had tacitly always assumed above that K is a field, this hypothesis is never used. Thus, the same conclusions hold if K is an arbitrary commutative ring (or even an arbitrary scheme).

5 The basic construction

5.1 The functor A_{E/K,N}

We now return to the Hurwitz functor H_{E/K,N} which was defined in section 3 and show that it is (finely) representable. To this end we shall generalize the “basic construction” of genus 2 covers presented in [FK], [Ka2] so as to obtain an (open) embedding of functors

Ψ : H_{E/K,N} ↪ X_{E/K,N,−1}.

Before defining this functor in full generality, let us briefly recall that this “basic construction” shows that any (minimal) genus 2 K-cover f : C → E of degree N determines a unique pair (E′_{f}, ψ_{f}) ∈ X_{E/K,n,−1}(K) where E′_{f}/K is the complementary elliptic curve (cf. Proposition 2.7) and ψ_{f} : E[N] →^{∼} E′[N] is an anti-isometry, i.e. an isomorphism of determinant −1, and that conversely for each such a pair (satisfying a suitable additional hypothesis) one can reconstruct the cover f : C → E.

We shall now see that the same statement holds for (normalized) genus 2 covers of E_{S}/S over an arbitrary base S. However, instead of proving this directly, it is more convenient to divide the above construction into two steps, which amounts to a factorization of Ψ as

Ψ = Ψ′ ∘ τ : H_{E/K,N} ↪ A_{E/K,N} →^{∼} X_{E/K,N,−1},

in which τ is (essentially) the Torelli map which associates to a curve its (polarized) Jacobian and A_{E/K,N} is the functor that classifies principally polarized abelian surfaces with an embedding “of degree N” of E. In order to explain these terms more precisely, we introduce the following definition and notation.

Definition. Let J/S be an abelian scheme with a principal polarization λ : J →^{∼} Ĵ. Then an injective homomorphism h : E_{S} ↪ J is said to have degree N if we have ĥ ∘ λ ∘ h = λ_{E_{S}/S} ∘ [N]_{E_{S}}. (16)

Notation. Let E/K be an elliptic curve E/K and N ≥ 2 be an integer. For any K-scheme S let

A_{E/K,N}(S) = {(J, λ, h)}/≃

denote the set of isomorphism classes of triples (J, λ, h) consisting of an abelian scheme J/S of relative dimension 2, a principal polarization λ : J →^{∼} Ĵ of J, and an injective homomorphism h : E_{S} ↪ J of degree N. Here, two such triples (J, λ, h) and (J′, λ′, h′) are called isomorphic if there exists an S-isomorphism α : J →^{∼} J′ such that λ′ = ĥ ∘ λ ∘ h and h′ = α ∘ h; we then write (J, λ, h) ≃ (J′, λ′, h′). It is immediate that if β : S′ → S is any K-morphism, then the triple β^{∗}(J, λ, h) := (J_{(S′)}, λ_{(S′)}, h_{(S′)}) obtained from (J, λ, h) by base-change induces an element in A_{E/K,N}(S′) and so we have a (functorial) map

A_{E/K,N}(β) : A_{E/K,N}(S) → A_{E/K,N}(S′).

We have thus defined a contravariant functor

A_{E/K,N} : \underline{Sch}_{/K} → \underline{Sets}.

As a first step of the basic construction we define the “Torelli map” τ which was mentioned above.

Proposition 5.1 If f : C → E_{S} is a minimal genus 2 cover of degree N, then h_{f} := f^{∗} ∘ λ_{E_{S}/S} : E_{S} → J is an injective homomorphism of degree N. Thus, the rule (f : C → E_{S}) ↦ (J_{C/S}, λ_{C/S}, h_{f}) defines a morphism of functors

τ : H_{E/K,N} → A_{E/K,N}.

Proof. Since f is minimal, f^{∗} : J_{E_{S}} → J_{C/S} is an injective homomorphism (by definition). Now by (44) we have [N]_{J_{E_{S} /S}} = f_{∗} ∘f^{∗} = λ_{E_{S}/S}^{−1} ∘(f^{∗})^{}ˆ ∘λ_{C/S} ∘f^{∗}, which shows that f^{∗} and hence h_{f} has degree N.

5.2 The isomorphism Ψ′ : A_{E/K,N} → X_{E/K,N,−1}

Our next aim is to construct the morphism Ψ′ : A_{E/K,N} → X_{E/K,N,−1}. For this, we first prove:

Proposition 5.2 Let J/S be an abelian scheme of relative dimension 2 with a principal polarization λ : J →^{∼} Ĵ, and let h : E_{S} → J be an injective homomorphism of degree N. If h^{∗} := ĥ ∘ λ : J → J_{E_{S}/S} denotes the “dual” of h, then E′_{h} := Ker(h^{∗}) is an elliptic curve over S, and there is a unique isomorphism

ψ_{h} : E_{S}[N] →^{∼} E′_{h}[N]

of the N-torsion subgroup schemes of E_{S}/S and E′_{h}/S such that h′ ∘ ψ_{h} = h_{|E_{S}[N]}, where h′ : E′_{h} = Ker(h^{∗}) ↪ J denotes the associated closed immersion.

Proof. Since h : E_{S} ↪ J is a closed immersion, its “dual” h^{∗} : J → J_{E_{S}/S} is surjective and has connected fibres by (42). Clearly, these fibres have dimension 2 − 1 = 1 and so E′_{h} := Ker(h^{∗}) is an elliptic curve over S which has a natural closed immersion h′ : E′_{h} ↪ J.

Consider the fibre product H := E_{S} ×_{J} E′_{h} via the closed immersions h and h′:

H →^{pr_{2}} E′_{h}
pr_{1} ↓  ↓ _{h′}
E_{S} →^{h} J

Since h and h′ are closed immersions, so are pr_{2} and pr_{1}. Moreover, since E′_{h} = Ker(h^{∗}) and h^{∗} ∘ h = λ_{E_{S}/S} ∘ [N]_{E_{S}}, the image of H with respect to pr_{1} is Ker(h^{∗}∘h) = Ker(λ_{E_{S}/S}∘[N]_{E_{S}}) = E_{S}[N] and so H is finite and flat of rank N^{2} and is annihilated by multiplication by N. Thus, the image of H under pr_{2} is contained in E′_{h}[N] and so is equal to E′[N] since both group schemes have rank N^{2}. Thus, if H′ denotes the common image of h ∘ pr_{1} = h′ ∘ pr_{2} in J, then the restrictions of h and h′ to the respective N-torsion subgroups induce isomorphisms

h_{|E_{S}[N]} : E_{S}[N] →^{∼} H′ and h′_{|E′_{h}[N]} : E′_{h}[N] →^{∼} H′,

and so ψ_{h} := (h′_{|E′_{h}[N]})^{−1} ∘ h_{|E_{S}[N]} : E_{S}[N] →^{∼} E′_{h}[N] is the desired isomorphism. Note that since h′ is a (closed) immersion (hence a monomorphism), ψ_{h} is uniquely determined by the indicated property.

Corollary 5.3 If (h′)^{∗} = (h′)^{}ˆ ∘ λ : J → J_{E′_{h}}, then we have

h(Ker(h^{∗} ∘ h)) = h′(Ker((h′)^{∗} ∘ h′)) = E_{S} ×_{J} E′_{h} ≃ E_{S}[N] (17)

and hence the injection h′ : E′_{h} ↪ J also has degree N.

Proof. The above proof shows that h(Ker(h^{∗}∘h)) = H′ = E_{S} ×_{J} E′_{h} ≃ E_{S}[N]. To prove that also h′(Ker((h′)^{∗} ∘ h′)) = E_{S} ×_{J} E′_{h}, we first note that by the definition of h′ we have the exact sequence (of abelian schemes)

0 → E′_{h} →^{h′} J →^{h^{∗}} J_{E_{S}} → 0. (18)

Dualizing this sequence and composing with λ and κ_{E_{S}} : E_{S} →^{∼} Ĵ_{E_{S}} yields the sequence

0 → E_{S} →^{h} J →^{(h′)^{∗}} J_{E_{S}} → 0 (19)

which is again exact by (43); here we have also used the fact that λ^{−1} ∘ (h^{∗})^{}ˆ ∘ κ_{E_{S}} = λ^{−1} ∘ λ̂ ∘ (ĥ)^{}ˆ ∘ κ_{E_{S}} = h. Thus we have h(E_{S}) = Ker((h′)^{∗}) and hence, since h′ is injective, we obtain h′(Ker((h′)^{∗}∘h′)) = h(E_{S})∩h′(E′_{h}) = E_{S}×_{J} E′_{h}, which proves (17).

To prove that h′ has degree N, i.e. that the equation (h′)^{∗} ∘h′ = λ_{E′_{h}} ∘[N] holds, it is enough (by rigidity) to prove this fibre-by-fibre, so assume S = Spec(F), where F is a field. Now since (h′)^{∗} ∘ h′ = ĥ′ ∘ λ ∘ h is a polarization on E′_{h} (cf. (37)), it follows that (h′)^{∗} ∘ h′ = λ_{E′_{h}} ∘ [n] for some n > 0 since all polarizations on E′_{h} are of this form. But since Ker((h′)^{∗} ∘ h′) ≃ E_{F} [N] has rank N^{2}, it follows that n = N, and so h′ has degree N, as claimed.

By the above proposition, each triple (J, λ, h) determines a pair (E′_{h}, ψ_{h}) and hence an element X_{E/K,N}(S). It is easy to see that this construction is compatible with the equivalence relation of covers; more precisely, we have:

Proposition 5.4 Suppose J_{i}/S are two abelian schemes of relative dimension 2 with principal polarizations λ_{i} : J_{i} →^{∼} Ĵ_{i} and that h_{i} : E_{S} → J_{i} are injective homomorphisms of degree N, for i = 1, 2. If α : J_{1} →^{∼} J_{2} is an isomorphism such that λ_{2} = α̂ ∘ λ_{1} ∘ α and h_{2} = α ∘ h_{1}, then there is an induced isomorphism α′ : E′_{h_{1}} →^{∼} E′_{h_{2}} such that α′ ∘ ψ_{h_{1}} = ψ_{h_{2}}, and α′ is uniquely characterized by this property if N ≥ 3. In particular, the assignment (J, λ, h) ↦ (E′_{h}, ψ_{h}) defines a morphism of functors

Ψ′ : A_{E/K,N} → X_{E/K,N}.

Proof. First note that α′ is uniquely determined by this property if N ≥ 3, for if α′′ : E′_{h_{1}} →^{∼} E′_{h_{2}} is another such isomorphism, then β := (α′)^{−1} ∘ α′′ ∈ Aut_{S}(E′_{h_{1}}) satisfies β_{|E′_{h_{1}}[N]} = id_{E′_{h_{1}}[N]}, and so β = id_{E′_{h_{1}}} (cf. [KM], p. 85), which means that α′′ = α′.

We now prove the existence of α′. Dualizing the relation h_{2} = α ∘ h_{1} (and composing with λ_{i}) yields h_{2}^{∗} = h_{1}^{∗} ∘ α^{∗}, where α^{∗} = λ_{2}^{−1} ∘ α̂ ∘ λ_{1} = α^{−1}, where the latter equality follows from the hypothesis λ_{2} = α̂ ∘ λ_{1} ∘ α. Thus h_{1}^{∗} = h_{2}^{∗} ∘α, and so by the universal property of kernels, α induces a (unique) isomorphism α′ : E′_{h_{1}} = Ker(h_{1}^{∗}) →^{∼} E′_{h_{2}} = Ker(h_{2}^{∗}) such that h′_{2} ∘ α′ = α ∘ h′_{1}, where h′_{i} : E′_{h_{i}} = Ker(h_{i}^{∗}) ↪ J_{i} are the canonical inclusions.

It is now easy to see that α′ ∘ ψ_{f_{1}} = ψ_{f_{2}}. Indeed, using the defining equation of ψ_{h_{1}} (cf. Prop. 5.2), we have h′_{2} ∘ α′ ∘ ψ_{h_{1}} = α ∘ h′_{1} ∘ ψ_{h_{1}} = α ∘ (h_{1})_{|E_{S}[N]} = (h_{2})_{|E_{S}[N]}, which means that α′ ∘ ψ_{h_{1}} satisfies the defining equation of ψ_{h_{2}}, and so ψ_{h_{2}} = α′ ∘ ψ_{h_{1}}, as claimed.

Thus the map (J, λ, h) ↦ (E′_{h}, ψ_{h}) is compatible with the equivalence relations and hence determines a map Ψ′_{S} : A_{E/K,N}(S) → X_{E/K,N}(S). Since this map is clearly compatible with base-change, Ψ′ = {Ψ′_{S}}_{S} defines a morphism of functors Ψ′ : A_{E/K,N} → X_{E/K,N}.

We next show that Ψ′ maps A_{E/K,N} into the component X_{E/K,N,−1} of X_{E/K,N} and that Ψ′ : A_{E/K,N} → X_{E/K,N,−1} is an isomorphism of functors. For this we shall first prove the following result.

Proposition 5.5 In the situation of Proposition 5.2, let A := E_{S} ×_{S} E′_{h} denote the product surface over S and let π : A → J be defined by

π = h ∘ pr_{E_{S}} + h′ ∘ pr_{E′_{h}} (20)

where pr_{E_{S}} : A → E_{S} and pr_{E′_{h}} : A → E′_{h} are the projections. Then π is an isogeny of degree N^{2} whose kernel is G_{ψ_{h}} = Graph(−ψ_{h}) ⊂ E_{S}[N] ×_{S} E′_{f}[N] ⊂ A, the graph of the isomorphism −ψ_{h}. Furthermore, if π′ : J → A is defined by

π′ := i_{E_{S}/S} ∘ λ_{E_{S}/S}^{−1} ∘ h^{∗} + i_{E′_{h}} ∘ λ_{E′_{h}/S}^{−1} ∘ (h′)^{∗} (21)

where i_{E_{S}} : E_{S} ↪ A and i_{E′_{h}} : E′_{h} ↪ A are the canonical embeddings and (h′)^{∗} := (h′)^{}ˆ ∘ λ, then we have π′ ∘ π = [N]_{A} and π ∘ π′ = [N]_{J}. Furthermore, if λ_{A} = λ_{E_{S}/S} ⊗ λ_{E′_{h}/S} : A →^{∼} Â denotes the product polarization, then we have the commutative diagram

\begin{matrix}A & →^{λ_{A}} & Â \\ ^{π} ↓ & & ↓ _{}_{π̂′} \\ J & →^{λ} & Ĵ \\ _{π′} ↓ & & ↓ π̂ \\ A & →^{λ_{A}} & Â . \end{matrix} (22)

Proof. Let k : K_{ψ} := Ker(π) ↪ A denote the closed immersion defined by the kernel of π and let p = (pr_{E_{S}})_{|K_{ψ}} = pr_{E_{S}} ∘ k : K_{ψ} → E_{S}, and p′ = (pr_{E′})_{|K_{ψ}} = pr_{E′} ∘ k : K_{ψ} → E′ = E′_{h} denote the restriction of the projection maps of A to K_{ψ}. Then by the universal property of kernels it follows from (20) that (K_{ψ}, p, −p′) is the fibre product E_{S} ×_{J} E′ with respect to the maps h and h′. Thus, by the proof of Prop. 5.2, it follows that p and p′ factor over the immersions j_{N} : E_{S}[N] ↪ E_{S} and j′_{N} : E′[N] ↪ E′ as p = j_{N} ∘ p_{N}

and p′ = j′_{N} ∘ p′_{N}, and that p_{N} : K_{ψ} →^{∼} E_{S}[N] and p′_{N} : K_{ψ} →^{∼} E′[N] are

isomorphisms. Thus, ψ_{h} = −p′_{N} ∘ p_{N}^{−1}, and so, if γ = γ_{−ψ_{h}} : E_{S}[N] → A[N] denotes the graph morphism, then we have

(j_{N} × j′_{N}) ∘ γ ∘ p_{N} = k : K_{ψ} ↪ A

because pr_{E_{S}} ∘(j_{N} ×j′_{N})∘γ∘p_{N} = j_{N} ∘pr_{E_{S}[N]}∘γ∘p_{N} = j_{N} ∘id_{E_{S}[N]}∘p_{N} = p = pr_{E_{S}} ∘k and pr_{E′} ∘(j_{N} ×j′_{N})∘γ ∘p_{N} = j′_{N} ∘pr_{E′[N]} ∘γ ∘p_{N} = j′_{N} ∘(−ψ_{h})∘p_{N} = j′_{N} ∘p′_{N} = p′ = pr_{E′} ∘k. Since p_{N} is an isomorphism, this equation means that K_{ψ} = Graph(−ψ_{h}) (as closed subschemes of A), as desired. Moreover, since Graph(−ψ_{h}) ≃ E_{S}[N] has rank N^{2}, we see that π is an isogeny (because A/S and J/S both have relative dimension 2).

In order to verify that π′ ∘ π = [N]_{A}, we first observe that

λ_{E_{S}/S}^{−1}∘h^{∗}∘h = [N]_{E_{S}}, h^{∗}∘h′ = 0, (h′)^{∗}∘h = 0, λ_{E′/S}^{−1}∘(h′)^{∗}∘h′ = [N]_{E′}. (23)

Indeed, the first and last equation of (23) are a restatement of (16) (since both h and h′ have degree N by Cor. 5.3), and the second and third equations follow from the exact sequences (18) and (19).

Now by (20), (21) and (23) we obtain π′ ∘ π = (i_{E_{S}} ∘ λ_{E_{S}/S}^{−1} ∘ h^{∗} + i_{E′_{h}} ∘ λ_{E′_{h}/S}^{−1} ∘ (h′)^{∗}) ∘ (h ∘ pr_{E_{S}} + h′ ∘ pr_{E′_{h}}) = i_{E_{S}} ∘ [N]_{E_{S}} ∘ pr_{E_{S}} + i_{E′_{h}} ∘ [N]_{E′_{h}} ∘ pr_{E′_{h}} = (i_{E_{S}} ∘ pr_{E_{S}} + i_{E′_{h}} ∘ pr_{E′_{h}}) ∘ [N]_{A} = [N]_{A}, as claimed. Furthermore, from this we get π ∘ π′ ∘ π = π ∘ [N]_{A} = [N]_{J} ∘ π, and so π ∘ π′ = [N]_{J} since π is an isogeny and hence an epimorphism in \underline{AbSch}_{/S}.

It remains to show that the diagram (22) commutes. For this we first observe that if φ = φ_{E_{S},E′} : Â →^{∼} J_{E_{S}/S} ×_{S} J_{E′/S} is the canonical isomorphism of (39), then φ ∘ π̂ ∘ λ = i_{J_{E_{S} /S}} ∘ h^{∗} + i_{J_{E′_{h}/S}} ∘ (h′)^{∗} (24)

because φ ∘ π̂ ∘ λ = φ ∘ (h ∘ pr_{E_{S}})^{}ˆ ∘ λ + φ ∘ (h′ ∘ pr_{E′_{h}})^{}ˆ ∘ λ = φ ∘ p̂r_{E_{S}} ∘ ĥ ∘ λ + φ ∘ p̂r_{E′_{h}} ∘ ĥ′ ∘ λ = i_{J_{E_{S} /S}} ∘ h^{∗} + i_{J_{E′_{h}/S}} ∘ (h′)^{∗}.

From (24), (20) and (23) we obtain φ∘π̂∘λ∘π = (i_{J_{E_{S} /S}} ∘h^{∗}+i_{J_{E′_{h}/S}} ∘(h′)^{∗})∘ (h∘pr_{E_{S}} +h′∘pr_{E′_{h}}) = i_{J_{E_{S} /S}} ∘λ_{E_{S}/S}∘[N]_{E_{S}} ∘pr_{E_{S}} +i_{J_{E′_{h}/S}} ∘λ_{E′_{h}/S}∘[N]_{E′_{h}}∘pr_{E′_{h}} = (λ_{E_{S}/S} × λ_{E′_{h}/S}) ∘ [N]_{A} = φ ∘ λ_{A} ∘ π′ ∘ π. Thus, φ ∘ π̂ ∘ λ ∘ π = φ ∘ λ_{A} ∘ π′ ∘ π, and hence π̂ ∘ λ = λ_{A} ∘ π′ because φ is an isomorphism and π is an isogeny. Thus, the bottom square of (22) commutes, and hence so does the top square since it is the dual of the bottom square.

Remarks. (a) The commutative diagram (22) is essentially the diagram on p. 157 of [FK], where it is proved by a slightly different method (in the case that S = Spec(K) and J = J_{C/K}).

(b) The maps π and π′ can also be characterized by the properties

(25)  π ∘ i_{E_{S}} = h and π ∘ i_{E′_{h}} = h′,
(26)  pr_{E_{S}} ∘ π′ = λ_{E_{S}/S}^{−1} ∘ h^{∗} and pr_{E′_{h}} ∘ π′ = λ_{E′_{h}/S}^{−1} ∘ (h′)^{∗}.

Corollary 5.6 We have Nλ_{A} = π̂ ∘ λ ∘ π, and so Graph(−ψ_{h}) ≤ A[N] is an isotropic subgroup of A[N] with respect to e^{Nλ_{A}}. Thus, ψ_{h} : E_{S}[N] →^{∼} E′[N] is an anti-isometry, i.e. e′_{N} ∘ (ψ_{h} ×_{S} ψ_{h}) = [−1]_{μ_{N}} ∘ e_{N}.

Proof. By commutativity of the bottom square of (22) we obtain (π̂ ∘λ)∘π = (λ_{A} ∘ π′) ∘ π = λ_{A} ∘ [N]_{A}, which proves the first assertion.

Thus, by the functorial property (38), we have for every S-scheme T and for all T -valued points x, y ∈ Graph(−ψ_{f}) = Ker(π) that e^{Nλ_{A}}(x, y) =

e^{π̂∘λ∘π}(x, y) =^{(38)} e^{λ}(π(x), π(y)) = e^{λ}(0, 0) = 1, which means that Graph(−ψ_{h}) is an isotropic subgroup of A[N] (with respect to e^{Nλ_{A}}).

Since λ_{A} = λ_{1} ⊗ λ_{2} is the product polarization (with λ_{1} = λ_{E_{S}/S} and λ_{2} = λ_{E′_{h}/S}), we have e^{Nλ_{A}} = (e^{Nλ_{1}} ∘ (pr_{1} × pr_{1})) · (e^{Nλ_{2}} ∘ (pr_{2} × pr_{2})). Thus, applying this to x = (x′, −ψ_{h}(x′)) and y = (y′, −ψ_{h}(y′)), where x′, y′ ∈ E_{S}[N](T), yields e^{Nλ_{1}}(x′, y′)e^{Nλ_{2}}(−ψ_{h}(x′), −ψ_{h}(y′)) = e^{Nλ_{A}}(x, y) = 1, which proves that −ψ_{h} (and hence also ψ_{h}) is an anti-isometry.

This therefore shows that Ψ′ maps A_{E/K,N} into the (−1)-component of X_{E/K,N}, i.e. into X_{E/K,N,−1}. In order to show that Ψ′ is an isomorphism, we shall construct an inverse morphism Ψ′′ : X_{E/K,N,−1} → A_{E/K,N}. To this end, we shall prove:

Proposition 5.7 Suppose E′/S is an elliptic curve with an anti-isometry ψ : E_{S}[N] →^{∼} E′[N]. Let A = E_{S} ×_{S} E′ and let G_{ψ} := Graph(−ψ) ≤ E_{S}[N] ×_{S} E′[N] = A[N] denote the graph of −ψ. Then the quotient J = J_{ψ} := A/G_{ψ} is a projective abelian scheme and the quotient map π = π_{ψ} : A → J = A/G_{ψ} is an isogeny of degree N^{2}. Furthermore, there exists a unique isogeny π′ : J → A such that π′ ∘ π = [N]_{A} and π ∘ π′ = [N]_{J}. In addition, J has a unique principal polarization λ_{J} : J →^{∼} Ĵ such that λ_{J} ∘ π = π̂′ ∘ λ_{A}, where λ_{A} = λ_{E_{S}/S} ⊗ λ_{E′/S} denotes the product polarization of A = E_{S} ×_{S} E′, and then we also have π̂ ∘ λ_{J} = λ_{A} ∘ π′.

Proof. Since G_{ψ} ≃ E_{S}[N] is a finite, flat closed subscheme of the (strongly) projective group scheme A/S, the quotient group scheme J := A/G_{ψ} exists and is quasi-projective; cf. Raynaud[Ra], Th. 1(iv) and/or [BLR], Th. 8.2/12. Furthermore, it is immediate that the quotient map π : A → J = A/G_{ψ} is finite, flat, and surjective and so J/S is projective; cf. [EGA], (II, 6.6.4). Since it is immediate that the fibres of J_{s} = A_{s}/(G_{ψ})_{s} are geometrically irreducible, we see that J_{s} is an abelian scheme.

It is thus clear that π : A → J is an isogeny of order N^{2} since G_{ψ} ≃ E_{S}[N] has rank N^{2}. Moreover, since G_{ψ} ≤ A[N] = Ker([N]_{A}), it follows by the universal property of quotients (cf. [Mu1], p. 3) that [N]_{A} = π′ ∘ π, for some homomorphism π′ : J → A. Clearly, π′ is an isogeny (since π and [N]_{A} are). By the same argument as in the proof of Prop. 5.5 one concludes that π ∘ π′ = [N]_{J} holds as well.

We now construct λ_{J}. For this, we first note that there can be at most one isomorphism λ : J →^{∼} Ĵ such that λ∘π = π̂′∘λ_{A}, for π is an epimorphism. Thus, it is enough to prove that λ exists after a faithfully flat base extension S′/S (and that it is a polarization).

Since ψ is an anti-isometry, G_{ψ} is an isotropic subgroup with respect to E^{Nλ_{A}} (see the proof of Cor. 5.6). Now λ_{A} = λ_{Θ}, where Θ = pr_{1}^{∗}O([0_{E_{S}/S}]) ⊗ pr_{2}^{∗}O([0_{E′/S}]). Thus, by Lemma 5.8 below we know that after a suitable finite, faithfully flat base extension (which, for ease of notation, will be suppressed in the sequel), there is an invertible sheaf M on J such that Θ^{N} ≃ π^{∗}M, and so λ_{A} ∘ [N]_{A} = λ_{Θ^{N}} = λ_{π^{∗}M} = π̂ ∘ λ_{M} ∘ π, the latter by (37). Thus π̂′ ∘ λ_{A} ∘ [N]_{A} = π̂′ ∘ π̂ ∘ λ_{M} ∘ π = λ_{M} ∘ π ∘ [N]_{A}, and so π̂′ ∘ λ_{A} = λ_{M} ∘ π since [N]_{A} is an epimorphism. Thus λ_{J} = λ_{M} is the desired polarization. Note that since deg(π) = deg(π′) = N^{2}, it follows that deg(λ_{J}) = deg(λ_{A}) = 1, i.e. that λ_{J} is a principal polarization.

Finally, to prove that the last equation also holds, we multiply the previous equation by π̂ to obtain π̂ ∘λ_{J} ∘π = π̂ ∘π̂′ ∘λ_{A} = [N]Â ∘λ_{A} = λ_{A} ∘[N]_{A} = λ_{A} ∘ π′ ∘ π, which implies that π̂ ∘ λ_{J} = λ_{A} ∘ π′ because π is an epimorphism.

In the above proof we had used the following basic fact about the descent of polarizations.

Lemma 5.8 Let π : A → B be an isogeny of abelian S-schemes and let L ∈ Pic(A) be an ample invertible sheaf on A. If Ker(π) is an isotropic subgroup of K(L) := Ker(λ_{L}) with respect to the pairing e^{L}, then there exists a finite, faithfully flat extension S′/S and an invertible sheaf M′ on B_{(S′)} such that L_{(S′)} ≃ π_{(S′)}^{∗}M′, where L_{(S′)} denotes the pullback of L to A_{(S′)}.

Proof. This can be proved by a modification of the proof of the Corollary on p. 231 of Mumford[Mu2] (see also [MB2], Cor. VI.1.3). As in [Mu2], §23 and/or [MB1]), let G(L) denote the theta-group associated to L; recall that G(L) fits into an exact sequence

0 → 𝔾_{m} → G(L) →^{p} K(L) → 0.

Let H = Ker(π) and G = p^{−1}(H). Since e^{L} is given by the commutator of G(L), the isotropy of H means that G is commutative. Now by [Mi1], Lemma III.4.17 (and its proof) there is finite faithfully flat extension S′/S such that G_{(S′)} ≃ 𝔾_{m} × H_{(S′)} (as group schemes). Thus, there is a homomorphism α : H_{(S′)} → G_{(S′)} such that p ∘ α = id, which means that L_{(S′)} = π^{∗}(M′) by [MB1], Th. 4.1.

Corollary 5.9 In the situation of Proposition 5.7, the maps h_{ψ} := π_{ψ} ∘ i_{E_{S}} : E_{S} → J_{ψ} and h′_{ψ} := π_{ψ} ∘i_{E′} : E′ → J_{ψ} are injective homomorphisms of degree N whose “duals” h_{ψ}^{∗} := ĥ_{ψ} ∘ λ_{J} and (h′_{ψ})^{∗} := ĥ′_{ψ} ∘ λ_{J} satisfy the relations

h_{ψ}^{∗} = λ_{E_{S}/S} ∘ pr_{E_{S}} ∘ π′_{ψ} and (h′_{ψ})^{∗} = λ_{E′/S} ∘ pr_{E′} ∘ π′_{ψ}, (27)

and hence fit into the exact sequences

0 → E′ →^{h′_{ψ}} J_{ψ} →^{h_{ψ}^{∗}} J_{E_{S}/S} → 0 and 0 → E_{S} →^{h_{ψ}} J_{ψ} →^{(h′_{ψ})^{∗}} J_{E′/S} → 0. (28)

In particular, the assignment (E′, ψ) ↦ (J_{ψ}, λ_{J_{ψ}}, h_{ψ}) defines a morphism of functors

Ψ′′ : X_{E/K,N,−1} → A_{E/K,N}.

Proof. Since ψ is an isomorphism, we have Ker(h_{ψ}) = i_{E_{S}}^{−1}(G_{ψ} ∩ i_{E_{S}}(E_{S})) = Ker(ψ) = {0}, and so h_{ψ} is an injective homomorphism. Similarly, h′_{ψ} is also an injective homomorphism.

We next verify (27). By definition, h_{ψ}^{∗} = ĥ_{ψ} ∘ λ_{J} = î_{E_{S}} ∘ π̂ ∘ λ_{J} = î_{E_{S}} ∘ λ_{A} ∘ π′ = λ_{E_{S}/S} ∘ pr_{E_{S}} ∘ π′, where the last equality used (40). This proves the first equation of (27), and the second is proved similarly.

The fact that h_{ψ} and h′_{ψ} have degree N follows immediately from (27)

because ĥ_{ψ} ∘ λ_{J} ∘ h_{ψ} = h_{ψ}^{∗} ∘ h_{ψ} =^{(27)} λ_{E_{S}/S} ∘ pr_{E_{S}} ∘ π′ ∘ π ∘ i_{E_{S}} = λ_{E_{S}/S} ∘

pr_{E_{S}} ∘ [N]_{A} ∘ i_{E_{S}} = λ_{E_{S}/S} ∘ [N]_{E_{S}}. Thus, h_{ψ} has degree N, and a similar computation shows that h′_{ψ} also has degree N.

Next we show that the sequences (28) are exact. By (27) we have h_{ψ}^{∗}∘h′_{ψ} = λ_{E_{S}/S} ∘ pr_{E_{S}} ∘ π′ ∘ π ∘ i_{E′} = λ_{E_{S}/S} ∘ pr_{E_{S}} ∘ i_{E′} ∘ [N]_{E′} = 0, and similarly, (h′_{ψ})^{∗} ∘ h_{ψ} = 0. Moreover, since h_{ψ} and h′_{ψ} are injective, their duals h_{ψ}^{∗} and (h′_{ψ})^{∗} are surjective and have connected fibres (cf. (42)), and so it follows easily that the sequences (28) are exact.

Finally, we verify that the assignment (E′, ψ) ↦ cl(J_{ψ}, λ_{J_{ψ}}, h_{ψ}) defines a functor. Indeed, by the above we know that the (isomorphism class of the) triple (J_{ψ}, λ_{J_{ψ}}, h_{ψ}) lies in A_{E/K,N}(S). Furthermore, it is easy to see that this assignment is compatible with the equivalence relation on X_{E/K,N,−1}(S), for if (E′′, ψ′) ≃ (E′, ψ) via α : E′ →^{∼} E′′ (with α ∘ ψ = ψ′), then α̃ := id_{E_{S}} × α : A = E_{S} ×_{S} E′ →^{∼} A′ := E_{S} ×_{S} E′′ is an isomorphism such that α̃(G_{ψ}) = G_{ψ′}, and so we have an induced isomorphism \overline{α} : J_{ψ} →^{∼} J_{ψ′} such that π_{ψ′} ∘ α̃ = \overline{α} ∘ π_{ψ}, from which one easily concludes that α defines an isomorphism (J_{ψ}, λ_{J_{ψ}}, h_{ψ}) ≃ (J_{ψ′}, λ_{J_{ψ′}}, h_{ψ′}).

Thus, we have a well-defined map Ψ′′_{S} : X_{E/K,N,−1}(S) → A_{E/K,N}(S). Since this map is clearly compatible with base-change, the collection Ψ′′ = {Ψ′′_{S}}_{S} defines the desired morphism of functors Ψ′′ : X_{E/K,N,−1} → A_{E/K,N}.

Theorem 5.10 The morphisms

Ψ′ : A_{E/K,N} → X_{E/K,N,−1} and Ψ′′ : X_{E/K,N,−1} → A_{E/K,N}

are inverses of each other and hence are isomorphisms.

Proof. Fix a K-scheme S, and let cl(J, λ, h) ∈ A_{E/K,N}(S). Then by definition Ψ′′_{S}(Ψ′_{S}(cl(J, λ, h))) = Ψ′′_{S}(cl(E′_{h}, ψ_{h})) = cl(J_{ψ_{h}}, λ_{J_{ψ_{h}}}, h_{ψ_{h}}), where E′_{h} = Ker(ĥ ∘ λ) and J_{ψ_{h}} = A_{h}/G_{ψ_{h}} with A_{h} = E_{S} ×_{S} E′_{h}. Now by Prop. 5.5 the map π : A_{h} → J is the quotient map with respect to G_{ψ_{h}}, and hence we have a canonical identification J = J_{ψ_{h}}. Furthermore, since λ ∘ π = π̂′ ∘ λ_{A_{h}} (by Prop. 5.5 again), we have (by definition) λ_{J_{ψ_{h}}} = λ. Finally, h_{ψ_{h}} = i_{E_{S}} ∘π = h by equation (25), and so Ψ′′_{S}(Ψ′_{S}(cl(J, λ, h))) = cl(J, λ, h).

Conversely, let cl(E′, ψ) ∈ X_{E/K,N,−1}(S). Then Ψ′_{S}(Ψ′′_{S}(cl(E′, ψ))) = Ψ′_{S}(cl(J_{ψ}, λ_{J_{ψ}}, h_{ψ})) = cl(E′_{h_{ψ}}, ψ_{h_{ψ}}). Here J_{ψ} = A/G_{ψ} and λ_{J} are as defined in Prop. 5.7 and h_{ψ} = π_{ψ} ∘ i_{E_{S}} as in Cor. 5.9. Furthermore, E′_{h_{ψ}} = Ker(h_{ψ}^{∗}), and ψ_{h_{ψ}} is the unique anti-isometry such that (h_{ψ})′ ∘ ψ_{h_{ψ}} = (h_{ψ})_{|E_{S}[N]}, where (h_{ψ})′ : E′_{h_{ψ}} ↪ J_{ψ} denotes the canonical inclusion.

From the first exact sequence in (28) we see that E′ = Ker(h_{ψ}^{∗}), and so there exists a unique isomorphism α : E′ →^{∼} E′_{h_{ψ}} such that (h_{ψ})′ ∘ α = h′_{ψ} := π_{ψ} ∘ i_{E′}.

We now claim that

(29) ψ_{h_{ψ}} = α ∘ ψ or, equivalently, (id × α)(G_{ψ}) = G_{ψ_{h_{ψ}}} ,

where, as above, G_{ψ} ≤ E_{S} ×_{S} E′ and G_{ψ_{h_{ψ}}} ≤ E_{S} ×_{S} E′_{h_{ψ}} denotes the graph of −ψ and of −ψ_{h_{ψ}}, respectively.

To prove (29), put π = h_{ψ}∘pr_{E_{S}} +(h_{ψ})′∘pr_{E′_{h_{ψ}}}. Then we have π∘(id×α) = π_{ψ} because (π ∘ (id × α)) ∘ i_{E_{S}} = (h_{ψ} ∘ pr_{E_{S}} ∘ i_{E_{S}}) + ((h_{ψ})′ ∘ α ∘ pr_{E′} ∘ i_{E_{S}} = h_{ψ} = π_{ψ} ∘i_{E_{S}} and (π ∘(id×α))∘i_{E′} = (h_{ψ} ∘pr_{E_{S}} ∘i_{E′})+((h_{ψ})′ ∘α∘pr_{E′} ∘i_{E′} = (h_{ψ})′ ∘ α = h′_{ψ} = π_{ψ} ∘ i_{E′}.

Now by Prop. 5.5 we have Ker(π) = G_{ψ_{h_{ψ}}}, and so G_{ψ_{h_{ψ}}} = Ker(π) = Ker((id × α) ∘ π_{ψ}) = (id × α)(Ker(π_{ψ})) = (id × α)(G_{ψ}) since Ker(π_{ψ}) = G_{ψ} by definition of π_{ψ}. This proves (29), and hence α defines an isomorphism (E′, ψ) ≃ (E′_{h_{ψ}}, ψ_{h_{ψ}}). This means that Ψ′_{S}(Ψ′′_{S}(cl(E′, ψ))) = cl(E′_{h_{ψ}}, ψ_{h_{ψ}}) = cl(E′, ψ), and so Ψ′_{S} and Ψ′′_{S} are inverse maps of each other.

Remarks. (a) Note that the above theorem does not require any hypotheses on the base field K, and hence is true even if char(K)|2N. In fact, the hypothesis that K is a field was never used, and so the same result holds if E/K is an elliptic curve over an arbitrary ring (or scheme) K.

(b) The above theorem is actually a special case of general result which is valid for arbitrary abelian varieties. More precisely, let E/K be an abelian variety (or abelian scheme) of (relative) dimension d and let λ : E → Ê be a polarization. Fix an integer g ≥ 1, and consider, for a K-scheme S, the sets

A_{E/K,λ}^{(g)}(S) = {cl(A, λ_{A}, h)} and X_{E/K,λ,−1}^{(g)}(S) = {cl(B, λ_{B}, ψ)}

in which A/S, B/S are abelian schemes of dimension g, λ_{A} : A →^{∼} Â is a principal polarization, h : E ↪ A an injective homomorphism of “type λ”, i.e. ĥ ∘ λ ∘ h = λ, λ_{B} : B → B̂ is a polarization and ψ : Ker(λ) →^{∼} Ker(λ_{B}) is an anti-isometry (with respect to the pairings e^{λ} and e^{λ_{B}}). These definitions lead to functors A_{E/K,λ}^{(g)} and X_{E/K,λ,−1}^{(g)} which generalize the functors A and X above; clearly A_{E/K,N} = A_{E/K,Nλ_{E/K}}^{(2)} and X_{E/K,N,−1} = X_{E/K,Nλ_{E/K},−1}^{(1)} (if d = 1). Now the above proof of Theorem 5.10 can be modified to show that we have in general an isomorphism of functors:

Ψ : A_{E/K,λ}^{(g)} →^{∼} X_{E/K,λ,−1}^{(g−d)}. (30)

By combining Theorem 5.10 with the fundamental representation result Corollary 4.3 we obtain

Corollary 5.11 If N ≥ 3 is invertible in K, then the smooth affine curve X_{E/K,N,−1} represents the functor A_{E/K,N}.

5.3 The Torelli map τ : H_{E/K,N} → A_{E/K,N}

We now turn to study the Torelli map τ : H_{E/K,N} → A_{E/K,N} in more detail. The following result may be viewed as a version of Torelli’s theorem for (special) genus 2 families of curves.

Proposition 5.12 If N ≥ 3, then the Torelli map τ : H_{E/K,N} → A_{E/K,N} is a monomorphism.

Proof. Fix a K-scheme S and suppose that f_{i} : C_{i} → E_{S}, i = 1, 2, are two normalized genus 2 covers of E_{S}/S and such that τ_{S}(cl(f_{1})) = τ_{S}(cl(f_{2})). We want to show that cl(f_{1}) = cl(f_{2}), i.e. that there exists an S-isomorphism φ : C_{1} → C_{2} such that f_{1} = f_{2} ∘ φ.

The hypothesis τ_{S}(cl(f_{1})) = τ_{S}(cl(f_{2})) means that there exists an isomor-

phism α : J_{C_{1}} →^{∼} J_{C_{2}} such that α̂ ∘ λ_{C_{2}/S} ∘ α = λ_{C_{1}/S} and α ∘ h_{1} = h_{2}, where

h_{i} := f_{i}^{∗} ∘ λ_{E_{S}/S}. Since (f_{i})_{∗} = λ_{J_{E_{S} /S}}^{−1} ∘ (f_{i}^{∗})^{}ˆ ∘ λ_{C_{i}/S} = h_{i}^{∗}, the proof of Prop. 5.4 shows that we also have (f_{1})_{∗} = (f_{2})_{∗} ∘ α. We now treat the case that N is odd or even separately.

(a) Suppose first that N is odd. Then there exists a sheaf L_{i} ∈ Pic(J_{C_{i}/S}) of relative degree 1 such that (f_{i})_{∗} ∘ j_{L_{i}} = λ_{E_{S}/S} ∘ f_{i}, for i = 1, 2; cf. Th. 3.2(d). Put θ_{i} = j_{L_{i}}(C_{i}), which is a symmetric theta-divisor of J_{C_{i}/S}, i.e. λ_{O(θ_{i})} = λ_{C_{i}/S} and [−1]θ_{i} = θ_{i}. Since λ_{O(α^{∗}θ_{2})} = α̂ ∘ λ_{O(θ_{2})} ∘ α = λ_{O(θ_{1})}, we see that θ′_{1} := α^{−1}θ_{2} is a theta divisor of J_{C_{1}/S}. Clearly, θ′_{1} is symmetric because [−1]θ′_{1} = [−1]α^{−1}θ_{2} = α^{−1}[−1]θ_{2} = θ′_{1}. Furthermore, by (13) we

have θ′_{1} ∩ Ker((f_{1})_{∗})[2] = α^{−1}(θ_{2} ∩ Ker((f_{2})_{∗})[2]) = α^{−1}(Ker((f_{2})_{∗})[2]^{#}) =^{(13)} Ker((f_{1})_{∗})[2]^{#}. Thus, θ_{1} and θ′_{1} are two symmetric theta-divisors satisfying (13), and so by the uniqueness assertion of Th. 3.2(e) it follows that θ_{1} = θ′_{1}. This means that α_{|θ_{2}}^{−1} : θ_{2} →^{∼} θ_{1} is an isomorphism, and hence there is a unique S-isomorphism φ : C_{1} →^{∼} C_{2} such that j_{L_{2}} ∘ φ = α ∘ j_{L_{1}}. Then f_{2} ∘ φ = (f_{2})_{∗} ∘ j_{L_{2}} ∘ φ = (f_{2})_{∗} ∘ α ∘ j_{L_{1}} = (f_{1})_{∗} ∘ j_{L_{1}} = f_{1}, as desired.

(b) Now suppose that N ≥ 4 is even. Prop. 3.3(b) it is enough to verify that the assertion is true after a finite faithfully flat base change S′/S which we can choose by Th. 3.2(d) in such a way that there exist sheaves L_{i} ∈ Pic((J_{C_{i}/S})_{(S′)}) of relative degree 1 such that (f_{i})_{∗} ∘ j_{L_{i}} = λ_{E_{S}/S} ∘ f_{i}, for i = 1, 2. (In addition, we can assume that C_{(S′)}/S′ has a section.) By a similar argument as for the odd case we conclude that there exists an x ∈ Ker(f_{1})_{∗}[2](S′) such that T_{x}α^{−1}θ_{2} = θ_{1}, where θ_{i} = j_{L_{i}}((C_{i})_{(S′)}), and so there is a unique S′-isomorphism φ : (C_{1})_{(S′)} →^{∼} (C_{2})_{(S′)} such that j_{L_{2}} ∘ φ = α ∘ T_{−x} ∘ j_{L_{1}}. Since −x ∈ Ker(f_{1})_{∗}[2], a similar computation as in the odd case shows that f_{2} ∘ φ = f_{1}, and so the assertion follows.

Corollary 5.13 If f : C → E_{S} is a normalized genus 2 cover of degree N, then E′_{f} = Ker(f_{∗}) is an elliptic curve over S and we have a unique anti-isometry ψ_{f} : E_{S}[N] →^{∼} E′_{f}[N] such that (f′)^{∗} ∘ ψ_{f} = (f_{∗})_{|E_{S}[N]}, where (f′)^{∗} : E′_{f} ↪ J_{C/S} denotes the canonical embedding. Furthermore, the rule (f : C → E_{S}) ↦ (E′_{f}, ψ_{f}) defines a functor

Ψ = Ψ′ ∘ τ : H_{E/K,N} → X_{E,N,−1}

which is a monomorphism if N ≥ 3.

Proof. Since Ψ := Ψ′ ∘ τ is a monomorphism for N ≥ 3 by Th. 5.10 and Prop. 5.12, all the assertions follow once we have shown that (E′_{f}, ψ_{f}) is a representative of Ψ′_{S}(τ_{S}(cl(f))). But this is clear, for by definition Ψ′_{S}τ_{S}(cl(f)) = Ψ′_{S}(cl(J_{C/S}, λ_{C/S}, h_{f})) = cl(E′_{h_{f}} , ψ_{h_{f}}), where h_{f} = f^{∗} ∘ λ_{E_{S}/S}, E′_{h_{f}} = Ker(h_{f}^{∗}) = Ker(f_{∗}) = E′_{f}, and ψ_{f} := ψ_{h_{f}} is uniquely determined by h′_{f} ∘ ψ_{f} = (h_{f})_{|E_{S}[N]}, with h′_{f} = (f′)^{∗} : E′_{f} ↪ J_{C/S}.

The final step in the basic construction is to identify the image of H_{E/K,N} in A_{E/K,N} with respect to the Torelli map τ. The criterion that will be given below amounts essentially to the condition that the theta-divisor(s) associated to the polarization λ_{J} of an abelian surface J/S be smooth over S. However, since a given polarization need not have a theta-divisor that is rational over S, this condition has to be suitably modified. To this end we introduce the following concept.

Definition. A principal polarization λ : J → Ĵ of an abelian scheme J/S is called theta-smooth at s ∈ S if the principal polarization

λ\overline{s} : J\overline{s} := J ⊗ \overline{κ(s)} →^{∼} Ĵ\overline{s}

obtained by base change with \overline{κ(s)}, the algebraic closure of κ(s), has a thetadivisor θ\overline{s} which is smooth over \overline{κ(s)}.

We now prove the following fundamental fact which is also of independent interest.

Proposition 5.14 If J/S is an abelian scheme of relative dimension 2 with a principal polarization λ : J → Ĵ, then following conditions are equivalent:

(i) (J, λ) is a Jacobian, i.e. there is a smooth curve C/S of genus 2 such that (J_{C/S}, λ_{C/S}) ≃ (J, λ).

(ii) λ is theta-smooth for all s ∈ S.

(iii) There exists a finite, faithfully flat base extension β : S′ → S such that λ_{(S′)} has an associated theta-divisor θ′ which is smooth over S′.

Proof. (i) ⇒ (ii): Let s ∈ S. Since C\overline{s} has genus 2 = dim(J\overline{s}), the image j_{x}(C\overline{s}) of j_{x} : C\overline{s} ↪ J\overline{s} (for any x ∈ C\overline{s}(\overline{κ(s)})) is a smooth theta-divisor of J\overline{s} associated to λ\overline{s}, and so λ is theta-smooth at s, for all s ∈ S.

(ii) ⇒ (iii): By Lemma 5.15 below we know that there exists a finite, faithfully flat cover β : S′ → S such that λ_{(S′)} has a theta-divisor θ′, and then θ′ ; \overline{s′} is a theta-divisor of λ\overline{s′}, for all s′ ∈ S′. Since θ′ ; \overline{s′} is unique up to a translation on J\overline{s′}, condition (ii) implies that θ′ ; \overline{s′} is smooth over \overline{κ(s′)}, and hence θ′_{s′} is smooth over κ(s′). Thus, θ′_{s′} is smooth for all s′ ∈ S′, and so it follows that θ′ is smooth over S′.

(iii) ⇒ (i): We first prove that there exists a smooth curve C/S such that C′ := C_{(S′)} ≃ θ′. For this we shall use the method of descent of Grothendieck (cf. [BLR], Th. 6.1/7) applied to the pair (θ′, L′), where L′ = ω_{θ′/S′}; note that ω_{θ′/S′} is an ample invertible sheaf on θ′ because θ′ is a smooth curve of genus 2 over S′.

To apply this method, consider S′′ := S′ ×_{S} S′ with the two projections p_{i} : S′′ → S′ and let θ′′_{i} := p_{i}^{∗}θ′ = θ′_{(p_{i})} be the base change of θ′ via p_{i}, for i = 1, 2. Now θ′′_{1} and θ′′_{2} are both theta-divisors on J′′ := J_{(S′′)} with respect to the principal polarization λ′′ := λ_{(S′′)}, and so by Lemma 7.1 there exists a unique section x ∈ J′′(S′′) such that T_{x}^{∗}(θ′′_{2}) = θ′′_{1}, i.e. t := (T_{x})_{|θ′′_{1}} : θ′′_{1} →^{∼} θ′′_{2} is an isomorphism of S′′-curves. Furthermore, since p_{i}^{∗}L′ = ω_{θ′′_{i}/S′′} = Ω_{θ′′_{i}/S′′}^{1}, it follows that we have a canonical sheaf isomorphism ω : t^{∗}p_{2}^{∗}L′ ≃ p_{1}^{∗}L′. Using the fact that x and ω are uniquely defined, it is now easy to check that the pair (t, ω) satisfies the cocyle condition and hence defines a descent datum on (θ′, L′). Thus, by Grothendieck’s theorem ([BLR], Th. 6.1/7), there exists a scheme C/S (and a sheaf L ∈ Pic(C)) such that C_{(S′)} ≃ θ′. Furthermore, since θ′/S′ is a smooth curve of genus 2, so is C/S (since S′/S is faithfully flat).

To show that J_{C/S} ≃ J, let j′ : C′ := C_{(S′)} ≃ θ′ ↪ J′ denote the embedding constructed above, and consider the morphism f′ : C′×_{S′} C′ → Ĵ′ defined by f′ = λ ∘ s ∘ j′ × j′, where s : J′ ×_{S′} J′ → J′ denotes the minus map (x, y) ↦ x − y. We claim that f′ = f_{(S′)}, for some f : C ×_{S} C → Ĵ. By descent theory ([BLR], Th. 6.1/6(a)), it is enough to show that p_{1}^{∗}f′ = p_{2}^{∗}f′, where as above p_{i} : S′′ = S′ ×_{S} S′ → S′ are the projections.

Suppose first that x′′, y′′ ∈ C(S′′) are two sections. Then (p_{i}^{∗}f′)(x′′, y′′) = λ(p_{i}^{∗}j′(x′′)−p_{i}^{∗}j′(y′′)) = cl((T_{p_{i}^{∗}j′(x′′)}^{∗}p_{i}^{∗}O(θ′)⊗(p_{i}^{∗}O(θ′))^{−1})⊗(T_{p_{i}^{∗}j′(y′′)}^{∗}p_{i}^{∗}O(θ′)⊗ (p_{i}^{∗}O(θ′))^{−1})^{−1}) = cl(T_{p_{i}^{∗}j′(x′′)}^{∗}p_{i}^{∗}O(θ′) ⊗ (T_{p_{i}^{∗}j′(y′′)}^{∗}p_{i}^{∗}O(θ′))^{−1}). Now by the above we know that there exists an x ∈ J(S′′) such that p_{2}^{∗}j′ = T_{x} ∘ p_{1}^{∗}j′ and T_{x}^{∗}p_{2}^{∗}O(θ′) ≃ p_{1}^{∗}O(θ′), and so T_{p_{2}^{∗}j′(x′′)}^{∗}p_{2}^{∗}O(θ′) ≃ T_{p_{1}^{∗}j′(x′′)}^{∗}T_{x}^{∗}p_{2}^{∗}O(θ′) ≃ T_{p_{1}^{∗}j′(x′′)}^{∗}p_{1}^{∗}O(θ′) (and similarly for x′′ replaced by y′′.) This therefore gives p_{1}^{∗}(f′)(x′′, y′′) = cl(T_{p_{1}^{∗}j′(x′′)}^{∗}p_{1}^{∗}O(θ′)⊗(T_{p_{1}^{∗}j′(y′′)}^{∗}p_{1}^{∗}O(θ′))^{−1})= cl(T_{p_{2}^{∗}j′(x′′)}^{∗}p_{2}^{∗}O(θ′) ⊗(T_{p_{2}^{∗}j′(y′′)}^{∗}p_{2}^{∗}O(θ′))^{−1}) = p_{2}^{∗}(f′)(x′′, y′′). Repeating the same argument with arbitrary T -valued points (for a scheme T/S′′) shows that we have p_{1}^{∗}(f′) = p_{2}^{∗}(f′).

Thus, by descent theory, there exists a unique morphism f : C ×_{S} C → Ĵ such that f_{(S′)} = f′. It is immediate from the definition that f ∘δ = 0, where δ is the diagonal map, and so by the Albanese property (∗ ∗ ∗) (applied to λ^{−1} ∘f : C ×C → J) there exists a unique homomorphism h : J_{C/S} → J such that h ∘ f_{C/S} = λ^{−1} ∘ f. Now h is an isomorphism because after base-change it is clear from the definition that f′ = λ ∘ f_{C′/S′}.

Above we had used:

Lemma 5.15 If λ : A → Â is a polarization of an abelian scheme A/S, then there exists a finite, faithfully flat base extension β : S′ → S such that λ_{(S′)} is defined by an invertible sheaf L′ ∈ Pic(A_{(S′)}). Furthermore, if λ is a principal polarization, then L′ can be chosen such that L′ ≃ O(θ′) for a theta-divisor θ′ on A_{(S′)}.

Proof. First note that by Mumford[Mu1], p. 121, there exists L ∈ Pic(A) such that λ_{L} = 2λ. Then A[2] ≤ K(L) is an isotropic subgroup, and so by Lemma 5.8 there exists a finite, faithfully flat base extension β_{0} : S′_{0} → S such that L_{(S′_{0})} ≃ [2]^{∗}(L′_{0}), for some L′_{0} ∈ Pic(A_{(S′_{0})}). Then by a similar argument as in [Mu2], p. 231, there exist, after a suitable finite, faithfully flat base-change β_{1} : S′ → S′_{0}, invertible sheaves L′ ∈ Pic(A′) and M′ ∈ Pic(S′) such that L_{(S′)} ≃ (L′)^{2} ⊗ p_{A′}^{∗}(M′), where A′ = A_{(S′)}. Then 2λ_{(S′)} = λ_{L_{(S′)}} = λ_{(L′)^{2}} = 2λ_{L′}, and so λ_{(S′)} = λ_{L}. This proves the first assertion.

To prove the second assertion, consider M := p_{∗}L′, where p : A_{(S′)} → S′ is the structure map. Then M is locally free of rank 1 cf. [Mu1], p. 123; recall that λ is now assumed to be a principal polarization. Thus M ∈ Pic(S), and hence L′′ := L′ ⊗ (p^{∗}M)^{−1} satisfies p_{∗}(L′′) ≃ O_{S}. Via this isomorphism, the global section id_{O_{S}} ∈ Hom(O_{S}, O_{S}) = Γ(S, O_{S}) gives rise to a global section of L′′ which defines a relative Cartier divisor θ′ representing L′′ (see the discussion on p. 212ff of [BLR]), and so the second assertion follows.

Corollary 5.16 If λ is a principal polarization of an abelian scheme J/S of relative dimension 2, then the subset S^{θ−sm}(λ) ⊂ S of those points of S at which λ is theta-smooth is an open subset of S.

Proof. By the lemma there exists a finite faithfully flat base extension β : S′ → S such that λ_{(S′)} has a theta-divisor θ ⊂ J_{(S′)}. Then by definition (cf. [EGA], (IV, 17.3.7)) the set θ^{sm} ⊂ θ of smooth points of θ/S′ is an open subset of θ and hence S′_{0} = S′ \ p(θ \ θ^{sm}) is also open (since the structure map p = p_{J_{(S′)}} : J_{(S′)} → S′ is proper). By definition (and [EGA], (IV.17.8.2)) S′_{0} is the set of points s ∈ S where θ_{s} is smooth, and so S′_{0} = β^{−1}(S^{θ−sm}(λ)). Since β is faithfully flat (and hence is open and surjective), it follows that S^{θ−sm}(λ) is open in S.

Notation. For each K-scheme S let

J_{E/K,N}(S) = A_{E/K,N}^{θ−sm}(S) ⊂ A_{E/K,N}(S)

denote the subset consisting of those classes cl(J, λ, h) such that J is thetasmooth (for all s ∈ S). Since any base-change of a theta-smooth abelian scheme is again theta-smooth, this defines a subfunctor J_{E/K,N} of A_{E/K,N}.

Proposition 5.17 The functor J_{E/K,N} is an open subfunctor of A_{A/K,N}. Furthermore, the Torelli morphism τ : H_{E/K,N} → A_{E/K,N} factors over J_{E/K,N} and defines a surjection

τ : H_{E/K,N} → J_{E/K,N}

which is an isomorphism for N ≥ 3. In particular, τ is an open embedding of functors (if N ≥ 3).

Proof. Let S be a K-scheme and let F : h_{S} → J_{E/K,N} be a morphism of functors. If we write F (id_{S}) = cl(J, λ, h) ∈ A_{E/K,N}(S), then by Cor. 5.16 the set U := S^{θ−sm}(λ) ⊂ S at which λ is theta-smooth is an open subset of S, and so F (id_{S})_{|U} := (J_{(U)}, λ_{(U)}, h_{(U)} ∈ J_{E/K,N}(U). Clearly, f(id_{S})_{|U} represents the fibre product J_{E/K,N} ×_{A_{E/K,N}} h_{S}, which means that J_{E/K,N} is an open subfunctor of A_{E/K,N}. This proves the first assertion.

Since each Jacobian is theta-smooth, it is immediate that τ maps into J_{E/K,N}. To prove that τ is surjective, let S be a K-scheme and let cl(J, λ, h)∈ J_{E/K,N}(S). Then by Prop. 5.14 there exists a smooth curve C/S of genus 2 such that (J_{C/S}, λ_{C/S}) ≃ (J, λ). Moreover, by Th. 3.2(f) there exists a normalized cover f : C → E_{S} of degree N such that f^{∗} = h. Thus τ_{S}(cl(f)) = cl(J, λ, h), and so τ is surjective. Moreover, if N ≥ 3 then τ is an isomorphism because τ_{S} is injective by Cor. 5.13.

This, therefore, completes the “basic construction”. We can now summarize our results as follows.

Theorem 5.18 The functor Ψ : H_{E/K,N} → X_{E/K,N,−1} is an open embedding of functors if N ≥ 3; in particular, Ψ is relatively representable. Thus, if N ≥ 3 is invertible in K, then H_{E/K,N} is represented by a smooth curve H_{E/K,N}/K which is an open subscheme of the (twisted) modular curve X_{E/K,N,−1}. In particular, the fibres of H_{E/K,N}/K are geometrically irreducible.

Proof. Recall that Ψ = Ψ′ ∘ τ (cf. Cor. 5.13). Since Ψ′ is an isomorphism by Theorem 5.10 and τ is an open embedding by Prop. 5.17 (if N ≥ 3), the first assertion follows.

Now if N ≥ 3 is invertible in K, then by Cor. 4.3 the twisted modular curve X_{E/K,N−1} represents the functor X_{E/K,N,−1}, and so by the first assertion H_{E/K,K} is represented by an open subscheme H_{E/K,N} of X_{E/K,N,−1}. Since X_{E/K,N,−1} is a smooth K-curve with geometrically connected fibres (cf.

Cor. 4.3), the same is true for H_{E/K,N} if (and only if) each geometric fibre (H_{E/K,N})\overline{s} is not empty.

To see that this is the case, write k = κ(\overline{s}) and let E′/k be an elliptic curve which is not isogenous to E_{k}. Since k is algebraically closed (and char(k)|̸ N), there exists an anti-isometry ψ : E_{k}[N] →^{∼} E′[N], which determines a triple cl(J, λ, h) = Ψ′′_{k}(cl(E_{k}, E′, ψ)) ∈ A_{E/K,N}(k). But then λ is automatically theta-smooth (cf. [FK], Prop. 1.4 or Theorem 6.1 below), and so H_{E/K,N}(k) ≠ ∅ by Prop. 5.17.

Remark. Note that the above Theorem 5.18 includes Theorem 1.1 of the introduction as a special case.

The “basic construction” itself is the following statement, which follows from the fact that τ : H_{E/K,N} → J_{E/K,N} is surjective (Proposition 5.17) and that Ψ′ is bijective (Theorem 5.10).

Corollary 5.19 (Basic Construction) If E′/S is an elliptic curve and ψ : E_{S}[N] → E′[N] is an anti-isometry which is “theta-smooth” (in the sense that the induced principal polarization λ_{J} on J_{ψ} = (E_{S} ×_{S} E′)/Graph(−ψ) is theta-smooth), then there is a normalized genus 2 cover f : C → E_{S} of degree N such that the associated anti-isometry ψ_{f} is equivalent to ψ. Moreover, every normalized genus 2 cover of degree N ≥ 2 arises in this way.

Remarks. (a) A small blemish in the above result is that the notion of a theta-smooth anti-isometry is defined in a round-about way. However, by using the results of [Ka2] it is possible to give a definition that is intrinsic to the data (E′, ψ) as will be explained in the next section; cf. Theorem 6.1.

(b) Although the basic idea and underlying steps of this “basic construction” are as in [FK], there is one notable difference in the method of proof. In [FK], the existence of the curve C associated to a (theta-smooth) antiisometry ψ was proved by using (for N odd) a characterization of a suitable symmetric theta-divisor in terms of its linear equivalence class; cf. [FK], Proposition 1.1. This was replaced here by a different characterization in terms of a condition involving certain 2-torsion points of J_{ψ}; cf. Theorem 3.2(e) (together with some descent arguments).

6 The degeneracy locus D_{E/K,N}

In the previous section we had constructed the moduli space H_{E/K,N} as an open subset of the twisted modular curve X_{E/K,N−1}. In order to complete the description of H_{E/K,N}, we need to analyze the “degeneracy locus”

D_{E/K,N} := X_{E/K,N,−1} \ H_{E/K,N}.

As a set^{1}, D_{E/K,N} is the disjoint union of its fibres (D_{E/K,N})_{s} = D_{E_{s}/κ(s),N}, and so it is enough to describe these. Thus, we may assume in the sequel that K = κ(s) is a field.

For an algebraically closed field K, the results of [Ka2] yield an explicit description of the set D_{E/K,N}. To be able connect these results with the set D_{E/K,N}, we first recall that Proposition 5.17 yields a natural identification

(31) D_{E/K,N} = {cl(E′, ψ) ∈ X_{E/K,N,−1}(K) : ψ is reducible}

where, as in [Ka2], p. 98, an anti-isometry ψ : E[N] → E′[N] is called reducible if its associated theta-divisor θ on J_{ψ} is reducible. Note that since the theta-divisor θ is reducible if and only if it is not smooth, the antiisometry ψ is reducible if and only if it is not theta-smooth in the sense of Corollary 5.19.

Several characterizations of reducible anti-isometries were given in [Ka2]; one of these is the following.

Theorem 6.1 An isometry ψ : E[N] →^{∼} E′[N] is reducible if and only if there exist four isogenies f_{i} : E → E_{i} and f′_{i} : E_{i} → E′, for i = 1, 2, satisfying the following conditions:

(32)  deg(f_{1}) + deg(f_{2}) = N,
(33)  Ker(f_{1}) ∩ Ker(f_{2}) = {0},
(34)  f′_{1} ∘ f_{1} = −f′_{2} ∘ f_{2},
(35)  (f′_{i})^{∗} ∘ ψ = (f_{i})_{|E[N]}, for i = 1, 2,

where (f′_{i})^{∗} = λ_{E_{i}}^{−1} ∘ f̂′_{i} ∘ λ_{E′} : E′ → E_{i} denotes the “dual” of f′_{i}.

1Although D_{E/K,N} is a priori just a closed subset of H_{E/K,N}, we can give it a unique subscheme structure by observing that it is the underlying set of the subscheme which represents a suitable closed subfunctor of X_{E/K,N,−1}. However, since this extra structure is not required in the sequel, we do not explain this in more detail here.

In particular, if N is prime, then ψ is reducible if and only if there exists an integer k with 0 < k < N and an isogeny f : E → E′ of degree k(N − k) such that

f_{|E[N]} = kψ.

Proof. The first assertion is Corollary 2.4 of [Ka2] (with f′_{2} replaced by −f′_{2}), and the second is explained in Remark 2.5.

In addition, the main results of [Ka2] and [Ka3] can be re-interpreted to yield a formula for the number of points in the degeneracy locus D_{E/K,N} (which is finite by Theorem 1.1).

Theorem 6.2 The number of points in D_{E/K,N} is

#D_{E/K,N} = \frac{1}{2} ∑_{k=1}^{N−1} σ(E, k(N − k), N) ≤ \frac{1}{24N} (5N − 6)#SL_{2}(ℤ/Nℤ), (36)

where σ(E, n, N) denotes the number of subgroup schemes H ≤ E of order n such that E[p] ≰ H, for all primes p|N. Furthermore, equality holds in (36) if and only if char(K)|̸ N!.

Proof. As in [Ka2], p. 107, let r(E, E′, N) denote the number of reducible anti-isometries ψ : E[N] → E′[N]. Then from formula (31) it follows that

#D_{E/K,N} = ∑_{E′} \frac{r(E, E′, N)}{#Aut(E′)},

where the sum extends over a system of representatives of the isomorphism classes of elliptic curves E′/K. But this sum is exactly the left hand side of the “mass formula” (4.1) of Theorem 4.1 of [Ka2], p. 115, and so the (first) equality of (36) is just a restatement of the formula (4.1). Similarly, the inequality of (36) is a restatement Theorem 6 of [Ka3], in which it is also asserted that equality holds if and only if char(K)|̸ N!.

Remark. The above result shows that if char(K) = 0, then D_{E/K,N} consists asymptotically of \frac{5}{4π^{2}} N^{3} points because

#SL_{2}(ℤ/Nℤ) = N^{3} ∏_{p|N} (1 − \frac{1}{p^{2}} ) ∼ \frac{1}{ζ(2)}N^{3} = \frac{6}{π^{2}} N^{3}.

7 Appendix: Jacobians of relative curves

The purpose of this appendix is review (and extend) some of the basic facts concerning the Jacobians of relative curves C/S. The main references for these facts are [Mu1], chapter 6, and [BLR], chapter 9. In addition, we review some facts concerning relative Cartier divisors.

Definition. If g ≥ 1, then a (relative) curve of genus g is a proper, smooth morphism p : C → S (of finite presentation) whose fibres C_{s} are geometrically connected curves of genus g. If g = 1, then we assume in addition that there exists a section η : S → C; in this case it follows from [Mu1], p. 124, that C/S is an abelian scheme of relative dimension 1 with zero-section 0_{C} = η.

Basic Facts: 1) Projectivity: Each relative curve C/S is strongly projective in the sense of Altman-Kleiman; cf. [BLR], p. 211. More precisely, if g ≥ 2, then the relative dualizing sheaf ω_{C/S} is S-ample (in fact, ω_{C/S}^{⊗n} is relatively very ample for n ≥ 3 by [DM], p. 78), and if g = 1 then the invertible sheaf O_{C}(η) associated to the section η is S-ample (cf. [BLR], Remark 9.3/2) (in fact, O_{C}(η)^{⊗n} is relatively very ample for n ≥ 3).

2) Jacobians: The (relative) Picard-functor Pic_{C/S} is representable by a smooth, separated S-scheme Pic_{C/S} which is locally of finite presentation, and we have a decomposition

Pic_{C/S} = ∐_{n∈ℤ}J_{C/S}^{(n)}

where J_{C/S}^{(n)} = (Pic_{C/S})^{n} denotes the open and closed subscheme of Pic_{C/S} consisting of all line bundles of degree n; cf. [BLR], Th. 9.3/1. Thus, for each S-scheme T we have a functorial injection

cl : Pic^{(n)}(X_{T})/p_{(T)}^{∗}Pic(T) ↪ J_{C/S}^{(n)}(T) = Hom(T, J_{C/S}^{(n)})

which is surjective if and only there exists a universal sheaf on C ×_{S} J^{(n)}; cf. [MB3], p. 34. Furthermore, J_{C/S}^{(0)} = (Pic_{C/S})^{0} coincides with the identity component J_{C/S} := Pic_{C/S}^{0} of Pic_{C/S} and is an abelian scheme over S ([BLR], loc. cit. and Prop. 9.4/4). In addition, each J_{C/S}^{(n)} is strongly projective over S and is an S-torsor under J_{C/S}; cf. [BLR], Th. 8.2/5, Th. 9.3/1 or [MB3], p. 34. In particular, if there is an invertible sheaf L ∈ Pic^{(n)}(C) of relative degree n, then the map g ↦ g · cl(L) defines an isomorphism T_{L} : J_{C/S} →^{∼} J_{C/S}^{(n)} of S-schemes.

3) Duality and Polarizations: The dual abelian scheme Ĵ_{C/S} := Pic_{J/S}^{0} of J = J_{C/S} exists and there is an explicit isomorphism λ_{C/S} : J_{C/S} →^{∼} Ĵ_{C/S}; which is described in more detail in 5) below.

More generally, for any (projective) abelian scheme A/S, its dual Â = Pic_{A/S}^{0} = Pic_{A/S}^{τ} exists and is a (projective) abelian scheme (cf. [Mu1], p. 117 and [FC], p. 3); here, Pic_{A/S}^{τ}(T) consists of those L ∈ Pic(A_{T})/Pic(T) such that (a power of) L_{t} is algebraically equivalent to 0, for all t ∈ T. By functoriality, any homomorphism f : A → B of abelian schemes induces a homomorphism f̂ = f^{∗} : B̂ → Â, called the dual homomorphism. Furthermore, the canonical map κ_{A} : A → ˆ Â is an isomorphism (cf. [FC], p. 3; this follows from [Mu2], p. 132), and we have f = κ_{B}^{−1} ∘ ˆ f̂∘ κ_{A}.

Each invertible sheaf L ∈ Pic(A) defines an S-homomorphism λ_{L} : A → Â by the rule x ↦ cl(T_{x}^{∗}L ⊗ L^{−1}) (functorially in S); cf. [Mu1], p. 120. Note that if h : B → A is any S-homomorphism, then we have

λ_{h^{∗}L} = ĥ ∘ λ_{L} ∘ h, (37)

as is immediate from the definition of ĥ.

If L is (relatively) ample, then λ_{L} is an isogeny and hence a polarization in the sense of Mumford[Mu1], p. 120. (However, not every polarization is of this form.) Note that if λ : A → Â is a polarization, then λ is symmetric in the sense that λ̂ ∘ κ_{A} = λ; cf. [FC], p. 4.

Each polarization λ = λ_{L} : A → Â induces a (non-degenerate) pairing

e^{λ} : K(λ) ×_{S} K(λ) → (𝔾_{m})_{/S},

where K(λ) = Ker(λ); cf. [Mu2], p. 228ff and [FC], p. 5. This pairing is related to the usual e_{N}-pairing e_{N} : A[N] ×_{S} Â[N] → μ_{N} by the formula e_{N} ∘ (id_{A[N]} × λ) = e^{Nλ}, if λ is a principal polarization (and N is invertible in S); cf. formula (4) on p.228 of [Mu2]. We also note that if π : B → A is an isogeny, then we have the formula

e^{π̂∘λ∘π} = e^{λ} ∘ π × π on π^{−1}K(λ) × π^{−1}K(λ); (38)

cf. property (1) of [Mu2], p. 228 (combined with the fact that π̂∘λ_{L}∘π = λ_{π^{∗}L}, as is evident from the definitions).

4) Products of Abelian Schemes: It is immediate that the category \underline{AbSch}_{/S} of abelian S-schemes (with group homomorphisms) is an additive category (in the sense of [HS], p. 75). Thus, if A_{1}, A_{2} are two abelian Sschemes, then A = A_{1} ×_{S} A_{2} is a co-product of A_{1} and A_{2} in \underline{AbSch}_{/S} with respect to the maps i_{A_{1}} = id_{A_{1}} × 0_{A_{2}} : A_{1} = A_{1} ×_{S} S → A and i_{A_{2}} = 0_{A_{2}} × id_{A_{2}} : A_{2} = S ×_{S} A_{2} → A; cf. [HS], Prop. II.9.1. Furthermore, by duality it then follows that Â is a product of Â_{1} and Â_{2} with respect to the morphisms î_{A_{1}} and î_{A_{2}}, and so there is a unique isomorphism

(39) φ_{A_{1},A_{2}} : Â →^{∼} Â_{1} × Â_{2} such that prÂ_{k} ∘ φ_{A_{1},A_{2}} = î_{A_{k}}, for k = 1, 2;

we then also have that φ_{A_{1},A_{2}} ∘ p̂r_{A_{k}} = iÂ_{k}, for k = 1, 2.

Note that if λ_{k} : A_{k} → Â_{k}, k = 1, 2 are two polarizations, then the unique map λ = λ_{1} ⊗ λ_{2} : A → Â such that φ_{A_{1},A_{2}} ∘ λ = λ_{1} ×_{S} λ_{2}, or, equivalently, such that î_{A_{k}} ∘ (λ_{1} ⊗ λ_{2}) = λ_{k} ∘ pr_{A_{k}}, for k = 1, 2, (40)

is again a polarization (since this is true fibre-by-fibre), called the product polarization of A = A_{1} ×_{S} A_{2} defined by λ_{1} and λ_{2}.

For later use we also note that if f : A → B is a homomorphism of abelian schemes then

(41) f is finite ⇔ f̂ is surjective.

(Indeed, since f is finite (respectively, surjective) if and only if f_{s} has this property for all s ∈ S (observe that f is proper and use [EGA], (IV, 18.12.4), respectively, (I, 3.6.1)), it is enough to verify this for S = Spec(K), K a field, in which case this is known; cf. [La], p. 125.) Furthermore we have

(42) f is a closed immersion ⇔ Ker(f̂) is an abelian subscheme of B̂ and f̂ is surjective.

(Again, this is known in the case that the base is a field (cf. [La], p. 216). The general case be reduced to the field case by noting that f is a closed immersion if and only if f_{s} : A_{s} → B_{s} has this property for all s ∈ S (use [EGA], (IV, 18.12.6)) and similarly, Ker(f̂) is an abelian subscheme if and only if Ker(f_{s}) = Ker(f)_{s} is an abelian subvariety (i.e. is geometrically reduced and irreducible).) Note that it follows easily from (42) that if

0 → A →^{f} B →^{g} C → 0

is an exact sequence of abelian S-schemes, then the dual sequence is also exact (and conversely):

0 → Ĉ →^{ĝ} B̂ →^{f̂} Â → 0. (43)

5) Theta-divisors: If J_{C/S} is the Jacobian of a relative curve C/S of genus g, then there is a canonical principal polarization λ_{C/S} : J_{C/S} →^{∼} Ĵ_{C/S}; cf. [Mu1], p. 118 or [MB3], p. 40. In general, λ_{C/S} is not of the form λ_{L}, for some L ∈ Pic(C), even if S = Spec(K), K a field. However, if C/S has a section or, more generally, if C/S has an invertible sheaf of relative degree g − 1, then there exits a theta-divisor θ on J_{C/S}, i.e. an effective relative Cartier divisor θ ⊂ J_{C/S} such that λ_{C/S} = λ_{O(θ)}; cf. [MB3], p. 40. More precisely, θ is constructed as follows. Let Θ = Im(C^{g−1} → J_{C/S}^{(g−1)}), which is an effective relative Cartier divisor on J^{(g−1)} (for any base S); cf. [MB3], p.

35. If we have an invertible sheaf L of relative degree g − 1, then it induces an isomorphism T_{L} : J_{C/S} →^{∼} J_{C/S}^{(g−1)}, and the inverse image θ_{L} = T_{L}^{∗}Θ of Θ under T_{L} is a theta-divisor on J_{C/S}.

Note that any two theta-divisors are translates of each other:

Lemma 7.1 Let D be an effective relative Cartier divisor on a strongly projective abelian scheme p : A → S and let L = O_{A}(D) be its associated invertible sheaf.

(a) If H^{0}(A_{s}, L_{s}) = 1 and H^{1}(A_{s}, L_{s}) = 0, for all s ∈ S, then the canonical map p_{∗}O_{A} = O_{S} → p_{∗}L is an isomorphism. Thus, if D′ is any effective relative Cartier divisor such that cl(O_{A}(D′)) = cl(L) in Pic_{A/S}(S), then D′ = D (as effective Cartier divisors).

(b) If L is ample and defines a principal polarization λ_{L} : A →^{∼} Â, and D′ is an effective relative Cartier divisor such that λ_{O_{A}(D′)} = λ_{L} (or such that cl(O_{A}(D′ − D)) ∈ Pic_{A/S}^{0}(S)), then D′ = T_{x}^{∗}(D), for some x ∈ A(S).

Proof. (a) Without loss of generality we may assume that S is locally noetherian. Since D is an effective divisor, it defines an injection f_{D} : O_{A} ↪ O_{A}(D) = L, which induces an injection p_{∗}(f_{D}) : p_{∗}(O_{A}) = O_{S} ↪ p_{∗}(L). Now the hypotheses on L imply that R^{1}p_{∗}L = 0 and that p_{∗}L is locally free and that p_{∗}L_{s} ⊗ κ(s) →^{∼} H^{0}(A_{s}, L_{s}) is an isomorphism; cf. [Ha], Th. III.12.11. Since H^{0}(A_{s}, O_{A_{s}}) →^{∼} H^{0}(A_{s}, L_{s}) is an isomorphism, it follows that p_{∗}(f_{D})_{s} : p_{∗}(O_{A})_{s} →^{∼} p_{∗}(L)_{s} is surjective, for all s ∈ S, and hence so is p_{∗}(f_{D}) : p_{∗}(O_{A}) → p_{∗}(L) by Nakayama. Thus, p_{∗}(f_{D}) is an isomorphism.

Moreover, since the above result is also true after an arbitrary basechange, it follows that p_{∗}L is cohomologically flat in dimension 0. Thus, by [BLR], Prop. 8.2/7, the set of effective relative Cartier divisors D′ ∈ Div^{+}(A/S) of A/S with cl(O_{A}(D′)) = cl(L) (in Pic_{A/S}(S)) is parametrized by ℙ(F), where F is the dual of p_{∗}(L) ≃ O_{S}. But then ℙ(F) = S, and so Hom_{S}(S, ℙ(F)) = {id_{S}}, which proves the assertion.

(b) First note that if λ_{L′} = λ_{L} then λ_{L⊗L^{−1}} = 0 and hence L′ ⊗ L^{−1} ∈ Pic^{0}(A/S) (by the definition of Pic^{0}(A/S)); in particular, cl(L′ ⊗ L^{−1}) ∈ Pic_{A/S}^{0}(S) = Pic^{0}(A/S)/p^{∗}Pic(S).

Now since λ_{L} : A →^{∼} Â = Pic_{A/S}^{0} is an isomorphism, there exists an x ∈ A(S) such that cl(O_{A}(D′ − D)) = cl(T_{x}^{∗}L ⊗ L^{−1}), and so cl(O_{A}(D′)) = cl(T_{x}^{∗}L) = cl(O(T_{x}^{∗}D)) in Pic_{A/S}(S). By the Riemann-Roch Theorem and the Vanishing Theorem ([Mu2], p. 150) L (and hence also T_{x}^{∗}L) satisfies the hypotheses of part (a), and hence it follows that D′ = T_{x}^{∗}(D), as desired.

6) The Autoduality of Jacobians: If a : S → C is a section of C/S, then there is an S-morphism j_{a} : C → J_{C/S} which represents the map z ↦ cl(O_{C}(z − a)); cf. [MB3], p. 43. Furthermore, the induced map j_{a}^{∗} : Ĵ_{C/S} = Pic_{J/S}^{0} → J_{C/S} = Pic_{C/S}^{0} is an isomorphism which satisfies λ_{C/S} ∘ j_{a}^{∗} = −id_{J} ([MB3], p. 43); in particular, j_{a}^{∗} does not depend on a. Note that j_{a} is a closed immersion since this is known to be true fibre-by-fibre and so we can apply [EGA] (IV, 18.12.6) again. Furthermore, j_{a} has the following universal property (usually called the Albanese property of J_{C/S}):

(∗) If A/S is an abelian scheme and f : C → A an S-morphism such that f(a) = 0_{A}, the zero-section of A/S, then there is a unique S- homomorphism f_{∗} : J_{C} → A such that f = f_{∗} ∘ j_{a}; in fact, f_{∗} is given by the formula f_{∗} = −κ_{A}^{−1} ∘ (f^{∗})^{∗} ∘ λ_{C/S}.

[Proof. Suppose first that there exists an f_{∗} with f = f_{∗} ∘ j_{a}. Then the map f^{∗} : Â = Pic_{A/S}^{0} → J_{C/S} = Pic_{C/S}^{0} induced by f satisfies λ_{C/S} ∘ f^{∗} = λ_{C/S} ∘ (f_{∗} ∘j_{a})^{∗} = λ_{C/S} ∘j_{a}^{∗} ∘(f_{∗})^{∗} = −(f_{∗})^{∗}. Dualizing this equation and composing it with κ_{A}^{−1} and κ_{J} yields f_{∗} = κ_{A}^{−1} ∘ (f_{∗})^{∗∗} ∘ κ_{J} = −κ_{A}^{−1} ∘ (λ_{C/S} ∘ f^{∗})^{∗} ∘ κ_{J} = −κ_{A}^{−1} ∘ (f^{∗})^{∗} ∘ λ_{C/S}^{∗} ∘ κ_{J} = −κ_{A}^{−1} ∘ (f^{∗})^{∗} ∘ λ_{C/S}, the latter by the symmetry of λ_{C/S}. This proves the uniqueness of f_{∗} and the asserted formula for f_{∗}.

To prove the existence of f_{∗}, define f_{∗} by this formula. Since this definition commutes with base-change, we have for each s ∈ S that (f_{∗})_{s} ∘ j_{a,s} = f_{s} because the autoduality property holds over a field (cf. [Mi2], p. 185) (and by uniqueness). But then f, f_{∗} ∘ j_{a} : C → A are two S-morphisms whose fibres agree, and so by rigidity ([Mu1], p. 116) there is a section η of p_{A} : A → S such that f = (η ∘ p_{C}) · (f_{∗} ∘ j_{a}). But since f(a) = 0_{A} = (f_{∗} ∘ j_{a})(a), it follows that η = 0_{A}, and so f = f_{∗} ∘ j_{a}, as claimed.]

More generally, if L ∈ Pic(C) has relative degree 1, then there is a closed immersion j_{L} : C → J_{C/S} which represents the map z ↦ cl(O_{C}(z) ⊗ L^{−1}), and we have λ_{C/S} ∘ j_{L}^{∗} = −id_{J}. Furthermore, if C/S has a section, then it follows from (∗) that j_{L} satisfies the following universal property:

(∗∗) If f : C → A any S-morphism to an abelian scheme A/S, then there is a unique S-homomorphism f_{∗} : J_{C/S} → A and section η ∈ A(S) such that f = T_{η} ∘ f_{∗} ∘ j_{L}.

There is another universal property (also called the Albanese property) which is sometimes more useful since it does not require the existence of sections and hence applies to an arbitrary base S. For this, let s_{C/S} : C ×_{S} C → J_{C/S} be the morphism defined by the rule s_{C/S}(x, y) = cl(O(x − y)), where x, y ∈ C(T) are T -valued points and T is any S-scheme. Note that if δ : C → C ×_{S} C denotes the diagonal morphism, then s_{C/S} ∘ δ = 0 (the constant map). Then by a similar argument as in [Mi2], we have the following analogue of [Mi2], Prop. 6.4:

(∗∗∗) If f : C ×_{S} C → A is any S-morphism to an abelian scheme A/S such that f ∘ δ = 0, then there is a unique S-homomorphism f_{∗} : J_{C/S} → A such that f = f_{∗} ∘ s_{C}.

7) Covers of Relative Curves: If N and g′ ≥ 1 are positive integers, then a genus g′ cover of C/S of degree N is an S-morphism f : C′ → C where C′/S is curve of genus g′ such that for each s ∈ S the induced morphism f_{s} : C′_{s} → C_{s} has degree N. Since each f_{s} is then automatically finite, surjective and flat (the latter since the local rings of C_{s} are discrete valuation rings), it follows (by using [EGA], (IV, 11.3.11)) that f a finite, flat and surjective morphism. Each such cover induces homomorphisms

f^{∗} : J_{C/S} → J_{C′/S} and f_{∗} := λ_{C/S}^{−1} ∘ (f^{∗})^{∗} ∘ λ_{C′/S} : J_{C′/S} → J_{C/S},

where first is given by functoriality and the second by dualizing the first (and composing with λ_{C′/S} and λ_{C/S}^{−1}). Note that if C′/S has a section a′ ∈ C′(S), then the homomorphism (j_{f(a)} ∘ f)_{∗} : J_{C′/S} → J_{C/S} defined by the Albanese property (∗) is the same as f_{∗}, i.e. we have f_{∗} = (j_{f(a)} ∘ f)_{∗}, as can be seen by using the formula for (j_{f(a)} ∘ f)_{∗} given in (∗) (and a short computation). Thus we have for any section x′ ∈ C′(S):

f_{∗}(cl(O_{C′}(x′ − a′))) = cl(O_{C}(f(x′) − f(a′))).

We also observe that we have the formula

f_{∗} ∘ f^{∗} = [N]_{J_{C/S}}, (44)

where [N]_{J_{C/S}} denotes the multiplication by N map on J_{C/S}. (Indeed, if S = Spec(K) is a field, then this is well-known (by using the previous formula). Thus, for an arbitrary base S, we see that the formula holds fibre-by-fibre, and hence is true over S by rigidity ([Mu1], p. 116) since both sides are homomorphisms of abelian schemes.)

Definition. An S-cover f : C′ → C (of degree N) is called minimal if any one of the following equivalent conditions holds:

(i) f^{∗} : J_{C/S} → J_{C′/S} is a closed immersion;

(ii) (f_{s})^{∗} : J_{C_{s}/κ(s)} → J_{C′_{s}/κ(s)} is a closed immersion for all s ∈ S; (iii) f_{∗} : J_{C′/S} → J_{C/S} is surjective and Ker(f_{∗}) is an abelian subscheme

of J_{C′/S}.

[To see that these conditions are equivalent, note that the following implications hold:

(iii) ⇒ (i): Use (42).

(i) ⇒ (ii): Clear, since (f_{s})^{∗} = (f^{∗})_{s} and since the base change of any closed immersion is again a closed immersion by [EGA], (I, 4.3.6).

(ii) ⇒ (iii): By (42) we know that (f_{∗})_{s} = (f_{s})_{∗} is surjective for all s ∈ S, and hence so is f_{∗}. Thus f_{∗} is flat (use [EGA] (IV, 11.3.11)), and hence so is Ker(f_{∗})/S. Furthermore, f_{∗} has connected fibres (since each (f_{∗})_{s} does by (42)), and so Ker(f_{∗}) is an abelian subscheme of J_{C/S}.]

Note that condition (i) shows that if f is minimal, then so is any base change f_{(T)} : C′_{(T)} → C_{(T)} (because the base change of any closed immersion is again a closed immersion).

For later use we also append here some basic facts about relative Cartier divisors.

8) Relative Cartier Divisors: These are defined for any scheme X/S which is flat and of finite presentation; cf. [EGA], §IV.21.15, [KM], §1.1 or [BLR], p. 212ff. For any base-change S′/S, the pull-back D_{S′} of a relative Cartier divisor D ∈ Div(X/S) is defined and is a relative Cartier divisor of X_{(S′)}/S′; cf. [EGA], (IV, 21.15.8).

If f : C′ → C is an S-cover of curves, then we have induced homomorphisms

f^{∗} : Div(C/S) → Div(C′/S) and f_{∗} : Div(C′/S) → Div(C/S)

between the groups Div(C/S) and Div(C′/S) of relative Cartier divisors, and these map effective divisors to effective divisors (cf. [EGA], §IV.21.15). In addition, we have f_{∗}f^{∗}D = ND, for all D ∈ Div(C/S); cf. [EGA], (IV, 21.5.6).

We note that the formation of f^{∗} commutes with base-change by [EGA], (IV, 21.15.9), and the same is true for f_{∗} by Lemma 7.2 below. In addition, for divisors D ∈ Div(C/S) and D′ ∈ Div(C′/S) we have

deg(f^{∗}D) = deg(f) deg(D) and deg(f_{∗}D′) = deg(D′).

Indeed, the first formula is Lemma 1.2.8 of [KM], and the second follows (by base-change) from the corresponding formula for curves over a field.

Furthermore, if a′ ∈ C′(S) is a section and [a′] ∈ Div(C′/S) denotes the associated relative Cartier divisor (cf. [KM], Lemma 1.2.2), then we have

f_{∗}[a′] = [f(a′)],

as can be seen by tracing through the definitions. (The discussion of [KM], p. 33, could also be used here.)

Lemma 7.2 Let p : X → S be flat and locally of finite presentation, and let f : X′ → X be finite and flat. Then for every relative Cartier divisor D′ ∈ Div(X′/S), the direct image f_{∗}D′ ∈ Div(X/S) is a relative Cartier divisor of X/S, and its formation commutes with every base-change g : S′ → S in the sense that we have the equality

(45) (f_{(S′)})_{∗}(g_{(X′)}^{∗}(D′) = g_{(X)}^{∗}(f_{∗}D′), for all D′ ∈ Div(X′/S).

Proof. The first assertion is proved in [EGA], (IV, 21.15.8). To prove the second, recall first that the pullbacks of relative Cartier divisors is always defined ([EGA], (IV, 21.15.9)), and so both sides of (45) are defined. Thus, the same proof of [EGA], Prop. (IV, 21.5.8) applies (where this formula is proven for arbitrary Cartier divisors but only for flat base-change so as to guarantee that the relevant pull-backs are defined).

Over general base S, equality of two relative Cartier divisors D, D′ ∈ Div^{+}(C/S) of C/S cannot be decided in the fibres C_{s}; indeed, this is already false for sections. However, if S is reduced, then it follows from [KM], Cor. 1.3.5 that D = D′ ⇔ D_{s} = D′_{s}, for all s ∈ S, (46)

where D_{s}, D′_{s} ∈ Div^{+}(C_{s}) denote the effective Cartier divisors induced by D and D′ on the fibre C_{s}.

The following lemma gives some other criteria for deciding equality (in a slightly more general setting):

Lemma 7.3 Let C/S be a relative curve and let Z ⊂ C be a closed subscheme which is finite over S and which has the property that d := deg(Z_{s}) is independent of s ∈ S.

(a) If there exists an effective Cartier divisor D ⊂ Z of degree d, then Z = D.

(b) If Z has d sections z_{1}, …, z_{d} ∈ Z(S) such that for every s ∈ S, the points z_{1}(s), …, z_{d}(s) ∈ Z_{s} are distinct, then Z is an effective relative Cartier divisor of C/S which is etale over S, and we have Z = [z_{1}] + … + [z_{d}].

(c) If there exists a finite, faithfully flat base extension S′/S such that Z ×_{S} S′ satisfies the hypothesis of (b), then Z ∈ Div^{+}(C/S) is etale over S.

Proof. (a) Let f : D ↪ Z denote the closed immersion. For any s ∈ S, the induced map f_{s} : D_{s} ≤ Z_{s} is an isomorphism because both are closed subschemes of degree d of smooth curve C_{s}. Now by hypothesis D is flat, of finite presentation over S, and so f is an isomorphism by [EGA], (IV, 17.9.5).

(b) Let D = [z_{1}] + [z_{2}] + … + [z_{d}] ∈ Div^{+}(C/S); clearly, z_{1}, …, z_{d} are sections of D, and D is etale over S; in fact, D = S ∐ … ∐ S (cf. [KM], Lemma 1.8.3). We thus have a natural morphism f : D → Z which induces isomorphisms f_{s} : D_{s} → Z_{s}, for all s ∈ S. Thus, by the argument in (a) we see that f : D ≃ Z is an isomorphism.

(c) By (b), Z ×_{S} S′ is etale over S′, hence Z is etale over S by [EGA], (IV, 17.7.3).

Remark. If, in the situation of Lemma 7.3, the base S is an integral noetherian scheme, then Z is automatically an effective relative Cartier divisor. (Indeed, since C/S and hence Z/S is projective, and the (constant) Hilbert polynomial of Z_{s} is d = deg(Z_{s}) which does not depend on s ∈ S by hypothesis, it follows that Z is flat over S by [Ha], Th. III.9.9.)

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