The number of genus 2 covers of an elliptic curve

Ernst Kani

Abstract: The main aim of this paper is to determine the number c_{N,D} of genus 2 covers of an elliptic curve E of fixed degree N ≥ 1 and fixed discriminant divisor D ∈ Div(E). In the case that D is reduced, this formula is due to Dijkgraaf.

The basic technique here for determining c_{N,D} is to exploit the geometry of a certain compactification C = C_{E,N} of the universal genus 2 curve over the Hurwitz space H_{E,N} which classifies (normalized) genus 2 covers of degree N of E. Thus, a secondary aim of this paper is to study the geometry of C. For example, the structure of its degenerate fibres is determined, and this yields formulae for the numerical invariants of C which are also of independent interest.

1 Introduction

Let E be an elliptic curve over an algebraically closed field K. The main aim of this paper is to compute the number c_{N,D} = #Cov_{E,N,D} of genus 2 covers of E of fixed degree N ≥ 1 and fixed discriminant divisor D ∈ Div(E). Since this number is closely related to the weighted number c̄_{N,D} := ∑_{f∈Cov_{E,N,D}} \frac{1}{|Aut(f)|} of such covers and since the latter leads to simpler formulae, we determine c̄_{N,D} as well.

Theorem 1 If char(K)|̸ N! and D ∈ Div(E) is an effective divisor of degree 2, then

c̄_{N,D} = \frac{N}{3μ_{D}} (σ_{3}(N) − Nσ_{1}(N)) − \frac{μ_{D} − 1}{24} (7σ_{3}(N) − (6N + 1)σ_{1}(N)) (1)

where μ_{D} = 1 if D is reduced and μ_{D} = 2 otherwise, and where σ_{k}(n) = ∑_{d|n} d^{k} denotes the sum of the kth powers of the divisors of n. Moreover, if we put σ_{1}(N/2) = 0 if N is odd, then the total number of genus 2 covers is given by

c_{N,D} = c̄_{N,D} + (\frac{N}{μ_{D}} − (μ_{D} − 1)) σ_{1}(N/2). (2)

Corollary 2 If char(K) = 0, then the generating function F_{D}(q) of the c̄_{N,D}’s is a quasi-modular form, i.e. F_{D} ∈ ℚ[E_{2}, E_{4}, E_{6}], where the E_{k}’s are the usual Eisenstein series: E_{k} = 1 + c_{k} ∑_{n≥1} σ_{k−1}(n)q^{n}. More precisely, we have

F_{D}(q) := ∑_{N≥1} c̄_{N,D}q^{N} = \frac{1}{25920μ_{D}} (5E_{2}^{3}−3E_{2}E_{4}−2E_{6})−\frac{μ_{D} − 1}{5760}(2E_{4}+5E_{2}^{2}+10E_{2}−17).

Note that in the case that μ_{D} = 1 (i.e. D is reduced), it follows that F_{D} is pure of weight 6, and so in this case the above formula for F_{D} reduces to that of Dijkgraaf[Di] (see also [KZ]), except that his weighted sum is defined as \frac{1}{2} c̄_{N,D}.

As will be shown in §2, the calculation of the numbers c_{N,D} and c̄_{N,D} can be reduced to the calculation of the number of such covers which are minimal, i.e. those covers which do not factor over a non-trivial isogeny of E.

Theorem 3 The number of minimal genus 2 covers of E of degree N ≥ 2 and discriminant D is given by

#Cov_{E,N,D}^{(min)} = (\frac{2}{3μ_{D}} (N − 1) − \frac{μ_{D} − 1}{12N}(7N − 6)) \overline{sl}(N), (3)

where \overline{sl}(N) = |SL_{2}(ℤ/Nℤ)/\{±1\}|.

The basic technique for proving this theorem is to study the geometry of the Hurwitz space H_{E,N} which classifies genus 2 covers of degree N of E which are normalized in the sense of [Ka3]. More precisely, in Corollary 14 below we shall relate the above number to the degree of a certain “discriminant map” δ = δ_{E,N} : H_{E,N} → ℙ^{1}.

The computation of deg(δ) will be achieved by relating it to the “basic numerical invariants” (such as the modular height) of a certain relative genus 2 curve C = C_{E,N} over the modular curve X(N) of level N. This relative curve, which is also of independent interest, is constructed in §4 as follows.

Since H = H_{E,N} is a fine moduli space (cf. [Ka3]), there exists a universal (normalized) genus 2 cover f_{univ} : C_{univ} → E_{H} = E × H of degree N; in particular, C_{univ}/H is a smooth relative curve of genus 2. Since H is an open subset of X = X(N), the minimal model C of C_{univ} is a relative genus 2 curve over X which contain C_{univ} as an open subset. Thus, by construction, the curve C has good reduction at all points of H, and it turns out that the set X \ H is the degenerate locus of C/X, i.e. the set of points of X where C/X has bad reduction.

The precise structure of the degenerate fibres of C/X is given in Theorem 22. To prove this theorem, we first calculate the modular height h_{C/X} of C/X (cf. Theorem 17) and then combine the “mass-formula” of [Ka1], [Ka2] with Mumford’s formula [Mu] (and with Noether’s formula). This also leads to a formula for the self-intersection number (ω_{C/X}^{0})^{2} of the relative dualizing sheaf ω_{C/X}^{0}:

Theorem 4 If char(K)|̸ N!, then the minimal model C of C_{univ} over X = X(N) is a stable curve of genus 2 which has \frac{1}{12N} (5N + 6)\overline{sl}(N) singular fibres, and each such singular fibre has a unique singular point. Moreover, the modular height h_{C/X} and the self-intersection number of the relative dualizing sheaf ω_{C/X}^{0} are given by

h_{C/X} = \frac{1}{12}\overline{sl}(N) and (ω_{C/X}^{0})^{2} = \frac{1}{12N} (7N − 6)\overline{sl}(N). (4)

Remark. The above theorem, which is a special case of Theorems 17, 21, 22 and Corollary 24, is also of independent interest because the above formula (4) shows that the relative curves C/X(N) are curves of “maximal height”, as is explained in more detail in Remark 18 below. In this connection one should also mention the recent preprint of E. Viehweg and K. Zuo[VZ] in which the authors give a remarkable characterization of abelian varieties (of arbitrary dimension) of “maximal height”.

Now the degree of δ : H_{E,N} → ℙ^{1} is related to the invariant (ω_{C/X}^{0})^{2} by the formula

deg(δ) = \frac{N}{36}(9(ω_{C/X}^{0})^{2} − W_{C/X}^{2}), (5)

in which W_{C/X} denotes the Weierstrass divisor (cf. §6). Since the self-intersection number W_{C/X}^{2} of the Weierstrass divisor can be calculated by the adjunction formula (cf. (38)), this, together with (4), yields the desired formula for deg(δ).

Finally, the proof of (5) (= equations (35) and (39) below) follows from a study of the different divisor D_{C/X} := Diff(f̄_{univ}) ∈ Div(C); cf. §5. In particular, we use the (generalized) Riemann-Hurwitz formula (cf. Corollary 35) and the degeneration structure of C/X to show that the sheaf L(D_{C/X}) associated to D_{C/X} is the relative dualizing sheaf ω_{C/X}^{0} of C/X (cf. Theorem 26). Formula (5) follows from this because the Weierstrass divisor W_{C/X} is closely related to the tri-canonical sheaf (ω_{C/X}^{0})^{⊗3}; cf. Proposition 29 and Corollary 30.

Acknowledgements. Many of the underlying ideas of this paper go back to the joint paper [FK], and I would like to thank Gerd Frey for the many stimulating discussions which we had about this topic over the years. Moreover, I would like to thank Barry Mazur for his letter[Ma] in which he pointed out that Corollary 4.2 of [FK] amounts to a weak form of (a metrized version of) Mumford’s formula. Indeed, this paper owes a lot to this observation, for by using the (geometric) version of Mumford’s formula[Mu], the arguments could be considerably shortened (cf. Remark 23(b) below). In addition, he drew my attention to the work of Dijkgraaf[Di]. I am also grateful to Mike Roth for the many conversations we had about this paper. Furthermore, I would like to thank Eckart Viehweg for several useful conversations and for explaining his interesting work with K. Zuo [VZ] to me.

Finally, I would like to gratefully acknowledge receipt of funding from the Natural Sciences and Engineering Research Council of Canada (NSERC).

2 Normalized Genus 2 Covers

Let E/K be an elliptic curve over an algebraically closed field K of characteristic ≠ 2, and let E[2] = {P_{0}, P_{1}, P_{2}, P_{3}} denote its group of 2-torsion points, where P_{0} = 0_{E} denotes the identity point. Moreover, let π_{E} : E → E/⟨[−1]_{E}⟩ ≃ ℙ^{1} denote the quotient map, and put P̄_{i} = π_{E}(P_{i}), for i = 0, …, 3.

If f : C → E is any genus 2 cover of E, then by Riemann-Hurwitz its discriminant Disc(f) = f_{∗}Diff(f) is an effective divisor of degree 2. As was mentioned in the introduction, we want to count the number of genus 2 covers f : C → E of fixed degree N = deg(f) and fixed discriminant D = Disc(f), i.e. we want to determine the cardinality of the set

Cov_{E,N,D} := {f : C → E : g_{C} = 2, deg(f) = N, Disc(f) = D}/≃

of isomorphism classes of genus 2 covers of E of degree N and discriminant D. Here, 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.

As we shall see presently, the study of this set can be reduced to the study of normalized genus 2 covers which were introduced in [Ka3]. To define these, recall first that f is called minimal if f does not factor over a non-trivial isogeny of E. Then we say that f is normalized if it minimal and if the norm (or direct image) f_{∗}W_{C} of the hyperelliptic (Weierstrass) divisor W_{C} has the form

(6) f_{∗}W_{C} = 3P_{0} + P_{1} + P_{2} + P_{3}, respectively f_{∗}W_{C} = 2(P_{1} + P_{2} + P_{3}),

if N is odd, respectively, if N is even; cf. [Ka3], §2. Recall from [Ka3], Proposition 2.2, that every minimal cover can be normalized by replacing f by a suitable translate T_{P} ∘ f (with P ∈ E(K)) and that each normalized cover f satisfies the relation

[−1]_{E} ∘ f = f ∘ σ_{C}, (7)

where σ_{C} denotes the hyperelliptic involution on C. Note that this equation shows in particular that Disc(f) is symmetric, i.e. that [−1]^{∗}Disc(f) = Disc(f). Thus Disc(f) = D_{P} := P + [−1]^{∗}P, for some P ∈ E(K); note that D_{P} is reduced if and only if P ∉ E[2].

Example 5 Let E be the elliptic curve defined by the equation y^{2} = (x−a)(x−b)(x− c), with abc ≠ 0 and let C be the genus 2 curve defined by s^{2} = (t^{2} −a)(t^{2} −b)(t^{2} −c). Then map f : C → E, defined by f^{∗}x = t^{2} and f^{∗}y = s, is clearly a normalized genus 2 cover of E of degree 2 with Disc(f) = D_{P}, where P = (0, \sqrt{−abc}) ∈ E(K).

It turns out that every normalized cover of degree 2 is of the above form; more precisely, we have:

Proposition 6 If P ∈ E(K) \ E[2], then there is (up to isomorphism) a unique normalized genus 2 cover f : C → E of degree 2 with Disc(f) = D_{P}. On the other hand, if P ∈ E[2], then there is no degree 2 cover f : C → E with Disc(f) = D_{P}.

Proof. First note that if f : C → E is any degree 2 cover, then each ramified fibre of f has a unique ramification point of ramification index 2, and hence Disc(f) is necessarily reduced. This proves the second assertion.

To prove the first, put P̄ = π_{E}(P), and choose a generator x of the function field of ℙ^{1} such that its principal divisor is (x) = P̄−P̄_{0}. Then E is given by an equation as in Example 5 for suitable a, b, c ≠ 0, and P = (0, \sqrt{−abc}). Thus, the above example shows that such a normalized degree 2 cover f : C → E exists.

To prove that f is unique up to isomorphism, suppose that f′ : C′ → E is another cover of this type. Now if π_{C} : C → C/⟨σ_{C}⟩ ≃ ℙ^{1} denotes the hyperelliptic cover of C, then equation (7) shows that π_{E} ∘ f factors over π_{C}, so that π_{E} ∘ f = f̄∘ π_{C}, for a unique double cover f̄: ℙ^{1} → ℙ^{1}. Similarly, we have π_{E} ∘ f′ = f̄′ ∘ π_{C′}, for another double cover f̄′ : ℙ^{1} → ℙ^{1}.

It is easy to see that both f̄ and f̄′ are ramified at P̄_{0} and P̄, and so it follows (since K is algebraically closed) that there exists an α ∈ Aut(ℙ^{1}) such that f̄ = f̄′ ∘ α. Now since f̄^{−1}({P̄_{1}, P̄_{2}, P̄_{3}}) = π_{C}(W_{C}), where W_{C} denotes the set of (six) Weierstrass points on C, and since similarly (f̄′)^{−1}({P̄_{1}, P̄_{2}, P̄_{3}}) = π_{C′}(W_{C′}), we see that α(π_{C}(W_{C})) = π_{C′}(W_{C′}), and this implies that α lifts to an isomorphism α̃ : C → C′ such that α ∘ π_{C} = π_{C′} ∘ α̃. Thus f̄′ ∘ π_{C′} ∘ α̃ = f̄ ∘ π_{C}, and so π_{E} ∘ f′ ∘ α̃ = π_{E} ∘ f.

We now show that this implies that either f = f′ ∘ α̃ or that f = f′ ∘ α̃ ∘ σ_{C}. For this, let τ and τ′ be the nontrivial involutions associated to f and f′, respectively. Then τ_{1} := α̃^{−1}τ′α̃ ∈ Gal(π_{E} ∘ f) = {1, σ_{C}, τ, σ_{C}τ}. Thus τ_{1} = τ or τ_{1} = σ_{C}τ and hence either f ∘α̃^{−1} or f ∘σ_{C} ∘α̃^{−1} is τ′-invariant, and so there exists an automorphism ρ ∈ Aut(E) such that ρ∘f′ = f ∘α̃^{−1} (or = f ∘σ_{C} ∘α̃^{−1}). Since π_{E} ∘f′ ∘α̃ = π_{E} ∘f, we see that π_{E} ∘ρ = π_{E}, and so ρ = id_{E} or ρ = [−1]_{E}. Thus, using (7), we see that either f = f′ ∘ α̃ or f = f′ ∘ α̃ ∘ σ_{C}. Thus f′ ≃ f, and so f is unique up to isomorphism.

Notation. Let Cov_{E,N,D}^{(norm)} (respectively, Cov_{E,N,D}^{(min)}) denote the subset of isomorphism classes cl(f) ∈ Cov_{E,N,D} for which f is normalized (respectively, for which f is minimal).

The connection between minimal and normalized covers is clarified by the following result.

Proposition 7 (a) For any P ∈ E(K), the map f ↦ T_{P} ∘ f induces a bijection

τ_{P} : Cov_{E,N,T_{P}^{∗} D} ⟶^{∼} Cov_{E,N,D}

which maps Cov_{E,N,T_{P}^{∗} D}^{(min)} onto Cov_{E,N,D}^{(min)}.

(b) For any effective divisor D ∈ Div^{(2)}(E) of degree 2, there is a point P ∈ E(K) such that T_{P}^{∗}D is symmetric.

(c) If D ∈ Div^{(2)}(E) is symmetric, then

Cov_{E,N,D}^{(min)} = ˙⋃_{P_{i}∈E[2]}τ_{P_{i}}(Cov_{E,N,T_{P_{i}}^{∗}D}^{(norm)}).

Proof. (a) Since T_{P}^{∗}(Disc(T_{P} ∘ f)) = Disc(f), this is immediate.

(b) Write D = P′ +P′′ with P′, P′′ ∈ E(K). Since K is algebraically closed, there is a point P ∈ E(K) such that 2P ∼ P′ + P′′. Then T_{P}^{∗}(P′′) ∼ (P′′ − P) + P_{0} ∼ P − P′ + P_{0} ∼ 2P_{0} − T_{P}^{∗}(P′) ∼ [−1]_{E}^{∗}T_{P}^{∗}(P′), and so T_{P}^{∗}(P′′) = [−1]_{E}^{∗}T_{P}^{∗}(P′). Thus T_{P}^{∗}D = T_{P}^{∗}(P′) + [−1]_{E}^{∗}T_{P}^{∗}(P′) is symmetric.

(c) It is clear by part (a) that the right hand side is contained in the left hand side. Conversely, suppose cl(f) ∈ Cov_{E,N,D}^{(min)}. Then by Proposition 2.2 of [Ka3], there exists a point P ∈ E(K) such that f′ = T_{P} ∘ f is normalized, and so we know by (7) that Disc(f′) = T_{P}^{∗}(Disc(f)) = T_{P}^{∗}(D) is symmetric. But D was also symmetric by hypothesis, so we must have that [−1]_{E}^{∗}P = P, i.e. that P ∈ E[2]. Thus f = T_{P} ∘ f′, with f′ ∈ Cov_{E,N,T_{P}^{∗} (D)}^{(norm)}, and so both sides are equal.

The study of arbitrary covers can be reduced to the study of minimal covers by means of the following (well-known) fact.

Proposition 8 If f : C → E is any cover and H = H_{f} := Ker(f^{∗} : J_{E} → J_{C}) is the kernel of the induced homomorphism of the Jacobians, then f factors as f = π_{H} ∘ f′, where π_{H} : E′_{H} → E is the (essentially) unique isogeny such that Ker(π_{H}^{∗}) = H, and where f′ : C → E′_{H} is minimal. Furthermore, if f = π_{H} ∘ f′′ is another factorization, then f′′ = T_{P′} ∘ f′, for some P′ ∈ Ker(π_{H}).

Proof. By the autoduality property of J_{C}, f : C → E is minimal if and only if f maximal in the sense of Serre[Se], VI, No. 13 (p. 129ff), and so the first assertion follows from Lemma 3 of [Se], (p. 130). The second assertion is an easy consequence of the uniqueness assertion of the autoduality property of J_{C}.

Corollary 9 Let H ≤ J_{E} be a finite subgroup of order n|N and let Cov_{E,N,D,H} ⊂ Cov_{E,N,D} denote the subset of all isomorphism classes cl(f) ∈ Cov_{E,N,D} with Ker(f^{∗}) = H. Fix a point Q′_{1} ∈ π_{H}^{−1}(Q_{1}), where D = Q_{1} + Q_{2}. Then the map f′ ↦ π_{H} ∘ f′ induces a surjection

τ_{H} : ˙ ⋃_{Q′_{2}∈π_{H}^{−1}(Q_{2})}Cov_{E′_{H}, \frac{N}{n} ,Q′_{1}+Q′_{2}}^{(min)} ⟶ Cov_{E,N,D,H}

which is a bijection if D is reduced. On the other hand, if D = 2Q_{1}, then for any P′ ∈ Ker(π_{H}) we have

τ_{H}^{−1}(τ_{H}(cl(f′_{1})) = \{cl(f′_{1}), cl(T_{P′} ∘ f′_{1})\}, for all cl(f′_{1}) ∈ Cov_{E′_{H}, \frac{N}{n} ,Q′_{1}+T_{P′}^{∗}(Q′_{1})}^{(min)}.

In particular, #τ_{H}^{−1}(τ_{H}(cl(f′_{1}))) = 2, if P′ ≠ 0, and #τ_{H}^{−1}(τ_{H}(cl(f′_{1}))) = 1, if P′ = 0.

Proof. It is immediate that the rule τ_{H}(cl(f′)) := cl(π_{H} ∘ f′) defines a map from the above union to Cov_{E,N,D,H}. To see that τ_{H} is surjective, let cl(f) ∈ Cov_{E,N,D,H}. Then by Proposition 8 we have f = π_{H} ∘ f′, for some minimal f′ : C → E′_{H}. Since (π_{H})_{∗}(Disc(f′)) = D, we have that Disc(f′) = T_{P′}^{∗}(Q′_{1} + Q′_{2}), for some P′ ∈ Ker(π_{N}) and Q′_{2} ∈ π_{H}^{−1}(Q_{2}). Put f′_{1} = T_{P′} ∘ f′. Then Disc(f′_{1}) = Q′_{1} + Q′_{2}, so cl(f′_{1}) ∈ Cov_{E′_{H},N/n,Q′_{1}+Q′_{2}}^{(min)} and π_{H} ∘ f′_{1} = π_{H} ∘ f′ = f, and hence τ_{H} is surjective.

Assume first that D is reduced. To show that τ_{H} is injective, suppose that f′_{i} : C_{i} → E′_{H}, i = 1, 2 are two covers with cl(f′_{i}) ∈ Cov_{E′_{H},N/n,D_{i}}^{(min)} such that π_{H}∘f′_{1} ≃ π_{H}∘f′_{2}. Then there is an isomorphism φ : C_{1} →^{∼} C_{2} such that π_{H} ∘ f′_{1} = π_{H} ∘ f′_{2} ∘ φ, and so by Proposition 8, there is a point P′ ∈ Ker(π_{H}) such that f′_{1} = T_{P′} ∘ f′_{2} ∘ φ. Then T_{P′}^{∗}Disc(f′_{1}) = Disc(f′_{2}). Write D_{i} = Disc(f′_{i}) = Q′_{1} + Q′_{2i}, where Q′_{2i} ∈ π_{H}^{−1}(Q_{2}). Now since Q′_{1}, T_{P′}^{∗}(Q′_{1}) ∈ π_{H}^{−1}(Q_{1}) and Q′_{22}, T_{P′}^{∗}(Q′_{21}) ∈ π_{H}^{−1}(Q_{2}), and since Q_{1} ≠ Q_{2}, we must have T_{P′}^{∗}(Q′_{1}) = Q′_{1} (and T_{P′}(Q′_{21}) = Q′_{22}). But this means that P′ = 0_{E′_{H}}, and so f′_{1} = f′_{2} ∘ φ, i.e. f′_{1} ≃ f′_{2}. Thus τ_{H} is injective (and hence bijective).

Now suppose that D = 2Q_{1}, and let cl(f′_{1}) ∈ Cov_{E′_{H},N/n,Q′_{1}+T_{P′}^{∗}Q′_{1}}^{(min)}. Then clearly {cl(f′_{1}), cl(T_{P′} ∘ f′_{1})} ⊂ τ_{H}^{−1}(τ_{H}(cl(f′_{1}))) because π_{H} ∘ f′_{1} = π_{H} ∘ T_{P′} ∘ f′_{1} and since Disc(T_{P′} ∘ f′_{1}) = T_{−P′}^{∗}(Disc(f′_{1}) = T_{−P′}^{∗}(Q′_{1}) + Q′_{1}.

To prove the opposite inclusion, let cl(f′_{2}) ∈ π_{H}^{−1}(π_{H}(cl(f′_{1}))). Then, as in the reduced case, there exists φ : C_{1} →^{∼} C_{2} and P′′ ∈ Ker(π_{H}) such that f′_{1} = T_{P′′} ∘ f′_{2} ∘ φ. Thus T_{P′′}^{∗}(Disc(f′_{1})) = Disc(f′_{2}) = Q′_{1} + T_{P′_{1}}^{∗}(Q′_{1}), for some P′_{1} ∈ Ker(π_{H}), and hence T_{P′′}^{∗}(Q′_{1}) + T_{P′′}^{∗}T_{P′}^{∗}(Q′_{1}) = Q′_{1} + T_{P′_{1}}^{∗}(Q′_{1}). If T_{P′′}^{∗}(Q′_{1}) = Q′_{1}, then as before P′′ = 0_{E′_{H}} and so cl(f′_{2}) = cl(f′_{1}). Thus, assume T_{P′′}^{∗}(Q′_{1}) ≠ Q′_{1}; then T_{P′′}^{∗}(Q′_{1}) = T_{P′_{1}}^{∗}(Q′_{1}) and T_{P′′}^{∗}T_{P′}^{∗}(Q′_{1}) = Q′_{1}. This implies that P′′ = P′_{1} and P′′ = −P′. Thus T_{P′} ∘ f′_{1} = f′_{2} ∘ φ, i.e. cl(f′_{2}) = cl(T_{P′} ∘ f′_{1}), and so the opposite inclusion holds.

The last assertion is clear from the previous identity, for cl(T_{P′} ∘ f′_{1}) = cl(f′_{1}) if and only if P′ = 0_{E′_{H}}.

To compare the total number c_{N,D} = #Cov_{E,N,D} of covers to the weighted number c̄_{N,D}, we prove

Lemma 10 If f : C → E is any genus 2 cover of degree N, then |Aut(f)| ≤ 2. Furthermore, |Aut(f)| = 2 if and only if |Ker(f^{∗})| = \frac{N}{2} . Thus c_{N,D} = c̄_{N,D}, if N is odd, whereas

c_{N,D} = c̄_{N,D} + \frac{1}{2} ∑_{#H= \frac{N}{2}} #Cov_{E,N,D,H}, if N is even.

Proof. Suppose G := Aut(f) ≠ {1}, and let f̄: C → C̄ := G\C denote the quotient map. Then C̄ is an elliptic curve and f factors as f = f̄′ ∘f̄. Thus Ker((f̄′)^{∗}) ≤ H := Ker(f^{∗}), and so π_{H} factors over f̄′, and hence f̄factors over the minimal map f′ : C → E′_{H} (cf. Proposition 8). But then f′ is (generically) Galois, so |Aut(f′)| = deg(f′). By rigidity (cf. [Ka3], Proposition 2.1), this happens if and only if deg(f′) = 2, or equivalently, if and only if |H| = N/2.

Proof of Theorem 1 (using Theorem 3): We first note that it follows from Theorem 3 that for any subgroup H ≤ J_{E} with #H = n|N (and n ≠ N) we have

#Cov_{E,N,D,H} = (\frac{1}{3μ_{D}} (N − n) − \frac{μ_{D} − 1}{12N}(7N − 6n)) \overline{sl}(N/n). (8)

To see this, put c(N, D) := #Cov_{E,N,D}^{(min)}, which depends only on N and μ_{D} by (3). Suppose first that D is reduced, i.e. μ_{D} = 1. Then by Corollary 9 and (3) we have #Cov_{E,N,D,H} = nc(N/n, D′) = \frac{2}{3}n(\frac{N}{n} − 1)\overline{sl}(N/n), which proves (8) in this case.

Now suppose that D = 2Q_{1} is not reduced, i.e. μ_{D} = 2. Then by Corollary 9 we have (with the notation there) that #Cov_{E,N,D,H} = c(N/n, 2Q′_{1}) + \frac{n−1}{2}c(N/n, D′) = (\frac{1}{3}n(\frac{N}{n} − 1) − \frac{1}{12N/n}(7\frac{N}{n} − 6))\overline{sl}(N/n), which proves (8) in this case as well.

We now verify formula (2). If N is odd, then c_{N,D} = c̄_{N,D} by Lemma 10. Thus, assume that N is even. Now if n = \frac{N}{2} , then formula (8) yields #Cov_{E,N,D,H} =

(\frac{N}{3μ_{D}} − \frac{μ_{D}−1}{3}) \overline{sl}(2) = 2(\frac{N}{μ_{D}} − (μ_{D} − 1)). Thus, since J_{E} has precisely σ_{1}(N/2) subgroups H of order N/2, we see that formula (2) follows from Lemma 10.

It thus remains to prove formula (1). For this, put sl(N) = #SL_{2}(ℤ/Nℤ). Since \overline{sl}(N) = \frac{1}{2}sl(N), if N ≥ 3, whereas \overline{sl}(2) = sl(2) = 6, we see from (8) combined with Lemma 10 that the weighted number of covers in Cov_{E,N,D,H} is

c̄_{N,D,H} := ∑_{f∈Cov_{E,N,D,H}} \frac{1}{|Aut(f)|} = (\frac{1}{6μ_{D}} (N − n) − \frac{μ_{D} − 1}{24N}(7N − 6n)) sl(N/n). (9)

Let c̄_{N,D,n} denote the right hand side of (9). Since this number only depends on n (and on N, D) and since J_{E} has precisely σ_{1}(n) subgroups H of order n, we see that

c̄_{N,D} = ∑_{{n|N \atop n ≠ N}} ∑_{{H ≤ J_{E} \atop |H| = n}} c̄_{N,D,H} = ∑_{{n|N \atop n ≠ N}} σ_{1}(n)c̄_{N,D,n} = ∑_{n|N} σ_{1}(n)c̄_{N,D,n} + \frac{μ_{D} − 1}{24}σ_{1}(N).

From this, formula (1) follows immediately once we have established the identities

(10)  ∑_{n|N} σ_{1}(n)sl(N/n) = σ_{3}(N),
(11)  ∑_{n|N} nσ_{1}(n)sl(N/n) = N^{2}σ_{1}(N).

To verify these identities, let f_{1}(N) and f_{2}(N) denote the left hand sides of (10) and (11), respectively. Since both sides of these equations are multiplicative, it is enough to consider the case that N = p^{r} is a prime power. For r = 1 these identities are immediate, and for r > 1 they can be verified by induction by using the recursion relations f_{1}(p^{r+1}) = p^{3}f_{1}(p^{r}) + 1 and f_{2}(p^{r+1}) = p^{3}f_{2}(p^{r}) + p^{2r+2}, which are also satisfied by σ_{3}(p^{r}) and p^{2r}σ(p^{r}), respectively.

Proof of Corollary 2 (using Theorem 1). Since E_{2} = 1 − 24 ∑_{n≥1} σ_{1}(n)q^{n} and E_{4} = 1 + 240 ∑_{n≥1} σ_{3}(n)q^{n}, it follows from (1) that F_{D}(q) = \frac{1}{3μ_{D}} (\frac{1}{240}θ(E_{4}) + \frac{1}{24}θ^{2}(E_{2})) − \frac{μ_{D}−1}{24} (\frac{7}{240}E_{4} + \frac{6}{24}θ(E_{2}) − \frac{1}{24}E_{2} − \frac{17}{240})), where θ = q \frac{d}{dq}. Thus, since θ(E_{2}) = \frac{1}{12}(E_{2}^{2} − E_{4}) and θ(E_{4}) = \frac{1}{3}(E_{2}E_{4} − E_{6}) (cf. [La], p. 161), we have θ^{2}(E_{2}) = θ(\frac{1}{12}(E_{2}^{2} − E_{4})) = \frac{1}{72}(E_{2}^{3} − 3E_{2}E_{4} + 2E_{6}), and so the given formula for F_{D}(q) follows.

3 The discriminant map

The main reason for working with normalized genus 2 covers in place of arbitrary covers is that normalized covers can be parameterized by a nice algebraic curve.

To explain this in more detail, consider for any extension field L/K the set H_{E/K,N}(L) = Cov_{E_{L},N}^{(norm)} = ⋃_{D} Cov_{E_{L},N,D}^{(norm)} of all isomorphism classes of normalized genus 2 covers of degree N of E_{L}/L. Then the assignment L ↦ H_{E/K,N}(L) extends in a natural way to a Hurwitz functor

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

cf. [Ka3], §3. Moreover, if N ≥ 3 is invertible in K (which we assume tacitly henceforth), then by Theorem 1.1 of [Ka3] we know that H_{E/K,N} is representable by an open subset of the (projective) modular curve X(N):

Theorem 11 If N ≥ 3, then the functor H_{E/K,N} is finely represented by an open subset H_{E,N} of the modular curve X(N)/K of level N.

It is interesting to note that this representability result was obtained by purely algebraic techniques; in particular, it did not use (not even implicitly) the Riemann Existence Theorem.

We next observe that the sets Cov_{E,N,D}^{(norm)} ⊂ H_{E/K,N}(K) of normalized genus 2 covers of fixed discriminant D can be identified as the fibres of the discriminant map δ_{E,N} : H_{E,N} → Div^{(2)}(E)^{sym} ≃ ℙ^{1}.

Proposition 12 (a) The rule cl(f) ↦ Disc(f) defines a morphism

δ_{E,N} : H_{E,N} → Div^{(2)}(E)^{sym} ≃ ℙ^{1}

such that Cov_{E,N,D}^{(norm)} ≃ δ_{E,N}^{−1}(D), for all D ∈ Div^{(2)}(E)^{sym}.

Proof. Let Div_{E/K}^{(2)} : \underline{Sch}_{/K} → \underline{Sets} denote the functor which associates to a scheme

S the set Div_{E/K}^{(2)}(S) = Div^{(2)}(E_{S}/S) of relative effective Cartier divisors of degree 2

of E_{S}/S (cf. [BLR], p. 214 and 237). Recall that Div_{E/K}^{(2)} is represented by E^{(2)}, the second symmetric power of E (cf. [Gr1], p. 21).

Now if S is any K-scheme and f : C → E_{S} is any normalized genus 2 cover of E_{S}/S (cf. [Ka3], §3), then Disc(f) = f_{∗}Diff(f) ∈ Div_{E/K}^{(2)}(S), so the assignment

cl(f) ↦ Disc(f) defines a map δ_{S} : H_{E/K,N}(S) → Div_{E/K}^{(2)}(S). We claim that δ_{S} commutes with base-chance. Indeed, the formation of Diff(f) commutes with basechange by general properties of differents (cf. Corollary 34) and hence Diff(f) = f_{∗}Diff(f) also commutes with base-change (since f_{∗} does; cf. [Ka3], Lemma 7.3). Thus, δ = {δ_{S}} : H_{E/K,N} → Div_{E/K}^{(2)} defines a natural transformation of functors, and hence is represented by a morphism δ = δ_{E,N} : H_{E,N} → E^{(2)}.

By (7) we know that Disc(f) is symmetric with respect to [−1], and so δ maps into the subset Div^{(2)}(E)^{sym} of symmetric divisors. Now if π_{E} : E → E/⟨[−1]⟩ ≃ ℙ^{1} denotes the quotient map, then the rule P̄ ↦ π_{E}^{∗}P̄ induces an isomorphism ℙ^{1}(K) →^{∼} Div^{(2)}(E)^{sym}.

The last assertion is clear from the construction since by definition Cov_{E,N,D}^{(norm)} is the fibre of δ_{K} : H_{E/K,N}(K) → Div_{E/K}^{(2)}(K) over D, and hence the set of K-rational

points of the geometric fibre can be identified with Cov_{E,N,D}^{(norm)}.

Notation. Let H_{E/K,N}^{sm} denote the subfunctor of H_{E/K,N} consisting of covers with smooth discriminants, i.e. H_{E/K,N}^{sm} is defined by the rule

H_{E/K,N}^{sm}(S) = {cl(f) ∈ H_{E/K,N}(S) : Disc(f) is smooth over S}, for all K-schemes S.

Theorem 13 (a) The functor H_{E/K,N}^{sm} is represented by an open subscheme H_{E,N}^{sm} of H_{E,N}, and the restriction δ_{E,N}^{sm} : H_{E,N}^{sm} → U := ℙ^{1} \ π_{E}(E[2]) of δ_{E,N} to H_{E,N}^{sm} is etale.

(b) If char(K)|̸ N!, then δ_{E,N}^{sm} is also finite, and hence #Cov_{E,N,D_{P}}^{(norm)} = deg(δ_{E,N}), for all P ∈ E(K) \ E[2].

Proof. (a) Let f_{univ} : C_{univ} → E_{H} denote the universal normalized genus 2 cover of degree N over H = H_{E,N} (which exists by Theorem 11). Since D := Disc(f_{univ}) ∈ Div(E_{H}/H) is an effective relative Cartier divisor, the set H^{sm} ⊂ H where D is smooth over H is an open subset of H, and it is immediate that H^{sm} represents the functor H_{E,N}^{sm}.

Since U represents the subfunctor of Div_{E/K}^{(2)}^{sym} of smooth divisors, the map δ_{E,N}^{sm} is etale if and only if the following deformation property holds (cf. [BLR], p. 288):

(*) If A a local Artinian K-algebra with residue field \overline{A} and if D ∈ Div_{E/K}^{(2)}(A)^{sym} is smooth symmetric divisor on E_{A} with image \overline{D} ∈ Div_{E/K}^{(2)}(\overline{A}), then each normalized genus 2 cover \overline{f} : \overline{C} → E\overline{A} with Disc(\overline{f}) = \overline{D} lifts uniquely to a normalized genus 2 cover f : C → E_{A} with Disc(f) = D.

To see that this property holds, we apply the deformation result of Fulton[Fu1], Theorem 4.8 (cf. also Wevers[Wev], Corollary 3.1.3) in the above situation to conclude that there exists a unique lift f : C → E_{A} of \overline{f} with Disc(f) = D. It remains to show that f is normalized.

We first show that f is minimal (in the sense of [Ka3], §7, p. 50). If not, then Ker(f^{∗}) is a finite flat group scheme over A of rank d > 1 (because f^{∗} : J_{E_{A}} → J_{C} is a homomorphism of abelian schemes), and so Ker(\overline{f}^{∗}) = Ker(f^{∗})_{s} also has rank d, where \{s\} = Spec(A). But this contradicts the fact that \overline{f} is minimal.

Thus, f is minimal. By [Ka3], Theorem 3.2(c) it is thus enough to show that f is pseudo-normalized, i.e. that (the analogue of) (7) holds. Now since f ∘ σ_{C} and [−1]_{E_{A}} ∘ f both lift \overline{f} ∘ σ\overline{C} = [−1]_{A} ∘ \overline{f} and have discriminant D = [−1]_{E_{A}}^{∗}D, there exists an isomorphism φ : C →^{∼} C such that \overline{φ} = id\overline{C} and f ∘ σ_{C} = [−1]_{E_{A}} ∘ f ∘ φ (cf. [Fu1], Remark after Theorem 5.8). However, then φ = id_{C} by [DM], Theorem (11.1), and so f ∘ σ_{C} = [−1]_{E_{A}} ∘ f, as desired.

(b) It is clear that the second assertion follows from the first together with part (a) (and Proposition 12). Now the first assertion could be proven by a method similar to that of the proof of Theorem 7.2 of Fulton[Fu1], but it seems simpler to deduce it from the results of the next section; cf. Theorem 26(e).

Corollary 14 If N ≥ 3, and D is an effective divisor of degree 2, then

(12) #Cov_{E,N,D}^{(min)} ≥ \frac{4}{μ_{D}} deg(\overline{δ}_{E,N}) − (μ_{D} − 1)(2g_{X(N)} − 2 + s_{E,N}),

where \overline{δ}_{E,N} : X(N) → ℙ^{1} denotes the unique extension of δ_{E,N} to X(N), g_{X(N)} denotes the genus of X(N), s_{E,N} = #(X(N) \ H_{E,N}), and μ_{D} = 1 if D is reduced and μ_{D} = 2 otherwise. Moreover, equality holds in (12) if and only if μ_{D} = 1 or if \overline{δ}_{E,N} is tamely ramified.

Proof. By Proposition 7(a),(b) we may assume that D is symmetric, so D = D_{P} with P ∈ E(K). Thus, by Propositions 7(c) and 12 we have

#Cov_{E,N,D_{P}}^{(min)} = ∑_{i=0}^{3} #δ_{E,N}^{−1}(D_{T_{P_{i}}^{∗}(P)}).

If μ_{D} = 1, i.e. if P ∉ E[2], then also T_{P_{i}}^{∗}(P) ∉ E[2] and so by Proposition 7 and Theorem 13(b) we have #Cov_{E,N,D_{P}}^{(min)} = 4 deg(δ_{E,N}) = 4 deg(\overline{δ}_{E,N}). This proves that equality holds in (12) in this case.

Now suppose that μ_{D} = 2, i.e. that P ∈ E[2]. Then by the above formula we have #Cov_{E,N,D_{P}}^{(min)} = ∑_{i=0}^{3} #δ_{E,N}^{−1}(D_{P_{i}}). To calculate this, we shall apply the Riemann-Hurwitz formula to \overline{δ} to obtain

4 deg(\overline{δ}_{E,N}) − ∑_{i=0}^{3} #\overline{δ}_{E,N}^{−1}(D_{P_{i}}) ≤ deg(Diff(\overline{δ}_{E,N})) = (2g_{X(N)} − 2) − deg(\overline{δ}_{E,N})(−2).

From this (12) follows immediately after a short computation because by Theorem

13 we know that ∪_{i}δ_{E,N}^{−1}(D_{P_{i}}) ∪^{.} S_{E,N} = ∪_{i}\overline{δ}_{E,N}^{−1}(D_{P_{i}}), where S_{E,N} = X(N) \ H_{E,N}.

Note since \overline{δ}_{E,N} is unramified outside the points π_{E}(E[2]) = \{D_{P_{0}}, …, D_{P_{3}}\} (cf. Theorem 13), we see that equality holds in the above equation (and hence in (12)) if and only if \overline{δ}_{E,N} is tamely ramified.

Remark.We shall see later (cf. Proposition 31) that \overline{δ}_{E,N} is always tamely ramified (provided that char(K) ∤ N!).

4 Compactification of the universal cover

In the previous section we saw that the number #Cov_{E,N,D_{P}}^{(min)} is (mainly) determined by the degree of the discriminant map δ : H = H_{E,N} → ℙ^{1}. As we shall see below in §5, this degree is closely related to the bi-degree of the discriminant divisor of the universal cover f_{H} : C_{H} → H. Since we want to use intersection theory to compute this degree, we first need to compactify the surface C_{H} as well as the cover f_{H}.

For this, recall first that by Theorem 11 we have a “universal cover” f_{H} = f_{univ} : C_{H} → E_{H} = E × H over H := H_{E,N} ⊂ X(N) with the property that all other normalized covers of degree N are obtained from this one by base-change. (Recall that we are always tacitly assuming that N ≥ 3 and that char(K)|̸ 2N.)

Let p : C = C_{X(N)} → X(N) denote the minimal model of the genus 2 curve C/F_{N} which is the generic fibre of C_{H}/H; here F_{N} = κ(X(N)) denotes the function field of X(N)/K. Thus, C is a projective, smooth surface over K, and p : C → X(N) is a genus 2 fibration whose fibres do not contain any rational (−1)-curves and whose generic fibre is C/F_{N}. Note that C_{H} := p^{−1}(H) ≃ C_{H} because C_{H}/H is smooth and hence is the minimal model of C over H. Thus, C/X(N) is a natural compactification of C_{H}/H.

In order to study the geometry of C/X(N), it is useful to generalize the above situation slightly by allowing base-extensions of X(N). Thus, let:

β : X → X(N) be a finite cover of degree d = deg(β), where

X is a smooth curve with function field F = κ(X), and let

C_{F} = C ⊗_{F_{N}} F be the base-change of the curve C/F_{N},

p = p_{X} : C = C_{X} → X be the minimal model of C_{F} over X, J = J_{F} the Jacobian variety of C_{F} /F, and J /X its Néron model. We first note that the universal cover f_{H} : C_{H} ≃ C_{H} → E_{H} naturally lifts and

extends to a morphism f : C → E_{X}; this follows from the following general result.

Proposition 15 Let p_{i} : Y_{i} → X be a two proper morphisms over a smooth curve X/K, and assume that Y_{1} is a regular irreducible surface and that Y_{2}/X is either an abelian scheme or a smooth relative curve of genus g ≥ 1. Then every X-rational map f : Y_{1} −−→ Y_{2} extends uniquely to an X-morphism f : Y_{1} → Y_{2}.

Proof. Here we shall use the following fact:

If π : Z → Y_{1} is a birational proper morphism and if C ⊂ Z is an irreducible curve such that π(C) is a point, then C is a rational curve.

(13)

[Indeed, if Z is also regular, then π is a sequence of blow-ups at smooth points (cf. [Ha], Corollary V.5.4), so in fact C ≃ ℙ^{1} for any such C. In the general case, we can find by the resolution of singularities theorem (cf. [Art]) a regular surface Z′ and a birational proper morphism π′ : Z′ → Z. Then there is an irreducible curve C′ ⊂ Z′ such that π′(C′) = C. By what was just said, C′ is rational, and hence so is C.]

Now let U = def(f) denote the largest open subset on which f is defined. Since p_{2} is proper and f is an X-rational map, f is a strictly rational map in the sense of Iitaka[I], p. 134; cf. [I], Lemma 2.23. Thus, since Y_{1} is normal, B := Y_{1} \ U has codimension ≥ 2 (cf. [I], Theorem 2.19), and hence is a finite set. Let Γ_{f} ⊂ Y_{1}×_{X} Y_{2} ⊂ Y_{1} ×Y_{2} denote the graph of the rational map f. Then p = (pr_{1})_{|Γ_{f}} : Γ_{f} → Y_{1} is proper and birational. Suppose y ∈ B. Then dim(q(p^{−1}(y))) > 0 by Zariski’s Main Theorem (cf. [I], Theorem 2.22), where q = (pr_{2})_{|Γ_{f}} : Γ_{f} → Y_{2}. Now every irreducible curve C ⊂ p^{−1}(y) is rational by (13). In addition, since C lies in a closed fibre of p_{1} ∘ p, we see that q(C) lies in a closed fibre of p_{2}. But the hypotheses on p_{2} imply that its fibres cannot contain any rational curves, so q(C) must be a point. Thus p(q^{−1}(y)) is 0-dimensional, contradiction, and hence B = ∅, i.e. def(f) = Y_{1}, which means that f extends (uniquely) to a morphism f : Y_{1} → Y_{2}, as claimed.

Corollary 16 Let f_{U} : C_{U} → E_{U} be the base-change of the above universal cover f_{H} : C_{H} → E_{H} with respect to β_{U} = β_{|U} : U → H, where U = β^{−1}(H). Then f_{U} extends uniquely to a proper, surjective morphism f = f_{X} : C → E_{X} := E × X, and we have β_{E_{X}} ∘ f_{X} = f_{X(N)} ∘ β_{C} (14)

where β_{E_{X}} = id_{E} × β : E_{X} → E_{X(N)} and β_{C} : C_{X} → C_{X(N)} are the morphisms induced by base-change. Moreover, if σ_{C} ∈ Aut_{X}(C) denotes the unique extension of the hyperelliptic involution σ_{C_{F}} to C, then we have

f ∘ σ_{C} = [−1]_{E_{X}} ∘ f. (15)

Proof. The first assertion follows immediately from Proposition 15, for E_{X} = E × X is a smooth relative curve of genus 1 over X. The second assertion (14) is clear, for by construction the two morphisms agree on the dense open set C_{U} = C_{H} ×_{H} U. (Note that β_{C} = β_{(C)} ∘ ν, where ν : C_{X} → C_{X(N)} ×_{X(N)} X is birational and β_{(C)} : C_{X(N)} ×_{X(N)} X → C_{X(N)} is the base-change map.)

The existence of σ_{C} follows from the universal property of minimal models (cf. [Li]). Now since f_{U} is normalized by construction, the two morphisms f ∘ σ_{C} and [−1] ∘ f agree on the dense open set p^{−1}(U) (cf. [Ka3], Theorem 3.2(c)) and hence are equal on C.

We now examine the geometry of C/X more closely. As a first step, let us compute its modular height h_{C/X}. This important invariant, which was first introduced by Parshin[Pa1],[Pa2] and Arakelov[Ar], is defined for any (regular) semi-stable curve C/X of genus g ≥ 1 by

h_{C/X} := deg_{X}(∧^{g}p_{∗}(ω_{C/X}^{0})) = − deg_{X}(∧^{g}R^{1}p_{∗}O_{C}).

where ω_{C/X}^{0} = ω_{C/K} ⊗ p^{∗}(ω_{X/K})^{−1} denotes the relative dualizing sheaf of C/X (cf. [BPV], p. 98 or [Kl]), and the equality follows from the relative duality isomorphism p_{∗}ω_{C/X}^{0} ≃ (R^{1}p_{∗}O_{C})^{∨}; cf. [BPV], p. 99 or [Kl], equation (1.1).

It turns out that the height of C/X is closely related to the modular height of E′/X, where E′ = E′_{X} is the minimal model of the “universal elliptic curve” E′_{F} = E′_{F_{N}} ⊗ F over X. (More precisely, E′_{F_{N}} is the generic fibre of the universal elliptic curve E′_{X′(N)} over X′(N) := X(N) \ X(N)_{∞}, where X(N)_{∞} = {cusps} denotes the set of cusps of X(N); i.e. E′_{X′(N)} (together with its level-N-structure) is the universal object of the functor parameterizing elliptic curves with level-N-structure of fixed determinant.)

Theorem 17 The relative curve p : C → X is semi-stable and has modular height

h_{C/X} = h_{E′/X} = \frac{N}{12}d#X(N)_{∞} = \frac{d}{12}\overline{sl}(N) := \frac{d}{12}# (SL_{2}(ℤ/Nℤ)/\{±1\}) . (16)

Proof. The “basic construction” of [FK] or [Ka3], Corollary 5.19, shows that the Jacobian J_{F_{N}} of C = C_{F_{N}} is isogenous to E_{F_{N}} ×E′_{F_{N}}, and so we also have J_{F} ∼ E_{F} ×E′_{F}. Now E′_{F} has semi-stable (or semi-abelian) reduction over F, i.e. the minimal model E′/X of E′_{F} is semi-stable: this follows either from the work of [DR] or, more simply, from Raynaud’s criterion (cf. [Gr2], Prop. 4.7) because the N-torsion points of E′_{F} are F -rational. Thus, since E_{F} has good reduction over F, we see that E_{F} ×E′_{F} and hence also J have semi-abelian reduction (use [BLR], 7.4/2 and 7.3/7). Thus, the Néron model J of J is semi-abelian, and so by [DM], Theorem (2.4), C/X is semi-stable.

To compute h_{C/X}, we shall use the fact (due to Parshin[Pa2], §3, Proposition 1 and/or Arakelov[Ar], Lemma 1.4) that

h_{C/X} = deg_{X}(λ_{J /X}), where λ_{J /X} = s^{∗}Ω_{J /X}^{1},

in which s : X → J denotes the zero-section of the Néron model J /X of the Jacobian J of C_{F}. Since here J ∼ E_{F} × E′_{F} (by an isogeny of degree N^{2}), we see that λ_{J /X} ≃ λ_{E_{X}/X} ⊗ λ_{E′/X} ≃ λ_{E′/X} where the last isomorphism follows from the fact that E_{X}/X is constant (so λ_{E_{X}/X} ≃ O_{X}).

Thus h_{C/X} = deg(λ_{J /X}) = h_{E′/X}, which proves the first equality of (16). To determine h_{E′/X}, we use the fact that for any (regular) semi-stable elliptic curve q : E′ → X we have h_{E′/X} = \frac{1}{12}δ_{E′/X} = \frac{1}{12} deg(j_{E′}), (17)

where δ_{E′/X} denotes the total number of singular points of all fibres of E′/X and j_{E′} : X → ℙ^{1} denotes the morphism defined by the j-invariant j_{E′} ∈ κ(X). [Indeed, the first equation of (17) follows from Noether’s formula (23) together with the fact that (ω_{E′/X}^{0})^{2} = 0 (which in turn follows from the formula ω_{E′/X}^{0} = q^{∗}q_{∗}ω_{E′/X}^{0}; cf. [DR], Proposition II.1.6(ii)). To prove the second equation, we first note that E′/X has bad reduction at x ∈ X if and only if v_{x}(j_{E′}) < 0, i.e. if and only if x ∈ j_{E′}^{−1}(∞), and that for such an x the fibre E′_{x} is a Néron polygon of length −v_{x}(j_{E′}); cf. [Ta]. Thus δ_{E′} = − ∑_{x} min(0, v_{x}(j_{E′})) = deg(j_{E′}), which proves the second equality of (17).]

Now for E′/X as above, the map j_{E′} is given by j_{E′} = p_{N} ∘ β, where p_{N} : X(N) → X(1) is the standard cover which is Galois with group SL_{2}(ℤ/Nℤ)/{±1}. Thus, since each x ∈ X(N)_{∞} = p_{N}^{−1}(∞) is ramified of degree N, we have deg(j_{E′}) = d · \overline{sl}(N) = dN#X(N)_{∞}, which therefore proves the second and third equation of (16).

Remark 18 For d = 1 the genus of X = X(N) is given by 2g_{X(N)}−2 = \overline{sl}(N)(\frac{1}{6} − \frac{1}{N}), and hence we can also write the above formula (16) in the form

h_{C/X(N)} = h_{J /X(N)} = \frac{1}{2}(2g_{X(N)} − 2 + #X(N)_{∞}).

Now since J has bad reduction precisely at X(N)_{∞} (cf. Proposition 19 below), this means that C and J have “maximal height”, because by a theorem of Faltings[Fa] the right hand side of this equation is the maximum value that the height of a 2dimensional abelian variety A/F can assume in characteristic 0 when A has good reduction outside of X(N)_{∞} (and A contains an elliptic curve defined over K).

In this connection it is interesting to note that in a recent preprint E. Viehweg and K. Zuo[VZ] give a beautiful structure theorem for abelian varieties (of arbitrary dimension) for which the Arakelov-Faltings inequality is an equality.

We next examine the singular locus of C/X, i.e. the set S(C/X) ⊂ X of points x ∈ X such that the fibre C_{x} is singular. Note that S(C/X) contains the set S_{0}(C/X) := S̃(J /X) of points x ∈ X where J has bad reduction, i.e. where the fibre J_{x} of the Néron model J is not proper (use [BLR], 9.4/4). Thus we can write

S(C/X) = S_{0}(C/X) ˙∪ S_{1}(C/X), with S_{1}(C/X) := S(C/X) \ S_{0}(C/X).

Proposition 19 (a) The set of points where the Jacobian J has bad reduction is the set S_{0}(C/X) = S(E′/X) = X_{∞} := β^{−1}(X(N)_{∞}). If x ∈ S_{0}(C/X), then C_{x} contains a unique component C_{x,0} whose normalization has genus 1, and the induced morphism f_{C_{x,0}} : C_{x,0} → E_{x} = E is finite of degree N. Furthermore, there is at least one singular point P_{x} ∈ C_{x} such that C_{x} \ {P_{x}} is connected.

(b) The set of points where the curve C has bad reduction (i.e. the singular locus of C/X) is S(C/X) = X \ U, where U = β^{−1}(H), and so S_{1}(C/X) = X′ \ U, where X′ = X\X_{∞}. If x ∈ S_{1}(C/X), then the fibre C_{x} of C at x is a chain C_{x} = C_{x,0}∪…∪C_{x,r} of smooth curves C_{x,i}. Furthermore, C_{x,0} = E_{x,1} and C_{x,r} = E_{x,2} are elliptic curves with self-intersection number E_{x,i}^{2} = −1, whereas all others components are rational curves with self-intersection number C_{x,j}^{2} = −2. In addition, the induced maps f_{x,i} : E_{x,i} → E_{x} = E are finite of total degree N, i.e. deg(f_{x,1}) + deg(f_{x,2}) = N.

Proof. Many of these assertions follow from the discussion of [FK], pp. 161–166, at least when N is odd and the Weierstrass points are F -rational. However, since we do not make these assumptions here, it seems better to give a direct proof of these facts.

(a) Since J ∼ E_{F} × E′_{F}, the connected component J_{x}^{0} of J_{x} is isogenous to E × (E′_{x})^{0} (cf. [BLR], 7.4/7), and so S_{0}(C/X) = S(E′/X) (use [BLR], 7.4/2 and 7.3/7). Furthermore, S(E′/X) = X_{∞}, as was shown in the proof of Theorem 17.

Now if x ∈ S(E′/X), then (E′_{x})^{0} ≃ 𝔾_{m}, and hence J_{x}^{0} has a 1-dimensional torus part and a 1-dimensional abelian part. Thus, since Pic_{C_{x}/K}^{0} ≃ J_{x}^{0} (cf. [BLR], 9.5/4), it follows from [BLR], 9.2/8, that C_{x} has a unique component whose normalization has genus 1 and that the graph of components of C_{x} has a loop, i.e. there exists a point P_{x} with the asserted property.

To verify the assertion about f_{C_{x,0}}, we will use the fact that since f is generically finite of degree N, we have f_{∗}f^{∗}D = ND, for every divisor D on E_{X}; cf. Fulton [Fu2], Example 8.1.7. Now for D = E_{x} we have f^{∗}E_{x} = C_{x} = C_{x,0} + C_{x,1} + … + C_{x,r}, where the C_{x,i} are the components of C_{x}. But since C_{x,i} is rational for i ≥ 1 and since E_{x} ≃ E is an elliptic curve, we see that f(C_{x,i}) is a point and so f_{∗}(C_{x,i}) = 0 for i ≥ 0. Thus f_{∗}C_{x,0} = f_{∗}f^{∗}E_{x} = NE_{x}, which means that f_{C_{x,0}} : C_{x,0} → E_{x} is finite and has degree N, as asserted.

(b) Since C_{X(N)} has semi-stable reduction, we have S(C/X) = β^{−1}S(C_{X(N)}/X(N)), and so it is enough to verify the first assertion for X = X(N).

Since C_{H}/H is smooth, we see that S := S(C/X) ⊂ X(N) \ H. To prove the opposite inclusion, we first recall some facts about the construction of H = H_{E/K,N} from [Ka3], §5.3:

1) Since K is algebraically closed, we have X_{E/K,K,−1} ≃ X′(N) = X(N) \ X(N)_{∞} (cf. [Ka3], Theorem 4.1). Moreover, there is a principally polarized abelian scheme (J′, λ′) over X′ := X′(N) whose restriction to H is the Jacobian scheme J_{C_{H}}/H of the universal genus 2 curve C_{H}/H (use [Ka3], Corollary 5.11, Proposition 5.12 and Theorem 5.18).

2) The set H is the open subscheme of points x ∈ X′(N) such that (J′_{x}, λ′_{x}) is theta-smooth (in the sense of [Ka3], §5.3); cf. [Ka3], Proposition 5.17.

Note that 1) shows that J′/X′ is the Néron model of J_{F} over X′ because J′/X′ is an abelian scheme with generic fibre J_{F} (use [BLR], 1.2/8). Thus J′ ≃ J_{|X′} and λ′ is the unique extention of the canonical polarization λ of the Jacobian J_{C_{H}}/H.

Now suppose that x ∈ X(N) \ S. Then C_{x} is smooth, and so C_{T} /T is a smooth relative curve over T = Spec(O_{X,x}). Then the principally polarized Jacobian (J_{C_{T}} , λ_{T}) of C_{T} is theta-smooth (cf. [Ka3], Proposition 5.14). But J_{C_{T}} ≃ J′_{|T} because both are Néron models of J_{F} over T, and also λ_{T} = λ′_{|J_{C_{T}}}, and so J′ is theta-smooth at x, i.e. x ∈ H. Thus X(N) \ S ⊂ H or S ⊃ X(N) \ H, and so S = X(N) \ H, as claimed.

Now let x ∈ S_{1} = S_{1}(C/X), where X is again a general cover of X(N). Then by definition C_{x} = C_{x,0} ∪ … ∪ C_{x,r} is singular whereas J_{x} is an abelian variety. Thus, by [BLR], 9.3/12 (together with [BLR], 9.5/4) it follows that all components of C_{x} are smooth and the configuration of components is tree-like. Moreover, since 2 = dim J_{x} = ∑_{i} g(C_{x,i}) by [BLR], 9.2/8, we see that C_{x} has precisely two components of genus 1. These have to be in fact the two ends of the graph, for any rational end component would be a (−1)-curve on C_{x}, which does not exist since C is minimal. Thus, C_{x} is a chain whose ends are the two elliptic curves.

The fact that the self-intersection numbers are as indicated is proved in Ogg[Ogg], p. 360; note that here we have type 13 in Ogg’s list (and type IV is Parshin’s list[Pa2]).

To prove the last assertion, we first observe that the same argument as in (a) shows that N = deg(f_{x,1}) + deg(f_{x,2}) (because all other components of C_{x} map to a point). Thus, at least one of the maps f_{x,i} is finite, but it seems more difficult to show that both are finite. To prove this, it is clearly enough to show that f(E_{x,i}) = E_{x}, for i = 1, 2 and to verify this, we may assume (in view of (14) and the fact that β_{E_{X}} is finite) that F is sufficiently large. In that case this follows immediately from the following result which is also of independent interest.

Proposition 20 Suppose that either N is odd or that the Weierstrass points of C_{F} are F -rational. Then there exists a morphism j′ : C′ := p^{−1}(X′) → J′ := p_{J}^{−1}(X′) ⊂ J such that j′ ∘ σ_{C′} = [−1]_{J′} ∘ j′ and λ_{E_{X′}} ∘ f_{|C′} = f_{∗} ∘ j′, (18)

where f_{∗} : J′ → J_{E_{X′}} = J_{E} ×X′ is the homomorphism on the abelian schemes induced by f, and λ_{E_{X′}} : E_{X′} →^{∼} J_{E_{X′}} is the canonical polarization. In addition, θ := j′(C′) is a theta-divisor of the principally polarized abelian scheme (J′, λ_{J′}) over X′, and (f_{∗})_{|θ} : θ → E_{X′} is finite. In particular, f(E_{x,i}) = E_{x}, for all x ∈ S_{1}(C/X) ⊂ X′.

Proof. By [Ka3], Proposition 2.4, there exists a morphism j_{F} : C_{F} ↪ J_{F} such that (18) holds for the generic fibres over F. Since J′/X′ is an abelian scheme, it follows from Proposition 15 that j_{F} extends uniquely to a morphism j′ : C′ → J′. Moreover, by the universal property of Néron models, (f_{F})∗ extends uniquely to a homomorphism f_{∗} : J′ → J_{E_{X′}}, and so (18) holds because it is true on the generic fibres.

Since j′ is proper, θ = j′(C′) is the closure of j_{F} (C_{F}) in J′, and so is an irreducible divisor on the smooth 3-fold J′/K. Thus θ cannot contain any component of a fibre of J′/X′ and so is flat over X′ (since X′ is a smooth curve). Thus θ_{x} = θ_{|J′_{x}} is an effective divisor on J′_{x}, for every x ∈ X′. Now since its generic fibre θ_{F} = θ_{|J_{F}} is the theta-divisor of J_{F} by [Ka3], Corollary 2.5, we see that θ_{x}^{2} = θ_{F}^{2} = 2 for all x ∈ X′, and so each θ_{x} is the theta-divisor of a principal polarization on J′_{x}, which means that λ := λ_{L(θ)} : J′ → Ĵ′ defines a principal polarization. But since λ agrees with the canonical polarization λ_{C} on the generic fibre, we have λ = λ_{J′}, and so θ is a theta-divisor associated to λ_{J′}.

To show that (f_{∗})_{|θ} is finite, we first observe that we have an exact sequence

0 → E′_{X′} →^{h′} J′ →^{f_{∗}} J_{E_{X′}} → 0, (19)

where h′ : E′_{X′} → J′ denotes the canonical extension (by the universal property of Néron models) of the map h′_{F} : E′_{F} → J_{F} constructed in [Ka3], Proposition 2.7. Thus (19) is exact when restricted to the generic fibres over F. Now since f_{F}^{∗} : J_{E_{F}} → J is an injection and (f_{∗})_{F} ∘ f_{F}^{∗} = [N], the hypotheses of [BLR], 7.5/3(a) are satisfied and so (19) is an exact sequence because J′/X′ is an abelian scheme.

Now let x ∈ X′. If θ_{x} is irreducible (hence also reduced), then (f_{∗})_{|θ_{x}} : θ_{x} → J_{E} is a surjective map f_{x} between integral proper curves and hence is finite. (Recall that f_{x} is surjective and hence also (f_{∗})_{|θ_{x}}.) Thus assume that θ_{x} is reducible, i.e. θ_{x} = θ_{x,1} + θ_{x,2}, where each θ_{x,i} is an elliptic curve on J′_{x}; cf. [We], Satz 2. Now if (f_{∗})_{|θ_{x}} is not finite, then f_{∗}(θ_{x,i}) is a point for some i, and then a translate of θ_{x,i} lies in E′_{x} := h′(E′_{x}) = Ker((f_{∗})_{x}). Since both are irreducible curves, we must have equality, and so (θ_{x}.E′_{x}) = (θ_{x}.θ_{x,i}) = 1. But (θ_{x}.E′_{x}) = (θ_{F} .E′_{F}) = deg(f_{F}) = N, contradiction. Thus (f_{∗})_{|θ_{x}} is finite for all x, and hence so is (f_{∗})_{θ}.

Finally, if x ∈ S_{1}, then j′(C_{x}) = j′(E_{x,1}) ∪ j′(E_{x,2}) = θ_{x}, and neither of the j′(E_{x,i}) can be a point for otherwise θ_{x} is an elliptic curve, which is impossible since θ_{x}^{2} = 2. Thus also f(E_{x,i}) = f_{∗}(j′(E_{x,i})) cannot be a point, and so f(E_{x,i}) = E, as claimed.

In the case that X = X(N), the cardinality of the singular locus given by the following result.

Theorem 21 If X = X(N), then the number of points in S_{1} = S_{1}(C/X) satisfies the inequality

#S_{1} ≤ \frac{1}{12}(5N − 6)#X(N)_{∞} = \frac{1}{12N} (5N − 6)\overline{sl}(N). (20)

Thus, the singular locus S = S(C/X) consists of at most \frac{1}{12N} (5N + 6)\overline{sl}(N) points.

Furthermore, equality holds in (20) if and only if either char(K) = 0 or char(K) > N, i.e. we have

#S_{1} = \frac{1}{12N} (5N − 6)\overline{sl}(N) ⇔ char(K)|̸ N!. (21)

Proof. By Proposition 19(b) we have S_{1} = X′(N)\H = X_{E/K,N,−1}\H_{E/K,N} = D_{E/K,N} (in the notation of [Ka3], §6). Since #X(N)_{∞} = \frac{1}{N} \overline{sl}(N) (cf. (16)), we see that the assertions follow directly from [Ka3], Theorem 6.2 (which is actually a restatement of the results of [Ka1],[Ka2]).

In the case that char(K)|̸ N!, the precise structure of the fibres of C/X can be determined as follows.

Theorem 22 If char(K)|̸ N! and X = X(N), then C/X is a stable curve which has precisely one singular point in each singular fibre C_{x}, and so δ_{C/X} = #S(C/X) = \frac{1}{12N} (5N + 6)\overline{sl}(N). Thus, the structure of the singular fibres is as follows:

(a) If x ∈ S_{1}, then C_{x} = E_{x,1} ∪ E_{x,2} is the union of two curves of genus 1 which meet transversely in a unique point P_{x}.

(b) If x ∈ S_{0} = S_{∞}, then the fibre C_{x} is an irreducible curve whose normalization is a curve of genus 1, and C_{x} has a unique singular point P_{x} ∈ C_{x}.

Proof. Let δ_{1} (respectively δ_{0}) denote the number of singular points of the fibres of C/X which disconnect (respectively, which do not disconnect) the fibre; thus δ := δ_{C/X} = δ_{0} + δ_{1} is the total number of singular points on all fibres of C/X. By Proposition 19 we have the inequalities

δ_{0} ≥ #S_{0} = #X(N)_{∞} and δ_{1} ≥ #S_{1}. (22)

There are two important relations which connect the four invariants h = h_{C/X}, ω^{2} = (ω_{C/X}^{0})^{2}, δ_{0}, and δ_{1}. The first of these is Noether’s formula

12h = ω^{2} + δ = ω^{2} + δ_{0} + δ_{1}, (23)

which is valid for any semi-stable curve of genus g ≥ 1; cf. e.g. Szpiro[Sz], Lemma 1 (p. 48). For genus 2 curves, however, we have in addition Mumford‘s formula

ω^{2} = \frac{1}{5}δ_{0} + \frac{7}{5}δ_{1}; (24)

cf. [Mu], formula (8.5) on p. 317 (together with the remarks of [Sa] on p. 237), or Ueno[U], formula (2.4), together with Theorem 2.9 (specialized to the semi-stable case). (See also Remark 23(b) below.) Combining these two relations yields

12h = \frac{6}{5}δ_{0} + \frac{12}{5}δ_{1}. (25)

In our situation, this identity, together with the inequalities (22), yields

12h ≥ \frac{6}{5}#S_{0} + \frac{12}{5}#S_{1} = \frac{6}{5}#S_{0} + \frac{1}{5}(5N − 6)#S_{0} = N#S_{0}, (26)

where the second last equality follows from (21). (Here we used the hypothesis that char(K)|̸ N!.) But since 12h = N#X(N)_{∞} = N#S_{0} by (16), it follows that all the inequalities in (26) and hence in (22) were equalities, i.e. we have

δ_{0} = #S_{0} = \frac{1}{N} \overline{sl}(N) and δ_{1} = #S_{1} = \frac{1}{12N} (5N − 6)\overline{sl}(N). (27)

We thus see that each singular fibre contains a unique singular point, as claimed. From this and Proposition 19, the assertions of parts (a) and (b) follow readily.

Remark 23 (a) Note that (16), (21) and (25) imply that the following converse of Theorem 22 holds: if each singular fibre C_{x} of C/X(N) contains a unique singular point, then char(K)|̸ N!.

(b) If we combine Noether’s formula (23) and Mumford’s formula (24) so as to eliminate δ_{1}, then we obtain the relation

h = \frac{1}{7}ω^{2} + \frac{1}{14}δ_{0}, (28)

which, via Noether’s formula, is equivalent to Mumford’s relation (24). Now in [FK], Corollary 4.2, a weak form of this equality was derived (in the arithmetic case) by studying the intersection numbers of the components of the Weierstrass divisor W. In the geometric case, this method can be refined to derive the above equality (28) directly.

Corollary 24 If char(K)|̸ N!, then the self-intersection number of ω_{C/X(N)}^{0} is given by

(ω_{C/X(N)}^{0})^{2} = \frac{1}{5}#S_{0} + \frac{7}{5}#S_{1} = \frac{1}{12N} (7N − 6)\overline{sl}(N). (29)

Proof. Substitute the values of δ_{0}, δ_{1} and h which are given by (27) and (16) in Noether’s formula (23).

Corollary 25 If char(K)|̸ N! and X = X(N), then f : C → E_{X} is finite and flat of degree N. Furthermore, we have f(P_{x}) ∈ E_{x}[2] = E[2], for each singular point P_{x} ∈ C_{x}.

Proof. By Proposition 19 and Theorem 22 we know that f does not map any irreducible component of C_{x} to a point, and so f is quasi-finite. Since f is proper, this means that f is finite (cf. [Ha], Ex. III.11.2). Moreover, since C and E_{X} are smooth surfaces over K, it follows that f is also flat; cf. [Ha], Ex. III.9.3(a).

The last assertion is immediate: since P_{x} is the unique singular point in its fibre C_{x}, it is fixed by the hyperelliptic involution σ_{C} and hence f(P_{x}) is fixed by [−1]_{E_{X}} by (15).

5 The different divisor

Let D_{F} = Diff(f_{F}) denote the different divisor of the generic cover f_{F} : C_{F} → E_{F}, and let D_{C/X} denote its closure in C. Then there is a close connection between D_{C/X(N)}, the discriminant map δ : H_{E,N} → ℙ^{1} (defined in §3), the different divisor Diff(f) (cf. Appendix) of the compactified morphism f : C → E_{X(N)}, and the relative dualizing sheaf ω_{C/X(N)}^{0}, provided that char(K)|̸ N!, which we assume from now on.

Theorem 26 Suppose X = X(N), and write D = D_{C/X(N)}.

(a) We have D = Diff(f) and hence ω_{C/X}^{0} ≃ L(D).

(b) The divisor D is an irreducible curve on C which meets each singular fibre C_{x} precisely at the unique singular point P_{x}, and f_{|D} : D → f(D) ⊂ E × X is birational.

(c) The morphism π_{D} := pr_{2} ∘ f_{|D} : D → X is a σ_{D}-invariant cover of X of degree 2, where σ_{D} = (σ_{C})_{|D} is the restriction of the hyperelliptic involution to D. In particular, D/⟨σ_{D}⟩ ≃ X.

(d) There exists a unique morphism δ_{D} : X → ℙ^{1} such that

π_{E} ∘ q = δ_{D} ∘ π_{D}, (30)

where q := pr_{2} ∘ f_{|D} : D → E and π_{E} : E → E/⟨[−1]⟩ = ℙ^{1} is the usual double subcover of E. Furthermore, δ_{D} restricts to the discriminant map δ_{E,N} : H_{E,N} → ℙ^{1} on H_{E,N} ⊂ X, and hence δ_{D} = \overline{δ}_{E,N}. In addition, we have

(31) deg(\overline{δ}_{E,N}) = deg(δ_{D}) = deg(q) = (ω_{C/X}^{0}.f^{∗}(P × X)).

(e) The restriction δ_{E,N}^{sm} : H_{E,N}^{sm} → ℙ^{1} \ π_{E}(E[2]) of δ_{E,N} to H_{E,N}^{sm} is finite and etale.

Proof. (a) Note first that Diff(f) exists by Corollary 35 of the Appendix. Now by Corollary 25 we know that for each component Γ of a fibre C_{x} of C/X, the morphism f_{|Γ} : Γ → E_{x} = E is finite of degree deg(f_{|Γ}) ≤ N. Since char(K)|̸ N!, it follows that f_{|Γ} is automatically separable. Thus by Corollary 35(b) we see that B(f) = 0, and hence Diff(f) = D. Now since E/K is an elliptic curve, we have ω_{E/K} ≃ O_{E}, and hence ω_{E_{X}/X}^{0} ≃ pr_{1}^{∗}ω_{E/K} ≃ O_{E_{X}}. Thus by (47) we obtain ω_{C/X}^{0} ≃ f^{∗}ω_{E_{X}/X}^{0} ⊗ L(D) ≃ L(D), as claimed.

(b) If Γ ⊂ C_{x} is an irreducible component of C_{x}, where x ∈ S = S(C/X), then Γ \ {P_{x}} is a smooth open subset of an elliptic curve, and so Γ \ {P_{x}} → E is etale (since separable). Thus supp(Diff(f_{x})) ⊂ {P_{x}}, and so D can meet C_{x} only in P_{x}; cf. Corollary 35(b). This proves the second assertion.

The first assertion is a consequence of the second. For if D = D_{1} + D_{2} were reducible, then both components D_{i} must meet C_{x} in P_{x}. Since P_{x} is a singular point on C_{x}, we have (D_{i}.C_{x}) ≥ 2, so 2 = (D.C_{x}) = (D_{1}.C_{x}) + (D_{2}.C_{x}) ≥ 4, contradiction. Thus D is is irreducible.

To see that f_{|D} is birational, we first observe that

(32) f(D)^{2} ≥ D^{2} > 0, and hence f(D) ≁ A × X, for any A ∈ Div(E).

Indeed, since (A × X)^{2} = 0, it is clear that the second follows assertion from the first. To prove the latter, write f_{∗}D = kf(D); note that k = 1 or 2 because (f_{∗}D.E × x) = (D.C_{x}) = 2. Since f^{∗}f(D) = 2D + D′, where D′ is an effective divisor not containing D, we have k(f(D))^{2} = (f_{∗}D.f(D)) = (D.(2D+D′)) ≥ 2D^{2}. Since D^{2} = (ω_{C/X}^{0})^{2} > 0 by (a) and (29), we thus have f(D)^{2} ≥ \frac{2}{k}D^{2} ≥ D^{2} > 0, as claimed.

We next observe that σ_{C}^{∗}(D) = Diff(f ∘ σ_{C}) = Diff([−1] ∘ f) = Diff(f) = D, i.e. σ_{C} maps D into itself, and hence [−1]f(D) = fσ_{C}(D) = f(D).

Suppose now that D → f(D) were not birational. Then k = 2 and (f(D).E×x) = 1, for all x ∈ X, so f(D)_{F} ∈ E(F). But since [−1]f(D) = f(σ_{C}(D)) = f(D), this forces f(D)_{F} = P_{i} ∈ E[2]. Thus f(D) = P_{i} × X, which is impossible by (32), and hence D → f(D) is birational.

(c) Since π_{D}^{−1} = D ∩ C_{x} and (D.C_{x}) = 2, for all x ∈ X, we see that π_{D} is a double cover of X. Moreover, since σ_{C} maps D into itself (cf. proof of part (b)), σ_{D} = (σ_{C})_{|D} is an involution of D. In addition, we have π_{D}∘σ_{D} = pr_{2}∘f ∘(σ_{C})_{|D} = pr_{2}∘[−1]∘f_{|D} = π_{D}, so π_{D} ∘ σ_{D} = π_{D}, i.e. π_{D} is σ_{D}-invariant.

We claim that σ_{D} ≠ id_{D}. Indeed, since f(D) ≠ A × X by (32), there exists a point (P, x) ∈ f(D) with P ∉ E[2]. Then f(π_{D}^{−1}(x)) = {(P, x), (P′, x)}, where P′ = [−1]P ≠ P, and so σ_{D} interchanges the two points of π^{−1}(x); in particular, σ_{D} ≠ 1. Thus the map D/⟨σ_{D}⟩ → X is birational and hence is an isomorphism as X is smooth.

(d) Since f(D_{F}) is a [−1]-invariant effective divisor of degree 2 on E_{F}, there is a point P ∈ ℙ_{F}^{1} (F) such that π_{E}^{∗}(P) = f(D_{F}). Thus, if Γ denotes the closure of P in ℙ^{1} × X, then we have (π_{E} × id_{X})^{∗}(Γ) = f(D). Now since deg(P) = 1, we see that (Γ.ℙ^{1} × x) = 1, for all x ∈ X, and so Γ = Γ_{δ}^{t} ⊂ ℙ^{1} × X is the transpose of the graph of a unique morphism δ : X → ℙ^{1}. Note that the equation (π_{E} × id_{X})^{∗}Γ_{δ}^{t} = f(D) implies that π_{E} ∘ (pr_{2})_{|f(D)} = δ ∘ (pr_{1})_{|f(D)}, and from this (30) follows immediately.

Since deg(π_{D}) = deg(π_{E}) = 2, the second equality of (31) follows directly from (30), and the third is clear because we have by definition that deg(q) = deg(q^{∗}(P)) = deg((f^{∗}pr_{1}^{∗}(P)_{|D}) = (D.f^{∗}(P × X)).

It thus remains to show that (δ_{D})_{|H} = δ_{E,N}. For this, we first note that by definition (cf. §3) we have:

δ_{E,N}(x) = \overline{P} ⇔ Disc(f_{x}) = π_{E}^{∗}(\overline{P}) ⇔ supp(Disc(f_{x})) ⊂ π_{E}^{−1}(\overline{P}),

for all x ∈ H_{E,N}(K) and \overline{P} ∈ ℙ^{1}(K), where the latter equivalence follows from the fact that supp(Disc(f_{x})) is [−1]_{E}-invariant (and π_{E}^{−1}(\overline{P}) contain no proper [−1]_{E}- invariant subset. On the other hand, for any x ∈ X(K) we have supp(Disc(f_{x})) = pr_{1}(f(D) ∩ (E × x)) = q(D ∩ C_{x}) = q(π_{D}^{−1}) ⊂ π_{E}^{−1}(δ_{D}(x)), where the inclusion follows from (30) because π_{D} is surjective. Thus we see that δ_{E,N}(x) = δ_{D}(x), ∀x ∈ H.

(e) By Theorem 13(a) we know that δ_{E,N}^{sm} is etale, hence quasi-finite, so it is enough to show that δ_{E,N}^{sm} is proper. Moreover, since δ_{D} : X → ℙ^{1} is proper, it is enough to show (in view of part (d)) that H_{E,N}^{sm} = δ_{D}^{−1}(ℙ^{1} \ π_{E}(E[2])). Now if x ∈ δ_{D}^{−1}(ℙ^{1} \ π_{E}(E[2])), then C_{x} is smooth by part (b), and so x ∈ H_{E,N}^{sm} by definition (and (30)). Thus H_{E,N}^{sm} = δ_{D}^{−1}(ℙ^{1} \ π_{E}(E[2])) since the opposite inclusion is trivial.

Remark 27 From the above theorem we see that the curve f(D) = f_{∗}(Diff(f)) is the discriminant of f. Moreover, formula (31) shows that (f(D).P × X) = deg(q), so f(D) ⊂ E × X has bi-degree (deg(q), 2).

6 The Weierstrass divisor W

In order to compute the degree of q, we shall exploit the properties of the Weierstrass divisor W_{C/X} on C. By definition, W_{C/X} is the closure in C of the (usual) Weierstrass divisor W_{F} on C_{F}; recall that the latter is the locus of fixed points of the hyperelliptic involution σ_{F} on C_{F}, and hence is an effective divisor of degree deg(W_{F}) = 6.

As in [Pa2] and [FK], the discussion of Weierstrass divisors becomes much easier when all the Weierstrass points of C_{F} are F -rational. We therefore study W_{C/X} first in the case that X = X(2N) and then relate it to that of X(N).

Proposition 28 Let X = X(2N). Then we have:

(a) The Weierstrass points of C_{F} are F -rational, and so W_{C/X} = W_{1} + … + W_{6}, where the W_{i}’s are disjoint sections of C/X.

(b) For each x ∈ S_{1}(C/X), the base-change map (β_{C})_{x} : C_{x} → C_{β(x)} is an isomorphism.

(c) For each x ∈ S_{0}(C/X) = X_{∞}, the fibre C_{x} = C_{x} ∪B_{x} is the union of an elliptic curve C_{x} and a rational curve B_{x} ≃ ℙ^{1}, and we have C_{x}^{2} = B_{x}^{2} = −(B_{x}.C_{x}) = −2. Furthermore, (β_{C})_{x}(B_{x}) = P_{β(x)}, the unique singular point of C_{β(x)}.

(d) If D ∈ Div(C_{X(N)}) is the closure of an effective divisor D_{F_{N}} on C, and if D^{∗} ∈ Div(C) denotes the closure of the divisor (β_{C_{F}})^{∗}(D_{F_{N}}) on C_{F}, then we have

β_{C}^{∗}(D) = D^{∗} + ∑_{x∈X_{∞}} n_{x}B_{x}, where n_{x} = \frac{1}{2}(D^{∗}.B_{x}). (33)

In particular, we have β_{C}^{∗}(W_{C/X(N)}) = W_{C/X} + B, where B := ∑_{x∈X_{∞}} B_{x}. (e) We have ω_{C/X}^{0} ≃ β_{C}^{∗}(ω_{C/X(N)}^{0}), and so

(ω_{C/X(N)}^{0}.W_{C/X(N)}) + W_{C/X(N)}^{2} = #X(N)_{∞} = \frac{1}{N} \overline{sl}(N). (34)

Proof. (a) The first assertion follows from the fact that the 2-torsion points of J_{F} are F -rational. To verify the latter claim, recall that J_{F} = A_{F} /H, where A_{F} = E_{F} × E′_{F} and H ≤ A_{F} [N] is a suitable subgroup (of order N^{2}); cf. [Ka3], Corollary 5.19. Since the 2N-torsion points E′_{F} are F -rational, it follows that the same is true for A_{F} = E_{F} × E′_{F}, and so the 2-torsion points of J_{F} = A_{F} /H are F -rational, as desired.

Thus we see that W_{C/X} = W_{1} + … + W_{6} is the sum of six sections. Now by the same argument as in [FK], these meet each fibre in different places (because for each x ∈ X the kernel of the reduction maps J_{C} → (J)_{x} does not contain any non-trivial 2-torsion points), and so it follows that these sections are disjoint.

(b) This is clear because β : X(2N) → X(N) is unramified outside of X(N)_{∞}.

(c) Since e_{x}(β) = 2N/N = 2, the singularity P_{β(x)} of C_{β(x)} (cf. Theorem 22) is resolved by a line B_{x} ≃ ℙ^{1} which meets the elliptic curve C_{x} in two distinct points. Thus we have case II of Parshin’s list[Pa2], and so the intersection numbers are as indicated.

(d) By parts (b) and (c) we see that β_{C}^{∗}(D) = D^{∗} + ∑_{x∈X_{∞}} n_{x}B_{x} for some integers n_{x} ∈ ℤ. Since (β_{C})_{∗}B_{x} = 0, we obtain from the projection formula that 0 = (β_{C}^{∗}(D).B_{x}) = (D^{∗}.B_{x}) + n_{x}B_{x}^{2}, and so n_{x} = \frac{1}{2}(D^{∗}.B_{x}), which proves (33).

By Parshin[Pa2], p. 80, we know that (W_{C/X}.B_{x}) = 2, for all x ∈ X_{∞}, and so the formula for β_{C}^{∗}(W_{C/X(N)}) follows from (33).

(e) The first assertion is a general fact about relative dualizing sheaves; cf. [Ar], Lemma 3.4 or [Sz], Lemma 3(b), p. 50.

To prove (34), write W = W_{C/X(N)}, ω = ω_{C/X(N)}^{0} and W̃ = W_{C/X(2N)}, ω̃ = ω_{C/X(2N)}^{0}. In addition, let d = deg(β) = \overline{sl}(2N)/\overline{sl}(N). (Note that d = 8 if N is even and d = 6 if N is odd.) Then (β_{C})_{∗}(W̃) = dW, and so by the projection formula and part (d) we have

d((ω + W).W) = (β_{C}^{∗}(ω + W).W̃) = ((ω̃ + W̃ + B).W̃).

Now by the adjunction formula we have for any section W_{i} of C/X that (ω̃.W_{i}) = −W_{i}^{2}, and so by part (a) we have ((ω̃+W̃).W̃) = 0 because W̃ = W_{1}+…+W_{6} consists of 6 disjoint sections (so W̃^{2} = W_{1}^{2} + … + W_{6}^{2}). Moreover, since ((W̃ + B).B) = (β_{C}^{∗}(W).B) = (W.(β_{C})_{∗}(B)) = 0, we have (W̃ .B) = −B^{2} = 2#X_{∞}, the latter by part (c). Now since X_{∞} = β^{−1}(X(N)_{∞}), and each x ∈ X_{∞} is ramified of degree e_{x}(β) = 2, we see that #X_{∞} = \frac{d}{2}#X(N)_{∞}, and so (W̃ .B) = 2#X_{∞} = d#X(N)_{∞}. Thus d((ω + W).W) = ((ω̃ + W̃).W̃) + (B.W̃) = 0 + d#X(N)_{∞}, which proves (34).

Proposition 29 Suppose that X = X(N) and write D = D_{C/X} and W = W_{C/X}. Then we have 6D ∼ 2W + p^{∗}A, for some divisor A ∈ Div(X) of degree

deg(A) = #X(N)_{∞} − \frac{4}{3}W^{2} = \frac{1}{6}(9(ω_{C/X}^{0})^{2} − W^{2}) = \frac{1}{N} (N − 1)\overline{sl}(N). (35)

Proof. We first note that

(W.E_{x,1}) = (W.E_{x,2}) = 3, for all x ∈ S_{1}. (36)

Indeed, by [Pa2], p. 81, this is true after base-change to X(2N), with W replaced by W_{C/X(2N)}, and so (36) follows from this together with Proposition 28(b),(d).

Now since 6D_{F} and 2W_{F} are both 6-canonical divisors, we have 6D_{F} ∼ 2W_{F}. Moreover, since the fibres over S_{1} are the only reducible fibres of C/X (cf. Theorem 22) and since for x ∈ S_{1}, both 6D and 2W meet each component E_{x,i} of C_{x} with multiplicity 6 (by Theorem 26(b) and by (36), respectively), we can conclude that

(37) 6D ∼ 2W + p^{∗}(A), for some A ∈ Div(X).

Here we have used the following general fact (which is valid for any regular proper curve C/X without multiple fibres): Fact. Suppose that D_{1}, D_{2} ∈ Div(C) are two divisors such that their restrictions (D_{i})_{F} to the generic fibre are linearly equivalent, i.e. (D_{1})_{F} ∼ (D_{2})_{F} and which have the property that (D_{1}.Γ) = (D_{2}.Γ), for all components Γ of reducible fibres C_{x} of C/X. Then D_{1} ∼ D_{2} + p^{∗}A, for some divisor A ∈ Div(X). [Indeed, the first hypothesis yields that D_{1} ∼ D_{2} + B, for some divisor B consisting entirely of fibre components, and the second shows that (B.Γ) = 0, for all fibre components Γ, and so B = p^{∗}A, for some A by Zariski’s Lemma (cf. [BPV], p. 90).]

We now compute the degree of A. For this, recall first from Theorem 26(a) that ω := ω_{C/X}^{0} ≃ L(D). Thus, by (34) and (37) we have #X(N)_{∞} − W^{2} = (ω.W) = \frac{1}{6}((2W + p^{∗}(A)).W) = \frac{1}{3}W^{2} + deg(A), and so deg(A) = #X(N)_{∞} − \frac{4}{3}W^{2}, which proves the first equality of (35).

Next, by (37) we have 36ω^{2} = (2W + p^{∗}A)^{2} = 4W^{2} + 4(W.p^{∗}A) + (p^{∗}A)^{2} = 4W^{2} + 24 deg(A) + 0, so deg(A) = \frac{1}{6}(9ω^{2} − W^{2}), which proves the second equality of (35).

Thus we have 6 deg(A) = (9ω^{2} − W^{2}) = 6#X(N)_{∞} − 8W^{2}, and so

W^{2} = \frac{6}{7}#X(N)_{∞} − \frac{9}{7}ω^{2} = −\frac{3}{4N} (N − 2)\overline{sl}(N), (38)

the latter by (29). From this, the last equation of (35) is immediate.

Corollary 30 The degree of q is given by

deg(q) = \frac{N}{6} deg(A) = \frac{1}{6}(N − 1)\overline{sl}(N). (39)

Proof. Since f is normalized, we have f_{∗}(W) = ∑_{i=0}^{3} k_{i}(P_{i} × X) with suitable multiplicities k_{i} ≥ 0 (cf. equation (6)). Now since 2P_{i} ∼ 2P_{0}, we thus have 2f_{∗}W ∼ 12(P_{0} × X), and so, since ω := ω_{C/X}^{0} ∼ D by Theorem 26(a), we obtain

6f_{∗}ω ∼ 6f_{∗}D ∼ 2f_{∗}W + f_{∗}(p^{∗}A) ∼ 12(P_{0} × X) + N(E × A).

Thus, by (31) and the projection formula we have 6 deg(q) = 6(ω.f^{∗}(P_{0} × X)) = ((12(P_{0} × X) + N(E × A)).(P_{0} × X)) = N deg(A), which proves the first equality of (39). The second follows immediately from (34).

By the above corollary we have computed the degree deg(δ_{E,N}) and hence also the right hand side of formula (12). Thus, the desired formula for Cov_{E,N,D}^{(min)} of Theorem 3 follows once we know that \overline{δ}_{E,N} is tamely ramified.

Proposition 31 If char(K) ∤ N!, then the map \overline{δ}_{E,N} : X(N) → ℙ^{1} is tamely ramified.

Proof. We shall use the criterion given in Corollary 14. For this, fix a point P ∈ E(K). Since deg(\overline{δ}_{E,N}) = deg(q) = \frac{1}{6}(N − 1)\overline{sl}(N) by (31) and (39) and since (2g_{X(N)} − 2 + s_{E,N}) = \frac{N−6}{6N} \overline{sl}(N) + \frac{5N+6}{12N} \overline{sl}(N) = \frac{7N−6}{12N} \overline{sl}(N) by Theorem 21, we see that (12) reduces to the inequality

#Cov_{E,N,2P}^{(min)} ≥ (\frac{N − 1}{3} − \frac{7N − 6}{12N}) \overline{sl}(N) = \frac{(4N − 3)(N − 2)}{12N}\overline{sl}(N), (40)

and that equality holds if and only if \overline{δ}_{E,N} is tamely ramified. In particular, we see that equality holds if char(K) = 0.

We now derive another formula for #Cov_{E,N,2P}^{(min)} by using the tame fundamental group π_{1}^{t}(E_{P} , x), where E_{P} = E \ {P }, and x ∈ E_{P} (K) is a fixed base point. For this, let p = char(K), if char(K) > 0, and otherwise let p be any prime with p > N. Now the hypothesis char(K) ∤ N! guarantees that each f ∈ Cov_{E,N,2P}^{(min)} is tamely ramified and that the degree of its Galois closure f̃: C̃ → E is not divisible by p. Thus, Gal(f̃) is a quotient of the group π_{1}^{t}(E_{P} , x)^{(p′)}, the prime-to-p fundamental group, and so we have a natural bijection

(41) Cov_{E,N,2P}^{(min)} →^{∼} Hom′(π_{1}^{t}(E_{P} , x)^{(p′)}, S_{N})/S_{N}

where the symmetric group S_{N} acts by conjugation on Hom′(…) and the prime denotes the subset of those homomorphisms h : π_{1}^{t}(E_{P} , x)^{(p′)} → S_{N} satisfying the following conditions: 1) Im(h) is a transitive and primitive subgroup of S_{N}; 2) the ramification generator t = t_{P} ∈ Hom′(π_{1}^{t}(E_{P} , x)^{(p′)} at P is mapped either to a 3-cycle or to a (2, 2)-cycle.

Now by a fundamental result of Grothendieck (cf. e.g. [Wev], Theorem 4.3.1), π_{1}^{t}(E_{P} , x)^{(p′)} ≃ Γ̂_{1,1}^{(p′)}, the prime-to-p quotient of the group Γ̂_{1,1} which is the pro-finite completion of the discrete group Γ_{1,1} = ⟨σ_{1}, σ_{2}, τ : σ_{1}σ_{2}σ_{1}σ_{2}τ = 1⟩. Thus, by applying the above results to an elliptic curve E_{0}/K_{0} with char(K_{0}) = 0, we see from (41) and (40) that

#Hom′(Γ_{1,1}^{(p′)}, S_{N})/S_{N} = \frac{(4N − 3)(N − 2)}{12N}\overline{sl}(N), for any p > N;

here we have used the fact that \overline{δ}_{E_{0},N} is tamely ramified because char(K_{0}) = 0. Thus, if we substitute this formula in (41), we see that equality holds in (40), and hence \overline{δ}_{E,N} is tamely ramified.

Proof of Theorem 3. We may assume N ≥ 3 because the case N = 2 is settled by Proposition 6. Since \overline{δ}_{E,N} is tamely ramified by Proposition 31, we have by Corollary 14 that equality holds in (12). This proves (3) because deg(\overline{δ}_{E,N}) = \frac{1}{6}(N − 1)\overline{sl}(N) and (2g_{X(N)} − 2 + s_{E,N}) = \frac{7N−6}{12N} \overline{sl}(N); cf. the proof of Proposition 31.

7 Appendix: The Kähler different divisor

In this appendix we gather together some basic facts concerning the Kähler different divisor which were used above but which are difficult to find explicitly in the literature.

If f : X → Y is any morphism of finite type, then the Kähler different ideal sheaf D_{f} ⊂ O_{X} is defined as the 0-th Fitting ideal sheaf D_{f} = F_{0}(Ω_{X/Y}^{1} ) of the sheaf Ω_{X/Y}^{1} of relative differentials; cf. Kunz[Ku], p. 159 or [LK], p. 102. If D_{f} is invertible, then there is a unique effective Cartier divisor Diff(f) such that L(Diff(f)) = D_{f}^{−1}, which we call the Kähler different divisor. We observe:

Remark 32 (a) If the Kähler different divisor Diff(f) exists, then by [Ku], Theorem 10.7, its support is the ramification locus of f:

(42) supp(Diff(f)) = Ram(f) := supp(Ω_{X/Y}^{1} ).

(b) If g : Ỹ → Y is any morphism of schemes such that g_{X}(X̃′) ⊈ Ram(f), for all irreducible components X̃′ of X̃ := X ×_{Y} Ỹ, then the Kähler different divisor of f̃ = fỸ : X̃ → Ỹ also exists and is given by

Diff(f̃) = g_{X}^{∗}Diff(f). (43)

[Indeed, the hypothesis on g_{X} implies in view of (42) that the pullback Cartier divisor g_{X}^{∗}Diff(f) is defined and corresponds to the invertible ideal sheaf g_{X}^{∗}F_{0}(Ω_{X/Y}^{1} ) = F_{0}(Ω_{X/Y}^{1} ) · OX̃. Since F_{0}(Ω_{X/Y}^{1} ) · OX̃ = F_{0}(Ω_{X̃/Ỹ}^{1} ) = Df̃ by [Ku], Rule 10.3a), we see

that Df̃ is invertible, and so Diff(f̃) exists and is given by g_{X}^{∗}Diff(f).]

In general, Diff(f) need not exist, but we do have the following result which suffices for our purposes (and which generalizes the discussion of [I], p. 202):

Proposition 33 If f : X → Y is a generically etale morphism between smooth varieties over a field K, then the Kähler different ideal sheaf D_{f} is invertible, and hence the Kähler different divisor Diff(f) ∈ Div(X) exists. Moreover, we have the Riemann-Hurwitz relation

ω_{X/K} ≃ f^{∗}ω_{Y/K} ⊗ L(Diff(f)), (44)

where ω_{X/K} and ω_{Y/K} denote the canonical sheaves of X and Y.

Proof. Since f is generically etale, we see that Ω_{X/Y}^{1} is a torsion O_{X}-module (use [Ku], Corollary 10.5). Moreover, since Ω_{Y/K}^{1} is locally free and f is generically etale, we have the exact sequence

0 → f^{∗}Ω_{Y/K}^{1} →^{u} Ω_{X/K}^{1} → Ω_{X/Y}^{1} → 0. (45)

(Indeed, to see that u is injective, note that the local freeness of Ω_{Y/K}^{1} implies that it is naturally a subsheaf of the sheaf M_{Y} (Ω_{Y/K}^{1}) = M_{Y} ⊗_{O_{Y}} Ω_{Y/K}^{1} of meromorphic differential forms, and that hence we have f^{∗}Ω_{Y/K}^{1} ⊂ f^{∗}M_{Y} (Ω_{Y/K}^{1}) →^{∼} M_{X}(Ω_{X/K}^{1}). Thus, the map f^{∗}Ω_{Y/K}^{1} → Ω_{X/K}^{1} is induced by the map f^{∗}M_{Y} (Ω_{Y/K}^{1})→M_{X}(Ω_{X/K}^{1}), which is an isomorphism since f is generically etale.)

Since u : f^{∗}Ω_{Y/K}^{1} → Ω_{X/K}^{1} is an injective map of locally free sheaves of rank n, the map ∧(u) : f^{∗}ω_{Y/K} ≃ ∧^{n}f^{∗}Ω_{Y/K}^{1} → ω_{X/K} = ∧^{n}Ω^{1}X/K is also injective, and so f^{∗}ω_{Y/K} ≃ Im(u) = D_{f}ω_{X/K}, where the latter equality follows from the definition of the Fitting ideal (and that of ∧(u)). Thus D_{f} is invertible and we have (44).

Corollary 34 Let S be an arbitrary scheme, and let X/S and Y/S be two smooth relative varieties of dimension n over S, i.e. X/S and Y/S are smooth of finite presentation over S with geometrically integral fibres of dimension n. If f : X → Y is any S-morphism such that f_{s} : X_{s} → Y_{s} is generically etale, for all s ∈ S, then D_{f} is invertible and the Kähler different divisor Diff(f) ∈ Div(X/S) is a relative Cartier divisor of X/S which satisfies the Riemann-Hurwitz relation, i.e. ω_{X/S} ≃ f^{∗}ω_{Y/S} ⊗ L(Diff(f)). Furthermore, the formation of Diff(f) commutes with arbitrary base-change in the sense that for any g : S′ → S we have Diff(f_{(S′)}) = g_{X}^{∗}(Diff(f)).

Proof. We first show that the sequence

0 → f^{∗}Ω_{Y/S}^{1} →^{u} Ω_{X/S}^{1} → Ω_{X/Y}^{1} → 0. (46)

is exact. As usual, this sequence is right-exact. To show that u is injective, we follow the method of [LK], p. 106. For this we first observe that Ω_{X/S}^{1} is S-flat because Ω_{X/S}^{1} is a locally free O_{X}-module (of rank n) and X/S is flat. Moreover, by the proof of Proposition 33 we know that u_{s} = u ⊗ κ(s) : f_{s}^{∗}Ω_{Y_{s}/κ(s)}^{1} → Ω_{X_{s}/κ(s)}^{1} is injective, for all s ∈ S. Thus, by Corollary A.2 of [LK], it follows that u is (universally) injective. (Note that this corollary is applicable because we may reduce to case that S is noetherian by using [EGA] (IV,8.9.1) since X/S and Y/S are of finite presentation.)

From the exact sequence (46) we conclude as before that f^{∗}ω_{Y/S} ≃ Im(u) = D_{f}ω_{X/S}, and that hence D_{f} is invertible and that the Riemann-Hurwitz relation holds. Furthermore, D_{f} is then S-flat, and so Diff(f) ∈ Div(X/S).

Finally, the formation of Diff(f) commutes with base-change because the hypothesis of Remark 32(b) is satisfied by [EGA], (IV, 21.15.9).

Corollary 35 Let Y/X and Z/X be two regular relative curves over a smooth base curve X/K, where K is a perfect field, and let f : Y → Z be a generically etale X-morphism. In addition, let D denote the closure in Y of the different divisor Diff(f_{F}) ∈ Div(Y_{F}) of the generic cover f_{F} : Y_{F} → Z_{F}, where F = κ(X). Then there is an effective Cartier divisor B = B(f) ≥ 0 consisting entirely of components of fibres of Y/X with following properties:

(a) The divisor Diff(f) := D + B ∈ Div(Y) is the different divisor of f and hence satisfies (42). Moreover, we have the relative Riemann-Hurwitz relation:

ω_{Y/X}^{0} ≃ f^{∗}ω_{Z/X}^{0} ⊗ L(Diff(f)). (47)

(b) Suppose that the fibre Y_{x} of Y at x ∈ X is reduced. Then a component Γ of Y_{x} does not appear in B if and only if the induced map f_{|Γ} : Γ → f(Γ) ⊂ Z_{x} is finite and separable. If this condition holds for all components of Y_{x}, then D_{x} = D_{|Y_{x}} is the different divisor of f_{x} : Y_{x} → Z_{x}, i.e. D_{x} = Diff(f_{x}).

(c) If X is complete, then for any a fibre component Γ of Y/X we have

(B.Γ) = 2p_{a}(Γ) − 2 − Γ^{2} − (D.Γ) − (ω_{Z/X}^{0}.f_{∗}Γ). (48)

Proof. By hypothesis, Y and Z are smooth irreducible surfaces over K, and so Diff(f) exists by Proposition 33. We have Diff(f)_{|Y_{F}} = Diff(f_{F}) (apply Remark 32(b) with Ỹ = Y_{F}), and so Diff(f) = D + B, where B consists entirely of fibre components. This proves the existence of B.

(a) The first assertion is clear from the construction. Now if we multiply both sides of (44) by p_{Y}^{∗} ω_{X/K}^{−1} = f^{∗}p_{Z}^{∗}ω_{X/K}^{−1}, where p_{Y} : Y → X and p_{Z} : Z → X denote the structure maps, then we obtain (47) because ω_{Y/X}^{0} = ω_{Y/K} ⊗ p_{Y}^{∗} ω_{X/K}^{−1} and f^{∗}ω_{Z/X}^{0} = f^{∗}(ω_{Z/K} ⊗ p_{Z}^{∗}ω_{X/K}^{−1}) = f^{∗}ω_{Z/K} ⊗ p_{Y}^{∗} ω_{X/K}^{−1}.

(b) Since Y_{x} is reduced, we have 𝔪_{Y,Γ} = 𝔪_{X,x}O_{Y,Γ} = 𝔪_{Z,f(Γ)}O_{Y,Γ}, and so Γ ⊂ Ram(f) ⇔ κ(Γ)/κ(f(Γ)) is a finite separable field extension; cf. [BLR], 2.2/2. In view of (42), this proves the first assertion.

To prove the second, let g : Z_{x} ↪ Z denote the canonical closed immersion. If no components Γ of appears in B, then its base-change g_{Y} : Y_{x} → Y satisfies the hypothesis of Remark 32(b) 33, and so the assertion follows.

(c) Since Γ is a component of a fibre of Y/X, we have (ω_{Y/X}^{0}.Γ) = (ω_{Y/K}.Γ) = 2p_{a}(Γ)−2−Γ^{2}, the latter by the adjunction formula. Now by (47) we have (ω_{Y/X}^{0}.Γ) = ((D +B).Γ)+(f^{∗}ω_{Z/X}^{0}.Γ) = ((D +B).Γ)+(ω_{Z/X}^{0}.f_{∗}Γ) by the projection formula, and so (48) follows.

References

[Ar] S. Ju. Arakelov, Families of algebraic curves with fixed degeneracies. Izv. Akad. Nauk SSSR 35 (1971) = Math. USSR Izv. 5 (1971), 1277–1302.

[Art] M. Artin, Lipman’s proof of resolution of singularities of surfaces. In: Arithmetic Geometry (G. Cornell, J. Silverman), Springer-Verlag, New York, 1986, pp. 267–287.

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

[BPV] W. Barth, C. Peters, A. Van de Ven, Compact Complex Surfaces. Springer-Verlag, Berlin, 1984.

[DM] P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus. Publ. Math. I.H.E.S. 36 (1969), 75–109.

[DR] P. Deligne, M. Rapoport, Les schémas de modules de courbes elliptiques. In: Modular functions of one variable, II, Lecture Notes in Math. 349, Springer-Verlag, Berlin, 1973, pp. 143–316.

[Di] R. Dijkgraaf, Mirror symmetry and elliptic curves. In: The Moduli of Curves (R. Dijkgraaf, C. Faber, G. van der Geer, eds), Birkhäuser, Basel, 1995, pp. 149–163.

[EGA] A. Grothendieck, J. Dieudonné, Eléments de Géométrie Algébrique I-IV. Ch. I: Springer-Verlag, Berlin, 1971; ch. II-IV: Publ. Math. I.H.E.S. 8, 11, 17, 20, 24, 28, 32 (1961-68).

[Fa] G. Faltings, Arakelov’s theorem for abelian varieties. Invent. math. 73 (1983), 337–348.

[FK] G. Frey, E. Kani, Curves of genus 2 covering elliptic curves and an arithmetical application. In: Arithmetic Algebraic Geometry (G. van der Geer, F. Oort, J. Steenbrink, eds.), Progress In Math. vol. 89, Birkhäuser, Boston, 1991, pp. 153-176.

[Fu1] W. Fulton, Hurwitz schemes and irreducibility of moduli of algebraic curves. Ann. Math. (2) 90 (1969), 542–575.

[Fu2] W. Fulton, Intersection Theory. Springer-Verlag, 1984.

[Gr1] A. Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique IV: Les schémas de Hilbert. Séminaire Bourbaki No. 221 (1960/61).

[Gr2] A. Grothendieck, Modeles de Néron et Monodromie. SGA 7-I, Exp. IX, Springer Lecture Notes 288 (1972), 313–523.

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

[I] S. Iitaka, Algebraic Geometry. Springer-Verlag, New York, 1982.

[KZ] M. Kaneko, D. Zagier, A generalized Jacobi theta function and quasi-modular forms. In: The Moduli of Curves (R. Dijkgraaf, C. Faber, G. van der Geer, eds), Birkhäuser, Basel, 1995, pp. 165–172.

[Ka1] E. Kani, The number of curves of genus 2 with elliptic differentials. J. reine angew. Math. 485 (1997), 93-121.

[Ka2] E. Kani, The existence of curves of genus 2 with elliptic differentials. J. Number Theory 64 (1997), 130-161.

[Ka3] E. Kani, Hurwitz spaces of genus 2 covers of an elliptic curve. Collect. Math. 54 (2003), 1-51.

[Kl] S. Kleiman, Relative duality for quasi-coherent sheaves. Comp. Math. 41 (1980), 39–60.

[Ku] E. Kunz, Kähler Differentials. Vieweg, Wiesbaden, 1986.

[La] S. Lang, Introduction to Modular Forms, Springer-Verlag, Berlin, 1976.

[Li] S. Lichtenbaum, Minimal models over discrete valuation rings. Am. J. Math. 90 (1968), 380–405.

[LK] K. Lønsted, S. Kleiman, Basics on families of hyperelliptic curves. Comp. Math. 38 (1979), 83–111.

[Ma] B. Mazur, Letter to the author, 9 December 1989.

[Mu] D. Mumford, Towards an enumerative geometry of the moduli space of curves. In: Arithmetic and Geometry - Papers dedicated to I.R. Shafarevich (M. Artin, J. Tate, eds.), Birkhäuser, Boston, 1983, pp. 271–328.

[Ogg] A. Ogg, On pencils of curves of genus 2. Topology 5 (1966), 355-362.

[Pa1] A. Parshin, Algebraic curves over function fields I. Izv. Akad. Nauk SSSR 32 (1968) = Math. USSR Izv. 2 (1968), 1145–1170.

[Pa2] A. Parshin, Minimal models of curves of genus 2. Izv. Akad. Nauk SSSR 36 (1972) = Math. USSR Izv. 6 (1972), 65–108.

[Sa] T. Saito, The discriminants of curves of genus 2. Comp. Math. 69 (1989), 229-240.

[Se] J.-P. Serre, Groupes algébriques et corps de classes. Hermann, Paris, 1959.

[Sz] L. Szpiro, Proprietés numériques du faiseaux dualisant relatif. In: Séminaire sur les pinceaux des courbes des genre aux moins deux (L. Szpiro, ed.), Astérisque 86 (1981), 44–78.

[Ta] J. Tate, Algorithm for determining the type of a singular fibre of an elliptic pencil. In: Modular Functions of One Variable IV, Springer Lecture Notes 476 (1975), 33-52.

[U] K. Ueno, Discriminants of curves of genus 2 and arithmetic surfaces. In: Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata (H. Hijikata et al, eds.), Academic Press, Tokyo, 1988, pp. 749-770.

[VZ] E. Viehweg, K. Zuo, A characterization of certain Shimura curves in the moduli stack of abelian varieties. Preprint, 2002.

[We] A. Weil, Zum Beweis des Torellischen Satzes. Göttinger Nachr. 1957, No. 2, 33-53 = Œuvres Sci./Collected Papers II, Springer-Verlag, New York, 1979, pp. 307–327.

[Wev] S. Wevers, Deformation of tame admissible covers of curves. In: Aspects of Galois Theory (H. Völklein et al, eds.), LMS Lecture Note Series No. 256 (1999), Cambridge U Press, pp. 239–282.

Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada. E-mail address: kani@mast.queensu.ca