Curves of genus 2 with elliptic differentials and associated Hurwitz spaces

Gerhard Frey & Ernst Kani ^{∗}

August 30, 2008

1 Motivation

In the whole paper K is a field of finite type over its prime field K_{0} of characteristic p ≥ 0 and not equal to 2. As typical case we can take K as number field or as function field of one variable over a finite field. With K_{s} we denote its separable closure. With n we denote an odd number larger than 5 and prime to p.

We begin by stating three motivations.

1.1 Rational Fundamental Groups

Let C be a smooth projective geometrically irreducible curve over K with function field F (C).

We are interested in unramified Galois extension U(C) of F (C) with the additional property that K is algebraically closed in U(C).

The finite quotients of the Galois group G(U(C)/F (C)) correspond to covers

D →^{f} C

where D is a smooth projective absolutely irreducible curve defined over K and f is Galois (and hence étale).

∗In this paper we shall discuss results presented at the Conference on Arithmetic, Geometry, Cryptography and Coding Theory (AGCT 11). We would like to thank the organizers for the opportunity to participate in this very interesting and inspiring conference.

Base Points Take P ∈ C(K) and consider a connected Galois cover D →^{f} C in which P splits completely. Then D is absolutely irreducible. Moreover, normalizations of fibre products of such covers (for P fixed) satisfy the same conditions, and so there is a maximal unramified extension of F (C) in which P is totally split, or equivalently, an absolutely irreducible étale (pro-)cover of C over K in which the fibre over P consists of K-rational points, which is Galois with group Π_{K}(C, P), the K-rational fundamental group of C with base point P.

1.1.1 Questions

Let g be a non-negative integer.

Q1: Are there finitely generated fields K, curves C of genus g defined over K, prime numbers ℓ and points P ∈ C(K) such that Π_{K}(C, P) has an ℓ-adic subrepresentation with infinite image?

Q2: Are there finitely generated fields K, curves C of genus g defined over K and points P ∈ C(K) such that Π_{K}(C, P) is infinite?

Q3: Are there finitely generated fields K and curves C of genus g defined over K such that there is a finite extension K_{1}/K and an infinite tower of projective absolutely irreducible Galois covers of C defined over K_{1}?

Q4: For given n ∈ N and fixed K and g are there curves C of genus g defined over K such that there are projective absolutely irreducible Galois covers of C of degree ≥ n?

Known Results

1. For g = 0 there are no unramified extensions of C and so the answer to all questions is negative.

2. The same result is true for g = 1, as was explained in [FKV], §2. However, in this case the negative answer to Q4 (which implies negative answers to all others) relies on the deep result of Merel on the uniform bounds of torsion points of elliptic curves over fixed number fields.

3. In [FK1] it is shown that there are examples for positive answers for Question Q1. In all these examples K has positive characteristic and the genus of C is larger than 2.^{1}

4. If char(K) ≡ 3 mod 4 and K contains a fourth root of unity then for all g ≥ 3 there is an explicitly given subvariety of positive dimension in the moduli space of curves of curves g and curves C such that Q1 is true for curves with moduli point in this set ([FK1]).

5. For any field K containing the fourth roots of unity and for all g ≥ 3 there is an explicitly given subvariety of positive dimension in the moduli space of curves of curves g and curves C such that Q2 is true for curves with moduli point in this set ([FKV]).

6. There are examples (see [I] and [Ki])for curves of genus 2 over finite fields for which Q1 is true.

We want to ask in this paper whether there is, as in the case of curves of higher genus, a subspace of positive dimension in the moduli space of curves of genus 2 for which some of the questions can be answered positively. As we will see we can get partial answers by restricting us to curves of genus 2 with elliptic differentials, i.e. with Jacobians isogenous to a product of elliptic curves. In particular, Q4 has a positive answer (Proposition 6.10) for any K, and Q3 is true in positive characteristic (Corollary 7.15), in both cases for non-constant curves.

Acknowledgement. The authors would like to thank Helmut Völklein very much for numerous inspiring discussions and his great help to clarify the interplay between group theory and geometry. In addition, the second author gratefully acknowledges receipt of funding from the Natural Sciences and Engineering Research Council of Canada (NSERC).

1.2 Isomorphic Galois Representations

One of the most efficient tools for studying the absolute Galois group G_{K} of K is the investigation of representations induced by the action on torsion groups of abelian varieties, or more generally, on cohomology groups attached to varieties. In fact the results mentioned in 1.1.1 are obtained in this way.

1We remark that if we replace C by a number field the Conjecture of Fontaine-Mazur implies the negative answer to Q1.

Conversely, it is natural to ask how much information about these representations is needed to characterize the isogeny class of abelian varieties. By celebrated results of Tate (for finite fields), Faltings (for number fields) and Zarhin (for fields of positive characteristic) we know that ℓ-adic representations attached to Tate modules as well as infinitely many representations on torsion points of different prime orders are enough. For a discussion and some generalizations we refer to [FJ].

This question is closely related to arithmetic properties of abelian varieties. A discussion in a rather general frame can be found in [Fr3].

In this paper we restrict ourselves to the case that we look at pairs of elliptic curves E and E′ defined over K.

The representations of G_{K} induced on the points E[n] respectively E[n′] of order dividing n of E respectively E′ are denoted by ρ_{E/K,n} respectively ρ_{E′/K,n}.

Conjecture 1 (Darmon[Da]) There is a number n_{0}(K) such that for all elliptic curves E, E′ over K and all n ≥ n_{0}(K) we get:

If ρ_{E/K,n} ≅ ρ_{E′/K,n} then E is isogenous to E′.

A variant is

Conjecture 2 (Darmon/Kani) There is a number n_{0} such that for every n ≥ n_{0} and every number field K there are, up to simultaneous twists, only finitely many pairs (E, E′) of elliptic curves over K which are not isogenous yet whose mod n Galois representations are isomorphic: ρ_{E/K,n} ≅ ρ_{E′/K,n}. Moreover, for prime numbers n the bound n_{0} can be chosen to be 23.

Much weaker is

Conjecture 3 (Frey) We fix an elliptic curve E_{0}/K. Let S(E_{0}) denote the set of primes ℓ for which there exists a K-rational cyclic isogeny of degree ℓ of E_{0}.

There is a number n_{0}(E, K) such that for all elliptic curves E over K and all n ≥ n_{0}(E, K) and prime to elements in S(E_{0}) we get:

E is isogenous to E_{0} iff ρ_{E/K,n} ≅ ρ_{E_{0}/K,n}.

There is a close relation of these conjectures with other conjectures in diophantine geometry. We shall give some diophantine motivation for Conjecture 2 in Subsection 6.4. There it is shown that the conjecture predicts properties of rational points on explicitly given surfaces of general type, and Lang’s conjecture together with a conjectural modular description of curves of genus ≤ 1 (cf. Conjecture 4) on these surfaces would imply Conjecture 2.

Conjecture 3 is true if the height conjecture for elliptic curves holds; cf. [Fr2]. In particular, it is true in the case that K is a function field over a finite field.

For K = ℚ the height conjecture is equivalent with the ABC-conjecture and with the degree conjecture for modular parameterizations of elliptic curves ([Fr2]). Moreover, by switching from n to 2n, we get an equivalence of Conjecture 3 for even numbers with the asymptotic Fermat conjecture.

1.3 A “Special” Hurwitz Space

Our third motivation comes from group theory. By Hurwitz spaces one parameterizes covers of a base curve with given monodromy group and given ramification type. Over ℂ as ground field and (mainly) ℙ^{1} as base curve this is classical theory due to Hurwitz[Hu]. By the work of Fulton, Fried–Völklein and many others, the theory of Hurwitz spaces is extended to a powerful tool in Arithmetic Geometry.

We want to study covers

φ : ℙ_{K}^{1} → ℙ_{K}^{1}

of degree n which are primitive (i.e. have no proper intermediate subcovers) and which are ramified at 5 points with a certain ramification structure (as described below). In order to classify such covers, we shall use the following terminology.

Definition 1.1 Let f_{i} : C_{i} → C be two covers of the curve C.

(a) f_{1} is isomorphic to f_{2} (f_{1} ≅ f_{2}) if there is an isomorphism α : C_{1} → C_{2} with f_{1} = f_{2} ∘ α.

(b) f_{1} is equivalent to f_{2} (f_{1} ∼ f_{2})if there is an isomorphim α_{1} : C_{1} → C_{2} and an automorphism α_{2} : C → C with α_{2} ∘ f_{1} = f_{2} ∘ α_{1}.

Thus, the set of equivalence classes of covers φ : ℙ_{K}^{1} → ℙ_{K}^{1} as above has two points which can move “freely” and hence the associated Hurwitz space is a surface.

Let us now assume in addition that the (different) points P_{1}, …, P_{5} ∈ ℙ^{1}(K_{s}) ramify with ramification order at most 2. We want to make sure that the monodromy group of φ is S_{n} and so we assume that P_{5} has as ramification cycle a transposition. Together with the condition that φ is primitive this implies by group theory ([Wi] or [Hup]) that the Galois closure of φ is a regular cover with Galois group S_{n}; cf. [Fr1] or Corollary 3.7 below for more detail. Note that this condition clearly implies that P_{5} has exactly one ramified extension in the cover.

For 1 ≤ i ≤ 5 let r_{i} denote the number of unramified extensions of P_{i} after base change to K_{s}. By assumption r_{5} = n − 2. The Hurwitz genus formula for φ yields:

−2 = −2n + 1 + ∑_{1≤i≤4} \frac{(n − r_{i})}{2} = 1 − ∑_{1≤i≤4} \frac{r_{i}}{2} .

Thus ∑ r_{i} = 6, and hence, since n is odd, r_{i} is odd and so there are three points, say P_{1}, P_{2}, P_{3}, with r_{i} = 1 and one point P_{4} with r_{4} = 3. Since this numbering convention will be in force throughout the paper, we formulate it here for easy reference

Convention 1.2 The points P_{1}, …, P_{4} have been numbered in such a way that r_{1} = r_{2} = r_{3} = 1 and r_{4} = 3. Thus, there is a unique unramified extension above P_{i} for i = 1, 2, 3, and there are precisely 3 unramified extensions above P_{4}.

Since φ : ℙ_{K}^{1} → ℙ_{K}^{1} is defined over K, the discriminant divisor disc(φ) = P_{1} + … + P_{5} is K-rational. Since n − 2 > 3 the points P_{5} and P_{4} have to be K-rational and P_{1}, P_{2}, P_{3} are permuted under the Galois action.

Summarizing, we have:

Proposition 1.3 Let φ be a primitive K-cover of degree n of the projective line by itself ramified in the following way.

(*) φ is ramified in five points P_{1}, …, P_{5} in ℙ^{1}(K_{s}) with ramification orders ≤ 2 and the ramification cycle of P_{5} in the monodromy group is a transposition.

Then the Galois closure \overline{φ} : \overline{C} → ℙ_{K}^{1} of φ has Galois group S_{n} and is regular over K. The ramification structure induces five involutions σ_{i} ∈ S_{n} which generate S_{n} with ∏ σ_{i} = 1, where σ_{5} is a transposition, σ_{4} is a product of \frac{n−3}{2} transpositions and each of σ_{i}, i = 1, 2, 3 is a product of \frac{n−1}{2} transpositions (after a suitable numeration).

Thus, the ramification structure of φ (or, more precisely, of \overline{φ}) is described by the tuple

C = (cl(σ_{1}), …, cl(σ_{5})) = ((2)^{\frac{n−1}{2}} , (2)^{\frac{n−1}{2}} , (2)^{\frac{n−1}{2}} , (2)^{\frac{n−3}{2}} , (2)),

where cl(σ_{i}) denotes the conjugacy class of σ_{i} in S_{n} (which is determined by its cycle decomposition), and the symbol (2)^{k} means that cycle decomposition of σ_{i} consists of k transpositions.

Conversely we have the following result from Galois theory.

Proposition 1.4 Suppose \overline{φ} : \overline{C} → ℙ^{1} is Galois cover with Galois group G ≃ S_{n} which has ramification structure given by C as above. Then \overline{φ} is the Galois closure of a (primitive) cover φ : ℙ^{1} = H_{x}\\overline{C} → ℙ^{1} of degree n which satisfies condition (∗). Here H_{x} ≃ S_{n−1} is the stabilizer subgroup of an element x ∈ {1, …, n}.

We can express the above in terms of the associated Hurwitz spaces as follows. Let P_{n}^{∗}(K) denote the set of isomorphism classes of primitive covers φ : ℙ_{K}^{1} → ℙ_{K}^{1} of degree n which satisfy (∗), and let H^{ab}(S_{n}, C)(K) denote the set of isomorphism classes of (regular) Galois covers \overline{φ} : \overline{C} → ℙ_{K}^{1} with group S_{n} and ramification structure C as above. Then the above Propositions 1.3 and 1.4 show that the map φ ↦ \overline{φ} induces a bijection P_{n}^{∗}(K) →^{∼} H^{ab}(S_{n}, C)(K). Now since S_{n} has no outer automorphisms (as n ≠ 6), it follows that H^{ab}(S_{n}, C)(K) can be identified with the set of Krational points of the Hurwitz space H^{in}(S_{n}, C) as defined in [V1], ch. 10.

Since S_{n} has trivial centre, the Hurwitz space H^{in}(S_{n}, C) finely represents the associated Hurwitz functor H_{ℙ^{1}}(S_{n}, C) which classifies covers φ of type C; cf. Wewers[We], p. 66, 70. However, this moduli space (which has dimension 5) is not quite the space we want because we want to study equivalence classes rather than isomorphism classes of covers. Thus, we want to consider instead the quotient space

H_{n}^{∗} := H^{in}(S_{n}, C)/Aut(ℙ^{1})

(which has dimension 2). It can be shown (cf. Proposition 4.8 below) that H_{n}^{∗} coarsely represents the functor H_{n}^{∗} which classifies equivalence classes of such covers.

Remark 1.5 (a) Since H_{n}^{∗} is only a coarse moduli space for the functor H_{n}^{∗} (which is the quotient functor H_{ℙ^{1}}(S_{n}, C)/Aut(ℙ^{1})), it is more difficult to characterize the K-rational points of H_{n}^{∗}. It is clear, however, that if φ : ℙ_{K}^{1} → ℙ_{K}^{1} is a K-cover of type (*), then φ defines a K-rational point on H^{in}(S_{n}, C) and hence also on H_{n}^{∗}. We thus have a natural map H_{n}^{∗}(K) → H_{n}^{∗}(K) (which is a bijection if K is algebraically closed).

(b) Since H^{in}(S_{n}, C) is finite and étale over the smooth affine space U_{5} ⊂ ℙ^{5} (cf. [We], p. 7), it follows that is also smooth and affine. Thus, H_{n}^{∗} is normal and affine. This argument, however, does not guarantee that H^{in}(S_{n}, C) and H_{n}^{∗} are connected; this will be established later by another method (cf. Theorem 4.9).

(c) The system (σ_{1}, ⋯ σ_{5}) is not rigid (in the sense of [V1], p. 38). Of course it is not difficult to see this by group theory but it is more difficult to compute the number of non-conjugate isomorphism classes. We shall get an answer by using geometrical arguments in Theorem 3.11.

We shall see in the next sections that points on H_{n}^{∗} carry much more arithmetical information. In fact, we shall associate to such points covers of elliptic curves by curves of genus 2, and so one is naturally led to a Hurwitz moduli functor (in the spirit of Fulton [Fu]). A detailed discussion is given in [Ka5].

By applying this we can approach some of the problems mentioned in Remark 1.5 and see that there is a close connection between Subsection 1.2 and Subsection 1.3.

2 Genus 2 Covers of Elliptic Curves

2.1 Normalized Covers

Let C be a curve of genus 2 over K with Weierstraß points W_{C} = {W_{1}, …, W_{6}} and Weierstraß divisor W_{C} = W_{1} + … + W_{6}. (Note that the divisor W_{C} is K-rational, whereas the individual points W_{i} need not be.) We assume that C has an elliptic differential over K. This means that there is an elliptic curve E over K and a non-constant K-morphism

f : C → E.

In addition we assume that f has odd^{2} degree n.

Definition 2.1 A cover

f : C → E

(of the above type) is called normalized ([Ka5]) if

(a) f is minimal, i.e. there is no non-trivial isogeny η : E′ → E and a cover f′ : C → E′ with f = η ∘ f′.

(b) The norm (direct image) of W_{C} has the form

f_{∗}W_{C} = 3 · 0_{E} + P′_{1} + P′_{2} + P′_{3}. (1)

where P′_{i} are the non-trivial points of order 2 of E.

Note that every minimal cover f_{0} : C → E is equivalent to a normalized cover; more precisely, we have (cf. [Ka5], Proposition 2.2):

Lemma 2.2 Assume that f_{0} : C → E is a minimal cover of an elliptic curve E by a curve of genus 2 of odd degree n defined over K. Then there is a unique translation τ : E → E such that f = τ ∘ f_{0} is normalized.

For our purposes it is important to note that a normalized cover also satisfies the condition f ∘ ω_{C} = [−1]_{E} ∘ f, (2)

where ω_{C} denotes the hyperelliptic involution of C; cf. [Ka5], Corollary 2.3. It follows that f factors over the hyperelliptic cover π′ : C → C/⟨ω⟩ = ℙ^{1}.

By the Riemann-Hurwitz genus formula we see that the different of f has to be a divisor of degree 2.

2.2 Associated ℙ^{1}-Covers

We return to covers φ : ℙ^{1} → ℙ^{1} of type (*) and discuss their relation to curves of genus 2 with elliptic differentials. Indeed, it is this connection which motivated the consideration of this specific group theoretical situation (*); cf. [Fr1].

2In fact, the case that n is even is also important (see Sections 3 and 6.4), and is analyzed in [Ka5] and [Di].

To explain this connection, note first that the points P_{1}, ⋯, P_{4} determine an elliptic curve E/K which is unique up to a quadratic twist. This curve is constructed as follows. Choose a coordinate x of ℙ_{K}^{1} such that (x)_{∞} = P_{4}, and a cubic polynomial f_{3}(x) ∈ K[x] with (f_{3})_{0} = P_{1} + P_{2} + P_{3}. Then the (Weierstraß) equation

y^{2} = f_{3}(x)

defines an elliptic curve E/K and a double cover π : E → ℙ^{1} which is ramified at P_{1}, …, P_{4} and which maps the zero-point 0_{E} ∈ E(K) to P_{4}. Thus, the group of 2-torsion points of E is E[2] = {0_{E}, P′_{1}, P′_{2}, P′_{3}}, where P′_{i} ∈ π^{−1}(P_{i}) is the (unique) point above P_{i}. Note that P′_{i} is K-rational if and only if P_{i} is K-rational.

Let C_{0} = E ×_{ℙ^{1}} ℙ^{1} be the fibre product of E and ℙ^{1} over ℙ^{1} (with respect to the morphisms π and φ). It is immediate that C_{0} is an irreducible curve with function field F = K(X, Y), with X and Y satisfying a hyperelliptic equation of the form

Y^{2} = f_{3}(X) · g_{3}(X)

where g_{3} is a polynomial of degree 3 corresponding to the 3 unramified extensions of P_{4} in the cover φ (cf. Convention 1.2). Thus, the normalization C = C̃_{0} of C_{0} is a curve of genus 2 with morphisms f : C → E and π′ : C → ℙ^{1}, and f_{∗}(W_{C}) = 3 · P′_{4} + P′_{1} + P′_{2} + P′_{3}.

We get the diagram of morphisms

C

↙^{π′}

ℙ^{1}  ↓ ^{f}
φ ↓  E

↙^{π}

ℙ^{1}

The point P_{5} has two distinct extensions P and P′ = [−1]_{E}P to E, and there is exactly one point Q resp. Q′ = ωQ over P resp. P′ which is ramified of order 2. Hence the discriminant divisor of f is equal to π^{∗}(P_{5}) and the different divisor of f is equal to Q + Q′.

If we assume that φ is primitive, then it follows that f is also primitive (or minimal). Thus, we have Proposition 2.3 Let φ : ℙ_{K}^{1} → ℙ_{K}^{1} be a primitive cover of odd degree n satisfying (∗) and Convention 1.2 with respect to the points P_{1}, …, P_{5}, and let E/K be an elliptic curve with a morphism π : E → ℙ^{1} of degree 2 which is ramified at P_{1}, …, P_{4} such that π(0_{E}) = P_{4}. Then the normalization C of the fibre product ℙ^{1} ×_{ℙ^{1}} E is a curve of genus 2 defined over K and the morphism

f = φ_{(π)} : C = (ℙ^{1} ×_{ℙ^{1}} E)^{∼} → E

is normalized and has discriminant Disc(f) = π^{∗}(P_{5}).

Conversely, assume that f_{0} : C → E is a minimal cover of an elliptic curve E by a curve of genus 2 of odd degree n defined over K. After an appropriate translation (Lemma 2.2) we get a normalized cover

f : C → E.

Since f satisfies (2), we can pass to the respective quotients and obtain a primitive cover

φ : C/⟨ω⟩ = ℙ^{1} → E/⟨−id_{E}⟩ = ℙ^{1}

of degree n such that φ ∘ π′ = π ∘ f.

The Weierstraß points of C are mapped onto E[2] and hence to the ramification points of π : E → ℙ^{1}. By our hypothesis, three of the Weierstraß points lie over the zero-point of E whose image in ℙ^{1}(K) is denoted by P_{4}. The three others are mapped one-to-one to the points of order 2 of E. Denote by P_{1}, ..., P_{3} the images of these points. The discriminant divisor of f is the conorm (or pullback) of a divisor of ℙ^{1} with support equal to one point P_{5} ∈ ℙ^{1}(K). It follows that φ is unramified outside of {P_{1}, ..., P_{5}}. Summarizing, we have

Proposition 2.4 Let f_{0} : C → E be a minimal genus 2 cover of odd degree n of an elliptic curve E/K. Then there is a unique translation τ : E → E such that f = τ ∘ f_{0} is normalized. Moreover, π ∘ f factors over the hyperelliptic cover π′ : C → C/⟨ω⟩ = ℙ^{1}, i.e. there is a unique cover φ : ℙ^{1} → ℙ^{1} such that φ ∘ π′ = π ∘ f. (In particular, φ is primitive and has degree n.) Let P_{1}, …, P_{3} denote the images of the non-zero 2-torsion points of E under π and put P_{4} = π(0_{E}). In addition, let P_{5} ∈ ℙ^{1}(K) be the unique point such that Disc(f) = π^{∗}(P_{5}). Then φ : ℙ^{1} → ℙ^{1} satisfies (∗) and Convention 1.2 if and only if Disc(f) is reduced, i.e. if and only P_{5} ∉ {P_{1}, …, P_{4}}.

Proof. Everything except the last statement has been explained above. Since Disc(f) is a divisor of degree 2 and invariant under −id_{E} it is of the form P +(−id_{E}(P)) and hence it is reduced iff P_{5} is unramified, i.e. not contained in {P_{1}, …, P_{4}}.

Note that the above construction guarantees that P_{1}, …, P_{4} are distinct points, but not necessarily P_{1}, …, P_{5}. Assume now that P_{5} is different from P_{i}, i = 1, ..., 4 (the “generic” case, see Subsection 3.2). Then φ defines a point in the Hurwitz space H_{n}^{∗}.

The previous propositions suggest that the moduli space H_{n}^{∗} is closely related to another moduli space H_{n}, which represents the moduli problem H_{n} classifying isomorphism classes of pairs (C, f) where C is a curve of genus 2 and f is a normalized covering map from C to an elliptic curve E of degree n. More precisely, let us look at the subproblem H′_{n} of H_{n} which classifies the covers for which Disc(f) is reduced. Then we shall see that H_{n} can be (coarsely) represented by a surface H_{n}, and H′_{n} by an open subset H′_{n} of H_{n}. Moreover, over fields K of characteristic 0, H′_{n} turns out to be isomorphic to the the Hurwitz space H_{n}^{∗}; cf. Theorem 4.9 below.

The advantage of the spaces H′_{n} and H_{n} is that we can embed them explicitly as open subsets into an explicitly given surface.

To describe this embedding we shall use torsion structures on elliptic curves.

2.3 The “Basic Construction”

For details of the following discussion we refer to [FK].

Let (C, f) be a (normalized) genus 2 cover of E, and let J_{C} denote the Jacobian variety of C. Then the map f induces homomorphisms f_{∗} : J_{C} → E and f^{∗} : E → J_{C}. Because of the minimality of f, the morphism f^{∗} is injective, and the kernel E_{∗} of f_{∗} is an elliptic curve intersecting f^{∗}(E) exactly in the points of order n. It follows that there is a G_{K}-module isomorphism

α : E[n] → E_{∗}[n]

which turns out to be anti-isometric with respect to the Weil-pairings on E[n] and on E_{∗}[n]. The abelian variety J_{C} together with its natural principal polarization is isomorphic to (E × E_{∗})/Γ_{α} where Γ_{α} ≤ E[n] × E_{∗}[n] is the graph of α.

Conversely, take a K-rational triple (E, E′, α) with E, E′ elliptic curves over K and α a K-rational isomorphism from E[n] to E′[n] whose graph Γ_{α} is isotropic in E[n] × E′[n]. Then (E × E′)/Γ_{α} is a principally polarized abelian variety, and there is a K-rational effective divisor C in the class of n(0_{E} × E′ + E × 0_{E′})/Γ_{α} which is either a curve of genus 2 or a union of two elliptic curves intersecting in one point. If C is irreducible, then the projection p_{1} of E × E′ to E induces a cover

f_{α} : C → E

of degree n and hence we get a K-rational solution of the moduli problem H_{n} defined above.

Moreover, if the discriminant divisor Disc(f_{α}) is reduced, then f_{α} ∈ H′_{n}(K), and the associated cover φ : ℙ^{1} → ℙ^{1} gives rise to a point in the moduli space H_{n}^{∗}.

The triples (E, E′, α) are parameterized by an open part of the modular diagonal quotient surface Z_{n,−1}/K (cf. [Ka7]) and Subsection 4.1). If K contains a primitive n-th root of unity ζ_{n} ∈ K, then Z_{n,−1} is a quotient of the product surface X(n) × X(n), where X(n) is the modular curve parameterizing elliptic curves with level-n-structure (of fixed determinant); explicitly,

Z_{n,−1} = (X(n) × X(n))/Δ_{−1}(SL_{2}(ℤ/nℤ)),

where Δ_{−1}(G_{n}) is the “twisted diagonal subgroup” of G_{n} = SL_{2}(ℤ/nℤ) with respect to τ_{−1} ∈ Aut(G_{n}), i.e. Δ_{−1}(G_{n}) = {(g, τ_{−1}(g)) : g ∈ G_{n}} where τ_{−1} is the automorphism of G_{n} given by conjugation with an element β ∈ GL_{2}(ℤ/nℤ) of determinant −1; cf. [KS] and [Fr2]. If ζ_{n} ∉ K, then the construction of Z_{n,−1} is more complicated; in this case Z_{N,−1} is still a quotient of X(n) × X(n), (where now X(n)/K denotes Shimura’s canonical model of X(n)/ℂ), but now the quotient has to be taken with respect to an etale group scheme rather than a (constant) group (scheme); cf. [Ka7] for more detail.

For us it is important that we have found a very explicit variety which is isomorphic to the Hurwitz space H_{n}^{∗} (as we shall see), and this allows us to study its geometric and diophantine properties. So we want to compare the moduli spaces H_{n}^{∗} and H′_{n} which were introduced in the previous subsections. Although the discussion in there shows that there is a relation between their sets of K-rational points, this does not suffice to show that their coarse moduli spaces are isomorphic. For this, we shall study the associated moduli functors and/or stacks in more detail. We begin to do this “fiberwise”.

3 Covers with Fixed Base Curve

3.1 The Hurwitz spaces H_{E/K,N} and H′_{E/K,N}

Before we discuss the moduli problem H_{n} (with moduli space H_{n}) in more detail, let us consider the simpler problem of classifying the (normalized) genus 2 covers f : C → E over a fixed elliptic curve E/K. Thus, in the “dictionary” of the previous subsection, this corresponds (roughly) to case that we study covers φ : ℙ^{1} → ℙ^{1} in which the points P_{1}, …, P_{4} are fixed.

In this discussion we can drop the condition that the degree of the cover is odd, and we shall indicate this by switching from n to N. Note, however, that in the case that N is even, the normalization condition (1) has to be replaced by the condition

f_{∗}W_{C} = 2(P′_{1} + P′_{2} + P′_{3}). (3)

Then, as in the case of odd degree covers, for each minimal cover f_{0} : C → E there is a unique translation τ : E → E such that f = τ ∘ f_{0} is normalized (cf. [Ka5], Proposition 2.2). Moreover, f satisfies condition (2).

For any extension field L of K, let E_{L} = E ⊗ L denote the elliptic curve E lifted to L. We now consider the set

Cov_{E/K,N}(L) := {f : C → E_{L} : f is a normalized genus 2 cover defined over L with deg(f) = N}/ ≃

of isomorphism classes of normalized covers. As is explained in [Ka5], the assignment L ↦ Cov_{E/K,N}(L) can be extended in a natural way to a functor H_{E/K,N} : \underline{Sch}_{/K} → \underline{Sets}, and by Theorem 1.1 of [Ka5] we have:

Theorem 3.1 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 with the property that H_{E/K,N} ⊗ K_{s} is an open subset of the modular curve X(N)_{/K_{s}}.

Remark 3.2 The fact that the curve H := H_{E/K,N} finely represents the functor H_{E/K,N} means that there exists a universal normalized genus 2 cover f_{H} : C_{H} → E ×H of degree N with the property that every normalized genus 2 cover f : C → E × S of degree N (where S is any K-scheme) is obtained uniquely from f_{H} by base-change. In particular, the set Cov_{E/K,N}(K) of covers can be identified with the set of fibres f_{x} := (f_{H})_{x} : C_{x} → E_{x} = E of f_{H}, where x ∈ H(K).

The main idea for proving this is to use the “basic construction” of genus 2 curves which was presented in [FK], [Ka3] (and which was sketched in Subsection 2.3). In [Ka5], this construction is generalized to families of covers of elliptic curves, and this yields an (open) embedding of functors

Ψ = Ψ_{E/K,N} : H_{E/K,N} ↪ X_{E/K,N,−1}, (4)

where X_{E/K,N,−1} is the functor which classifies (isomorphism classes) of pairs (E′, ψ) consisting of an elliptic curve E′/K and an anti-isometry ψ : E[N] →^{∼} E′[N] (with respect to the e_{N}-pairings). Since this latter functor is finely representable by an affine curve X_{E/K,N,−1}, it follows that H_{E/K,N} is represented by an open subset H_{E/K,N} ⊂ X_{E/K,N,−1}. Note that if E[N] ≃ (ℤ/Nℤ)^{2} as a G_{K}-module, then the curve X_{E/K,N,−1} is isomorphic to the usual affine modular curve X′(N) = X(N) \ {cusps} of level N whose K-rational points correspond bijectively to isomorphism classes (E′, α), where E′/K is an elliptic curve and α : E′[N] →^{∼} (ℤ/Nℤ)^{2} is a level-N-structure (of fixed “determinant” ζ_{N}). Since this is always true over a finite separable extension of K, we see that H_{E/K,N} ⊗ K_{s} is (isomorphic to) a subset of X′(N)_{/K_{s}}.

Remark 3.3 (a) It is interesting to note that this representability result is obtained by purely algebraic techniques and hence does not use (not even implicitly) the Riemann Existence Theorem (RET).

(b) Note that while the moduli scheme X_{E/K,N,−1} depends only on the structure of the K-group scheme E[N], the subscheme H_{E/K,N} does depend on the choice of E/K; cf. [Ka5]. Moreover, if we replace E/K by a nonisomorphic twist E′/K of E/K, then E[N] ≄ E′[N] (as G_{K}-modules), and hence the functors X_{E/K,N,−1} and X_{E′/K,N,−1} are not isomorphic (as functors on \underline{Sch}_{/K}), and the same is true for H_{E/K,N} and H_{E′/K,N}. This shows that the moduli problem H_{E/K,N} is more refined than the corresponding one for covers φ : ℙ^{1} → ℙ^{1} satisfying condition (∗) with fixed ramification points P_{1}, …, P_{4}.

In view of the correspondence between genus 2 covers of elliptic curves and covers of ℙ^{1} (cf. Proposition 2.4), we shall also be interested in the subfunctor H′_{E/K,N} of H_{E/K,N} which classifies those (normalized) genus 2 covers f : C → E for which Disc(f) is reduced. More precisely, for any K-scheme S, let

H′_{E/K,N}(S) = {(C →^{f} E_{S}) ∈ H_{E/K,N}(S) : Disc(f_{s}) is reduced , ∀s ∈ S}.

It is clear that this defines a subfunctor of H_{E/K,N}, and so we should expect that it is represented by an open subscheme H′_{E/K,N} of H_{E/K,N}. This follows from the following more precise assertion.

Proposition 3.4 The rule f ↦ Disc(f) defines a morphism

δ_{E/K,N} : H_{E/K,N} → ℙ^{1}

such that Disc(f_{x}) = π^{∗}(δ_{E/K,N}(x)), for every x ∈ H_{E/K,N}(K). In particular, for every P̄ ∈ ℙ^{1}(K) we have

δ_{E/K,N}^{−1}(P̄) = Cov_{E/K,N,P̄}(K) := {f ∈ Cov_{E/K,N}(K) : Disc(f) = π^{∗}(P̄)},

and hence the functor H′_{E/K,N} is represented by the open subset H′_{E/K,N} := δ_{E/K,N}^{−1}(ℙ_{K}^{1} \ π(E[2])) of H_{E/K,N}.

Proof. The first two assertions are proved in [Ka6], Proposition 12, and the last is an immediate consequence of this and Theorem 3.1.

Remark 3.5 Let \underline{P} = (P_{1}, …, P_{5}) ∈ (ℙ_{K}^{1})^{5}(K) be a tuple of 5 distinct points with P_{4} = ∞. For an odd integer n, let Cov_{\underline{P} ,n}(K) denote the set of isomorphism classes of K-covers φ : ℙ_{K}^{1} → ℙ_{K}^{1} of deg(φ) = n satisfying condition (∗) and Convention 1.2. If E/K is an elliptic curve such that π_{E}(E[2]) = {P_{1}, …, P_{4}} (and π_{E}(0_{E}) = P_{4}), then the discussion of subsection 2.2 shows that we have a bijection

Cov_{\underline{P} ,n}(K) →^{∼} Cov_{E/K,n,P_{5}}(K).

Thus, by allowing the point P_{5} to move, we obtain a bijection

H_{P_{1},...,P_{4},n}(K) →^{∼} H′_{E/K,n}(K),

where H_{P_{1},...,P_{4},n} ⊂ H^{in}(S_{n}, C) is the Hurwitz space with fixed ramification points P_{1}, …, P_{4}. We therefore see that the set

H_{E/K,n}^{∂} := H_{E/K,n} \ H′_{E/K,n} = δ_{E/K,n}^{−1}(π(E[2]))

is a “boundary” of the Hurwitz space parametrizing ℙ^{1}-covers, even though it is contained in the Hurwitz space H_{E/K,N} parametrizing genus 2 covers of E.

3.2 The relative boundary H_{E/K,n}^{∂} = H_{E/K,n} \ H′_{E/K,n}

We now describe the genus 2 covers which correspond to the points of the “relative boundary” H_{E/K,n}^{∂} = H_{E/K,n} \ H′_{E/K,n} (where n is odd) in more detail. By Theorem 3.1 and Proposition 3.4 we know that each such point corresponds to a unique normalized genus 2 cover f : C → E of degree n such that its discriminant has the form Disc(f) = 2P′_{k}, where P′_{k} ∈ E[2]; such covers are classified as Type II in [Fr1], p. 85.

By Proposition 2.4 we still have an associated ℙ^{1}-cover φ : ℙ^{1} → ℙ^{1}, but now the ramification structure of φ no longer satisfies the “generic” condition (∗) of Subsection 1.3. Indeed, since these special covers may be viewed as the “limits” of the generic case when we let the point P_{5} move to coincide with one of the points P_{1}, …, P_{4}, we obtain new ramification types, i.e. new cycle decompositions (σ_{i}), where, as before, σ_{i} ∈ G_{φ} = Gal(\overline{φ}) denotes a generator of a ramification group above P_{i}.

Here we shall work out all the ramification types for these special covers. (For comparison purposes we also include the generic case as case 0.) It turns out that the ramification type C = C_{φ} of φ is completely determined by the different D_{f} = Diff(f) of f.

Proposition 3.6 Put m = \frac{n−1}{2} and let i, k ∈ \{1, 2, 3\}. Then the ramification structure of φ is given by the following table:

Case Condition  (σ_{i}), i ≠ k  (σ_{k})  (σ_{4})  (σ_{5})
0 Disc(f) ≠ 2P  (2)^{m}  −  (2)^{m−1} (2)
1 Disc(f) = 2P′_{k}, D_{f} ≠ 2P  (2)^{m}  (2)^{m−2}(4) (2)^{m−1} −
2 Disc(f) = 2P′_{k}, D_{f} = 2P  (2)^{m}  (2)^{m−1}(3) (2)^{m−1} −
3 Disc(f) = 2P′_{4}, D_{f} ≠ 2P  (2)^{m}  −  (2)^{m−3}(4) −
4 Disc(f) = 2P′_{4}, D_{f} = 2P  (2)^{m}  −  (2)^{m−2}(3) −

Proof. Put f_{1} = π ∘ f = φ ∘ π′ : C → ℙ^{1}. In each of the cases, we shall first work out the ramification structure of the points in f_{1}^{−1}(P_{j}), for j = 1, …, 5, and then deduce that of φ^{−1}(P_{j}) from this. From this we can then read of the cycle structure of σ_{j} ∈ G_{φ}. (Here G_{φ} is the monomdromy group of φ, i.e. the Galois group of the Galois closure \overline{φ} of φ.) Case 0. Here Disc(f) = P′_{5} + [−1]P′_{5}, and P′_{5} ∉ E[2] = {P′_{1}, P′_{2}, P′_{3}, P′_{4}}, where P′_{4} = 0_{E}. Thus, P′_{1}, …, P′_{4} are unramified in f, but are ramified of order 2 for π_{E}. It follows that there are n points in f_{1}^{−1}(P_{j}), j = 1, …, 4, and all have ramification index 2 with respect to f_{1}. Now for j = 1, 2, 3, precisely one point (the Weierstrass point) W_{j} ∈ f_{1}^{−1}(P_{j}) is ramified in π′, whereas the others are not. Thus, π′(W_{j}) is unramified wrt. φ , whereas the other m = \frac{n−1}{2} points in φ^{−1}(P_{j}) are each ramified of order 2. It follows that σ_{j} is a product of m transpositions, as claimed. On the other hand, since f_{1}^{−1}(P_{4}) contains 3 Weierstrass points, there are 3 unramified points in φ^{−1}(P_{4}), and the other m − 1 = \frac{n−3}{2} points are ramified of order 2. Thus, (σ_{4}) = (2)^{m−1}. Finally, since f_{1}^{−1}(P_{5}) contains precisely two points with ramification index 2 (which are interchanged by the hyperelliptic involution), whereas the others are unramified, it follows that φ^{−1}(P_{5}) has a unique point of index 2, and so σ_{5} is a transposition. Case 1. Here D_{f} = P_{k1} + P_{k2} with P_{k1} ≠ P_{k2} and f(P_{k1}) = f(P_{k2}) = P′_{k}, so the ramification structure above P_{j} for j = 1, …, 4, j ≠ k is as in case 0. Now above P′_{k} the ramification structure is (2)^{2} (wrt. f), so above P_{k} the ramification is (2)^{n−4}(4)^{2}. Now P_{kj} cannot be a Weierstrass point, for otherwise the ramification indices wrt. π′ and f are both 2, which is impossible since all ramification groups are cyclic; cf. Abhyankar’s Lemma. Thus, the unique Weierstrass point W_{k} ∈ f_{1}^{−1}(P_{k}) has index 2 wrt. f_{1} and hence π(W_{k}) is unramified wrt. φ. Thus, σ_{k} has type (2)^{\frac{n−5}{2}} (4). Case 2. Here D_{f} = 2W_{k}, the unique Weierstrass point above P′_{k}. Thus, above P′_{k} the ramification structure is (3) and hence above P_{k} it is (2)^{n−3}(6) wrt. f_{1}. It follows that σ_{k} has type (2)^{\frac{n−3}{2}} (3). Case 3. This is similar to case 1. Here D_{f} = P_{41} + P_{42} with P_{41} ≠ P_{42} and f(P_{41}) = f(P_{42}) = P′_{4} = 0_{E}, so the ramification structure above P_{j} for j = 1, …, 3, is as in case 0. Now above 0_{E} the ramification structure is (2)^{2} (wrt. f), so above P_{4} the ramification wrt. f_{1} is (2)^{n−4}(4)^{2}. As in case 1, P_{4j} cannot be a Weierstrass point, so we see that σ_{4} has type (2)^{\frac{n−7}{2}} (4) because the images of the 3 Weierstrass points are unramified wrt. φ. Case 4. Here D_{f} = 2W′_{k}, where W′_{k} is one of the 3 Weierstrass points above 0_{E}. Thus, above 0_{E} the ramification structure is (3) and hence above P_{k} it is (2)^{n−3}(6) wrt. f_{1}. It follows (cf. case 3) that σ_{4} has type (2)^{\frac{n−5}{2}} (3).

Corollary 3.7 (a) In cases 0 and 2, the monodromy group of φ and f is G_{φ} = G_{f} = S_{n}.

(b) In case 4, we have G_{f} = A_{n}. Moreover, G_{φ} = S_{n} when n ≡ 1 (mod 4), and G_{φ} = A_{n} when n ≡ 3 (mod 4).

Proof. Let F̃_{0} (respectively, F̃) be the Galois hull of κ(ℙ^{1})/φ^{∗}κ(ℙ^{1}) (respectively, of κ(C)/f^{∗}κ(E). Since F̃ = F̃_{0}κ(E) by Galois theory, we see that G_{φ} = G_{f} if and only if κ(E) ⊄ F̃_{0}.

Moreover, since f is minimal, we see that G_{φ} = Gal(F̃_{0}/κ(ℙ^{1})) and G_{f} = Gal(F̃/κ(E)) are primitive permutation groups of degree n; cf. [Hup], II.1.4. In case 0, σ_{5} is a 2-cycle, so G_{φ} = S_{n} by [Hup], II.4.5b). In case 2 (respectively, case 4) we have from Proposition 3.6 that σ_{4}^{2} (respectively, σ_{k}^{2}) is a 3-cycle, so G_{φ} ≥ A_{n} by [Hup], II.4.5c).

(a) Since Proposition 3.6 shows that G_{φ} ≰ A_{n}, it follows that G_{φ} = S_{n} in these cases. Moreover, suppose κ(E) ⊂ F̃_{0}. Then κ(E) = F_{1} := (F̃_{0})^{A_{n}}, because A_{n} is the unique subgroup of index 2 of S_{n}. Now in case 0 and m even we see from Proposition 3.6 that σ_{4} and σ_{5} are odd and the others are even permutations, so F_{1}/κ(ℙ^{1}) is ramified precisely at P_{4} and P_{5}. Thus, F_{1} ≠ κ(E) (which is ramified at P_{1}, … P_{4}). Similarly, we have in the other cases that F_{1} ≠ κ(E), and so G_{f} = G_{φ}.

(b) If m is even (i.e. n ≡ 1(4)), then G_{φ} ≰ A_{n} by Proposition 3.6 and so G_{φ} = S_{n}. Moreover, F_{1}/κ(ℙ^{1}) is ramified precisely at P_{4}, P, so F_{1} ≠ κ(E) and G_{f} = G_{φ} = S_{n}.

If m is odd, then by Proposition 3.6 we have σ_{i} ∈ A_{n}, for all i = 1, …, 4, and so G_{φ} ≤ A_{n} (because G_{φ} is generated by the σ_{i}’s and their conjugates). Thus G_{φ} = A_{n}. Since A_{n} has no subgroup of index 2, it follows that κ(E) ⊄ F̃_{0} and so G_{f} = G_{φ}.

Remark 3.8 The cases 2 and 4 have studied in detail from a different perspective by authors studying problems in billiard dynamics and square tilings; cf. [Mc1], [HL1] and the references therein. More precisely, if K = ℂ, and if f : C → E is a genus 2 cover of type 2 or 4, and ω_{E} is a holomorphic differential on E, then (C, f^{∗}ω_{E}) defines a point in the moduli space H(2) of genus 2 curves with a differential 1-form having a double zero (cf. [Mc1]).

By [Mc1], these two cases can be distinguished by their “spin invariant”: those in case 2 have spin 0, and those in case 4 have spin 1. Indeed, in case 2 (respectively, case 4) we have 1 (respectively, 3) Weierstrass points above the ramification point P′_{k} (respectively, P′_{4}), so the assertion follows from [Mc1], Theorem 6.1. (Note that [Mc1] and [HL1] assume that the ramification takes place over the origin of E, so in case 2 they consider the cover τ_{P′_{k}} ∘f in place of the cover f.)

3.3 The boundary of H_{E/K,N}

Since the moduli space H := H_{E/K,N} is a smooth, irreducible (affine) curve, it has a unique normal compactification X. Here we shall study the “boundary” ∂H = X \ H of H and how the correspondence between covers f_{x} : C_{x} → E and points of H (cf. Remark 3.2) extends to ∂H.

The first step towards this goal is to observe that ∂H = S(C), the set of places of bad reduction of the curve C/F which is the generic fibre of the relative universal curve C_{H}/H (cf. Remark 3.2). As a consequence, ∂H decomposes as

∂H = S(C) = S_{0} ∪ S_{1}

where S_{0} is the set of points of X where the Jacobian J_{C} of C/F has bad reduction and where S_{1} is the set of places where C has bad reduction yet J_{C} had good reduction.

To explain how the points of ∂H correspond to (degenerate) covers, we shall use the theory of minimal models to construct a canonical compactification f : C → E × X of the universal cover f_{H} : C_{H} → E × H (cf. Remark 3.2) and then study its fibres.

To explain this in more detail, assume for simplicity that K = \overline{K} is algebraically closed. In this case we know that H is an open subset of the modular curve X(N), which is therefore its canonical compactification, i.e. X = X(N). In addition, we shall assume that char(K) ∤ N! (i.e. that either char(K) = 0 or that char(K) > N); in the contrary case the structure of the boundary (and of the corresponding covers) is much more complicated.

Let p : C → X denote the minimal model of C/F over X. Thus, C is a projective, smooth surface over K, and p : C → X is a genus 2 fibration whose fibres do not contain any rational (−1)-curves. Moreover, the restriction of p to p^{−1}(H) is canonically isomorphic to p_{H} := pr_{2} ∘ f_{H} : C_{H} → H (which is the minimal model of C/F over H) and so we can identify C_{H} with the open subscheme p^{−1}(H) of C. Thus, C is a canonical compactification of C_{H}.

It turns out that f_{H} extends to a finite morphism f : C → E_{X} = E × X. Moreover, the precise structure of the fibres of C/X is as follows:

Theorem 3.9 If char(K) ∤ N!, then C/X(N) is a stable curve which has precisely one singular point in each singular fibre C_{x}, and the universal cover f_{H} extends to a finite morphism f : C → E_{X(N)} := E × X(N). Furthermore, a fibre C_{x} is singular if and only if x ∈ ∂H, i.e. S = ∂H, and the set S_{0} (as defined above) coincides with the set X(N)_{∞} of cusps of X(N). Moreover, the structure of the singular fibres of C/X(N) is as follows:

(a) If x ∈ S_{0}, then the fibre C_{x} is an irreducible curve whose normalization is a curve of genus 1. Furthermore, C_{x} has a unique singular point P_{x}, and the restriction f_{x} = f_{C_{x}} : C_{x} → (E_{X})_{x} ≃ E is a cover of degree N such that f_{x}(P_{x}) = 0_{E}.

(b) 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}. Furthermore, f_{x}(P_{x}) ∈ E[2] and the cover f_{x} : C_{x} → E has degree N in the sense that d_{x,1} + d_{x,2} = N, where d_{x,i} = deg(f_{x,i}) is the degree of f_{x,i} := (f_{x})_{|E_{x,i}} : E_{x,i} → E.

Moreover, the modular height of C/X(N) is h_{C/X} = \frac{1}{24}sl(N), where sl(N) = #SL_{2}(ℤ/Nℤ), and the self-intersection number of the relative dualizing sheaf ω_{C/X}^{0} is given by

(ω_{C/X(N)}^{0})^{2} = \frac{7}{5}#S_{1} + \frac{1}{5}#S_{0} = \frac{1}{24N} (7N − 6)sl(N). (5)

Remark 3.10 The proof of this proposition is based on the ideas and techniques of [FK], pp. 161–166. Note, however, that the conclusions obtained here are much stronger than those of [FK], for here we are able to determine the number of irreducible components of each fibre. In addition, in this situation one of the cases (i.e. the case β_{2}) of [FK] does not occur.

The reason for obtaining such strong results is due to two facts. First of all, by using the results of [Ka3], [Ka4] (cf. Step 8 below), we are able to determine precisely the total number of singular fibres in each case. Secondly, we can use a result of Mumford to determine certain invariants which give information about the individual degeneration types. It is interesting to note that such a formula does not seem to be available in the arithmetic case; indeed, a considerable portion of the paper [FK] was devoted to deriving a weak version of such a formula. (This was pointed out to us by Barry Mazur.)

Proof of Theorem 3.9 (Sketch). For convenience of the reader, we outline the main ideas of the proof of Theorem 3.9 and refer to [Ka6] for details. (The above theorem is contained in Corollary 16, Theorem 17, Proposition 19, Theorem 22, Corollary 24 and Corollary 25 of [Ka6].) Step 1: f_{H} extends to a morphism f : C → E_{X}.

Since E_{X} is a smooth proper curve over X = X(N), this follows from Zariski’s Main Theorem; cf. [Ka6], Proposition 15 and Corollary 16. Step 2: C/X is semistable and S_{0} = X(N)_{∞}. By the basic construction, J_{C} ∼ E_{F} × E′, where E_{F} = E ⊗ F and E′/F is an elliptic curve. Now since E′[N] ≃ E[N] ≃ (ℤ/Nℤ)^{2} (as G_{F} -modules), it follows that both E′ and J_{C} are semi-stable abelian varieties (as E has good reduction), and so C is a semi-stable curve (cf. [Ka6], proof of Theorem 17).

Clearly, J_{C} has bad reduction at x ∈ X if and only if E′ does. Since E′/X(N) is the universal elliptic curve with level-N-structure (of fixed determinant), it follows that S_{0} = X(N)_{∞}. Step 3: S(C) = ∂H. Since p_{H} : C_{H} → H has smooth fibres, it is clear that S(C) ⊂ ∂H. The opposite inclusion follows from the construction of H in [Ka5]; cf. [Ka6], Proposition 19 for a more detailed explanation. Step 4: The preliminary structure of C_{x} for x ∈ S_{0}. Here we are in case β) of [FK], p. 163ff. By the argument given there (or by the one presented in [Ka6], proof of Proposition 19), we conclude that C_{x} has a unique component C_{x,0} whose normalization has genus 1, and that there is at least one singular point P_{x} ∈ C_{x} such that C_{x} \ {P_{x}} is connected. Step 5: The preliminary structure of C_{x} for x ∈ S_{1}. Here we are in case α) of [FK], p. 161ff. Thus, if δ_{x} denotes the number of singular points of C_{x}, then C_{x} has δ_{x} + 1 irreducible components: two elliptic curves E_{x,1}, E_{x,2} which are connected by a chain of ℙ^{1}’s. Step 6: Computation of the modular height h_{C/X}. Recall that if p : C → X is any semi-stable of genus g ≥ 1, then its modular height is defined by h_{C/X} = deg(λ), where λ = λ_{C/X} = ∧^{g}p_{∗}ω_{C/X}^{0} ∈ Pic(X). Note that we also have λ ≃ λ_{J /X} := s^{∗}Ω_{J /X}^{1}, where J /X denote the Néron model of the Jacobian J_{C} and s : X → J denotes the zero-section of J /X; this is a formula due to Parshin and Arakelov (cf. [Ka6], proof of Theorem 17).

Since J_{C} ∼ E_{F} × E′_{F} (by an etale isogeny), and since E_{F} /F is constant, we see (by using the second description of λ) that λ_{C/X} ≃ λ_{E′/X}. Now for any (semi-stable) elliptic curve E′/X we have (by Noether’s formula) that h_{E′/X} = \frac{1}{12}δ_{E′/X}, where E′ denotes the Néron model of E′ over X. But δ_{E′/X} = N#S_{0} (because each singular fibre E′_{x} of E′/X lies over X(N)_{∞} = S_{0} (cf. Step 2) and is a Néron polygon of length N), and so we obtain

h_{C/X} = h_{E′/X} = \frac{N}{12}#S_{0} = \frac{1}{24}sl(N), (6)

where the latter equation follows from the first equality of (11) below. Step 7: The invariants δ_{0}, δ_{1}, ω^{2} and Mumford’s formula. 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. By Steps 4 and 5 we see that

δ_{0} ≥ #S_{0} and δ_{1} ≥ #S_{1}. (7)

Indeed, if x ∈ S_{0}, then by Step 4 there is at least one singular point P_{x} of C_{x} which does not disconnect the fibre, and if x ∈ S_{1}, then by Step 5 each of the δ_{x} ≥ 1 singular points in C_{x} disconnects the fibre.

There is a remarkable relation which connects the invariants h = h_{C/X}, δ_{0}, and δ_{1}:

12h = \frac{6}{5}δ_{0} + \frac{12}{5}δ_{1}. (8)

This equation is an immediate consequence of the following two relations which also involve the invariant ω^{2} = (ω_{C/X}^{0})^{2}, the self-intersection number of the relative dualizing sheaf ω_{C/X}^{0} of C/X:

12h = ω^{2} + δ_{0} + δ_{1}, and ω^{2} = \frac{1}{5}δ_{0} + \frac{7}{5}δ_{1}, (9)

which are called Noether’s formula and Mumford‘s formula respectively; cf. [Ka6], proof of Theorem 22 for the relevant references.

By combining equation (8) with the inequalities (7), we obtain the inequality

#S_{1} ≤ \frac{1}{12}(5N − 6)#S_{0}. (10)

Indeed, since h = \frac{N}{12}#S_{0} by (6), we thus obtain from (8) and (7) that N#S_{0} = 12h ≥ \frac{6}{5}#S_{0} + \frac{12}{5}#S_{1}, which is (10). Step 8: Computation of #S_{0} and #S_{1}. Whereas the computation of #S_{0} is easy (because S_{0} = X(N)_{∞} by step 2), the determination of #S_{1} is much harder. Indeed, as is explained in the proof of [Ka6], Theorem 21 (see also [Ka5], Theorem 6.2), this is essentially the “mass formula” of [Ka3] , [Ka4]. (The hypothesis char(K) ∤ N! is crucial here.) One obtains:

#S_{0} = \frac{sl(N)}{2N} and #S_{1} = \frac{1}{24N} (5N − 6)sl(N). (11)

Step 9: δ_{x} = 1, for all x ∈ S = S_{0} ∪ S_{1}. By equation (11) we see that we have equality in (10), and so we must have equality in (7) as well. Thus, δ_{0} = #S_{0} and δ_{1} = #S_{1}, which means that δ_{x} = 1, for all x ∈ S = S_{0} ∪ S_{1}, i.e. that C_{x} has a unique singular point. From this (together with Steps 4, 5) it is clear that the structure of C_{x} for x ∈ S_{0} ∪ S_{1} is as claimed in parts (a), (b), and so C/X is a stable curve. Step 10: f is finite. Since f is proper, it is enough to show that f is quasi-finite, i.e. that none of the components of a fibre C_{x} of p is mapped to a point under f. For the points x ∉ S_{1}, this easy (since C_{x} is irreducible). If x ∈ S_{1}, it is clear that at least one of the two components is not mapped to a point. To see that both have this property is a bit more difficult, particularly if N even; see [Ka6], Proposition 20, for the proof.

Since the rest of the assertions (concerning f_{x}(P_{x})) are verified easily, and since (5) follows from (9) (because δ_{0} = #S_{0} and δ_{1} = #S_{1} by step 9), this concludes the proof (sketch) of the theorem.

3.4 Application: the Rigidity Number

As an application of the above geometric ideas and constructions, we shall now compute the number #Cov_{E/K,N,P̄}(K) of covers with fixed discriminant π^{∗}(P̄); this number may be viewed (via the identification of Remark 3.5) as a measure of non-rigidity of the Hurwitz space H_{n}^{∗} because for rigid systems this number is 1; cf. [V1], p. 39.

Although this “rigidity number” can be defined via group theory, it seems very difficult to compute this number directly by counting tuples of conjugacy classes (cf. Remark 3.17). Here instead we shall use the geometric results of the previous subsections to calculate this number.

Theorem 3.11 Let N ≥ 3 be an integer, and suppose that K is an algebraically closed field. If char(K) ∤ N!, then for every P̄ ∈ ℙ^{1}(K) \ π(E[2]) we have

#Cov_{E/K,N,P̄}(K) = \frac{1}{12}(N − 1)sl(N),

where, as before, sl(N) := #SL_{2}(ℤ/Nℤ).

We briefly sketch the main ideas involved in the proof of Theorem 3.11. (For more detail, cf. [Ka6].) The basic idea is to relate the non-rigidity number #Cov_{E/K,N,P̄}(K) to the degree of the discriminant map δ_{E/K,N} (cf. Proposition 3.4) which in turn can be computed by using the intersection theory on compactification C of the universal genus 2 cover f : C → E × X(N). The key ingredients for this latter computation are the precise knowledge of the degenerate fibres of C (cf. Theorem 3.9) and the formula (5) for the self-intersection number (ω_{C/X}^{0})^{2} of the relative dualizing sheaf.

Step 1: #Cov_{E/K,N,P̄}(K) = deg(δ_{E/K,N}). By Proposition 3.4 we know that #Cov_{E/K,N,P̄}(K) = #δ_{E/K,N}(P̄)). On the other hand, since δ_{E/K,N} is finite and etale over P̄ ∉ π(E[2]) (cf. [Ka6], Theorem 13), the assertion follows.

Step 2: Analysis of the different divisor D on the compactification C. We next want to interpret deg(δ) as an intersection number. For this, we shall study the (Kähler) different divisor Diff(f) of the compactification f : C → E × X(N) of the universal cover f_{H}; cf. subsection 3.3.

Let D_{F} denote the different divisor of the generic (universal) cover f_{F} : C = C_{F} → E_{F}, and let D be its closure in the compactification C. By using the explicit structure of the fibres of p : C → X(N) (cf. Theorem 3.9) together with the Riemann-Hurwitz formula (for relative curves), we obtain

Proposition 3.12 The divisor D is an irreducible curve on C which is a degree 2 cover of X(N) via the map π_{D} := pr_{2}∘f_{|D}. If δ_{D} := pr_{1}∘f_{|D} : D → E denotes the projection onto the first factor, then

π_{E} ∘ δ_{D} = \overline{δ}_{E/K,N} ∘ π_{D}.

In addition, D = Diff(f), the different divisor of f : C → E_{X}, and hence we have ω_{C/X(N)}^{0} ∼ D. Thus, for any P ∈ E(K) we have

deg(δ_{E/K,N}) = deg(δ_{D}) = (ω_{C/X(N)}^{0}.f^{∗}(P × X)). (12)

Proof. [Ka6], Theorem 26. Note that this makes heavy use of the Structure Theorem 3.9.

Step 3: Analysis of the Weierstraß divisor W.

By using the above proposition we see that the theorem will follow once we have computed the intersection number (ω_{C/X}^{0}.f^{∗}(P × X)). As we shall see next, this number is closely related to the self-intersection number of the Weierstraß divisor W on C. This divisor W is the closure in C of the (usual) Weierstraß divisor W_{F} on C = C_{F}.

Proposition 3.13 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}{2N} (N − 1)sl(N), (13)

and hence

deg(δ_{D}) = \frac{N}{6} deg(A) = \frac{1}{12}(N − 1)sl(N). (14)

Proof. (Sketch; cf. [Ka6], Proposition 29). Since 6D_{F} and 2W_{F} are both 6canonical divisors on C, we have 6D_{F} ∼ 2W_{F}. Moreover, since the fibres over S_{1} are the only reducible fibres of C/X and since for x ∈ S_{1}, both 6D and 2W meet each component E_{x,i} of C_{x} with multiplicity 6 (cf. the diagram on p. 162 of [FK]), we can conclude that 6D − 2W ∼ p^{∗}(A), for some A ∈ Div(X).

To calculate the degree of A, we shall use formula (34) of [Ka6]:

(ω_{C/X(N)}^{0}.W) + W^{2} = #X(N)_{∞} = \frac{1}{2N} sl(N). (15)

This formula is established by applying the adjunction formula to the divisor W (after base-changing to X(2N)).

From (15) and the equivalence 6D ∼ 2W + p^{∗}A, the first two equalities of (13) follow readily. To deduce the third, note that the first two yield that 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}{8N} (N − 2)sl(N), (16)

the latter by the important formula (5). From this, the last equation of (13) is immediate.

Finally, since it easy to see that f_{∗}(2W) ∼ 12(0_{E} × X) (and since ω ∼ D by Proposition 3.12) we have 6f_{∗}ω ∼ 12(0_{E} ×X)+N(E ×A), and so by (12) and the projection formula we obtain

6 deg(δ_{D}) = 6(ω.f^{∗}(0_{E}×X)) = ((12(0_{E}×X)+N(E×A)).0_{E}×X) = N deg(A), which proves the first equality of (14). The second follows immediately from (13).

Step 4: Conclusion. Combining the formula of Step 1 with equations (12) and (14) yields the desired formula of Theorem 3.11.

From the above result (and its proof) we can also compute the number of points in the “interior boundary” H_{E/K,N}^{∂} (cf. Remark 3.5(b)).

Corollary 3.14 In the above situation we have

#H_{E/K,N}^{∂} = \frac{(4N − 3)(N − 2)}{24N}sl(N). (17)

Proof. We apply the Riemann-Hurwitz formula to the cover \overline{δ} = \overline{δ}_{E/K,N} : X = X(N) → ℙ^{1}. Since \overline{δ} is tamely ramified (cf. [Ka6], Proposition 31) and since \overline{δ} is unramified outside of π(E[2]) (cf. Step 1 of the proof of Theorem 3.11), we have by Riemann-Hurwitz

2g_{X} − 2 = deg(\overline{δ})(−2) + 4 deg(\overline{δ}) − #(\overline{δ}^{−1}(π(E[2]))).

It is easy to see that \overline{δ}^{−1}(π(E[2])) = ∂H ∪ H_{E/K,N}^{∂}. Thus, since 2g_{X} − 2 = \frac{N−6}{12N}sl(N), since #∂H = \frac{1}{24N} (5N +6)sl(N) by (11), and since deg \overline{δ} = \frac{1}{12}(N − 1)sl(N) (cf. (14) and (12)), we obtain that #H_{E/K,N}^{∂} = 2 deg(\overline{δ}) − #∂H − (2g_{X}−2) = \frac{1}{6}(N−1)sl(N)−\frac{1}{24N} (5N+6)sl(N)−\frac{N−6}{12N}sl(N) = \frac{4N^{2}−11N+6}{24N}sl(N), and so (17) follows.

Remark 3.15 If we compare the above formula (17) with (11), we see that there are more points in the relative boundary H_{E/K,N}^{∂} than in the boundary ∂H_{E/K,N}; more precisely, we have

#H_{E/K,N}^{∂} − #∂H_{E/K,N} = \frac{1}{6}(N − 6)sl(N).

As yet another application of Theorem 3.11, we compute the number c_{N,D} = #Cov_{E,N,D}^{all} of all genus 2 covers of E of fixed degree N ≥ 1 and fixed discriminant divisor D ∈ Div(E). (Thus, here we no longer assume that the cover is normalized.) Since this number is closely related to the weighted number c̄_{N,D} := ∑_{f∈Cov_{E,N,D}^{all}} \frac{1}{|Aut(f)|} of such covers and since the latter leads to simpler formulae, we determine c̄_{N,D} as well.

Corollary 3.16 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)) (18)

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). (19)

Proof. [Ka6], Theorem 1. As is explained in section 2 of that paper, this theorem can be deduced from Theorem 3.11.

Remark 3.17 It is interesting to note that the number c̄_{N,D} (for D reduced) can be computed by a method that is essentially group-theoretic (and which goes back to Hurwitz[Hu]); cf. [Dij] and also [Ma]. Thus, by reversing the reasoning of the proof of the above corollary (by using a Moebius inversion formula), one can obtain a group-theoretical proof of Theorem 3.11.

4 Comparison of the Moduli Spaces

4.1 The Moduli Spaces H_{N}, H′_{N} and Z_{N,−1}

We now turn to the study of the moduli problems H_{N} and H′_{N} which were (briefly) introduced at the end of Subsection 2.2. For this, we need to extend the previous Hurwitz functor H_{E/K,N} (cf. Subsection 3.1) for a fixed elliptic curve E/K to a functor over a variable elliptic curve E/K.

A convenient framework for studying such questions is the concept of a moduli problem for elliptic curves as introduced by Katz-Mazur[KM], p. 107: by definition, such a moduli problem is a contravariant functor

P : \underline{Ell}_{/R} → \underline{Sets}

from the category \underline{Ell}_{/R} to the category of sets. Here, as in [KM], p. 122, the moduli stack \underline{Ell}_{/R} of all elliptic curves over a ring R is the category whose objects are (relative) elliptic curves E/S where S is any R-scheme, and whose morphisms from E/S to E′/S′ consist of cartesian diagrams (with vertices E/S and E′/S′) over R.

As is explained in [KM], pp. 108 and 125, each moduli problem P on \underline{Ell}_{/R} gives rise to a contravariant functor

P̃ : \underline{Sch}_{/R} → \underline{Sets}

which classifies isomorphism classes of P-structures. This functor P̃ gives us a more naive interpretation of our moduli problem. However, although the study of P̃ suffices for the discussion of coarse moduli spaces, it is the study of P which yields the most powerful (and natural) results.

Here we shall be interested in two (or three) specific moduli problems on \underline{Ell}_{/R_{N}}, where throughout R_{N} = ℤ[1/(2N)]; these are defined by the rules

H_{N}(E/S) = H_{E/S,N}(S) and Z_{N,−1}(E/S) = X_{E/S,N,−1}(S),

for E/S ∈ ob(\underline{Ell}_{/R}), where the functors H_{N}(E/S) and Z_{N,−1} are as in Subsection 3.1. Clearly, each morphism in \underline{Ell}_{/R} induces a natural map (by base-change) between the relevant sets, so that we obtain a moduli problem on \underline{Ell}_{/R} as claimed.

In addition, we shall be interested in the moduli problem H′_{N} which is defined by rule H′_{N}(E/S) = H′_{E/S,N}(S) (where H′_{E/S,N} is the subfunctor of H_{E/S,N} defined at the end of Subsection 3.1). Since this extra condition is obviously compatible with base change, we see that H′_{N} is a subfunctor of H_{N} and hence a moduli problem on \underline{Ell}_{/R}.

Note that the functor H̃_{N} : \underline{Sch}_{/R} → \underline{Sets} associated to H_{N} has the following natural interpretation:

H̃_{N}(S) = {(C →^{f} E) ∈ H_{N}(E/S) : E/S is a (relative) elliptic curve}/ ∼ .

Moreover, H̃′_{N} and Z̃_{N,−1} have similar interpretations. Thus, although the moduli problems H_{N} and H′_{N} “classify” isomorphism classes of covers, their associated functors H̃_{N} and H̃′_{N} naturally “classify” equivalence classes of covers (in the sense of Definition 1.1).

Theorem 4.1 If N ≥ 3, then the moduli problems H_{N}, H′_{N} and Z_{N,−1} are relatively representable, (quasi-) affine, smooth and geometrically connected of relative dimension 1 over \underline{Ell}_{/R_{N}}. Furthermore, we have open embeddings (of functors)

Ψ_{N} : H_{N} ↪ Z_{N,−1} and Ψ′_{N} : H′_{N} ↪ H_{N} ↪ Z_{N,−1}.

Proof. For a fixed elliptic curve E/S, the “fibre functor” of H_{N} at E/S is by construction just the functor H_{E/S,N}, i.e.

(H_{N})_{E/S} = H_{E/S,N} : \underline{Sch}_{/S} → \underline{Sets}.

Now by Theorem 5.18 of [Ka5] the functor H_{E/S,N} is representable for every E/S, and this means precisely that H_{N} is relatively representable (cf. [KM], p. 108). (Actually, in [Ka5] the representability of H_{E/S,N} is asserted only for an affine base S, but from this the assertion follows (by gluing) for any base S.) Moreover, since the scheme H_{E/S,N} representing H_{E/S,N} is quasiaffine, smooth and geometrically connected of relative dimension 1 over S, this means that functor H_{N} also has these properties by definition (cf. [KM], p. 109).

By construction, the fibre functor (H′_{N})_{E/S} = H′_{E/S,N} is a subfunctor of (H_{N})_{E/S}, and it is easy to see that (H′_{N})_{E/S} is represented by an open subscheme H′ = H′_{E/S,N} of H = H_{E/S,N}. If S is a field, then this was proven in Proposition 3.4. In the general case we have H′ = H \ Z, where Z is the closed subscheme of H which is universal for the relation Disc(f_{H}) ≤ E_{H}[2]^{#} in the sense of [KM], Key Lemma 1.3.4. Here, f_{H} : C → E_{H} denotes the universal normalized genus 2 cover of degree N over E_{H} and E[2]^{#} = E[2] − 0_{E_{H}} (viewed as a Cartier divisor as in [Ka5]).

The proof of the assertions about Z_{N,−1} are entirely analogous to those for H_{N}; here we use [Ka5], Corollary 4.3 in place of Theorem 5.18. (Note that X_{E/S,N,−1} is the fibre functor of Z_{N,−1} at E/S.) Finally, the open embedding Ψ_{N} is induced by the (open) embeddings Ψ_{E/S,N} : H_{E/S,N} → X_{E/S,N,−1}; cf.

(4) or [Ka5], Theorem 5.18.

Remark 4.2 The above moduli problems H′_{N}, H_{N} and Z_{N,−1} are in fact affine and not just quasi-affine. For Z_{N,−1}, this is immediate from the above construction, but not for H′_{N} and H_{N}. Since we don’t need this here, we omit the proof.

It is clear that none of the functors H_{N}, H′_{N} and Z_{N,−1} is representable, for none is rigid (because the [−1] map acts trivially on each). However, since all are (quasi-) affine and relatively representable, there exists a coarse moduli scheme H_{N} := M(H_{N}), and H′_{N} := M(H′_{N}) and Z_{N,−1} = M(Z_{N,−1}) for each; cf. [KM], p. 224. Note that Z_{N,−1} (and hence also H_{N} and H′_{N}) comes equipped with a natural morphism

p_{N} : Z_{N,−1} → M_{1} := Spec(R_{N}[j]) = M([Γ(1)]),

to the j-line over R_{N} which is the coarse moduli space M_{1} = M([Γ(1)]) attached to the moduli problem [Γ(1)] classifying isomorphism classes of elliptic curves; cf. [KM], p. 229. Thus we have:

Corollary 4.3 The coarse moduli scheme H_{N} := M(H_{N}) of H_{N} is an open subscheme of the coarse moduli scheme Z_{N,−1} = M(Z_{N,−1}) of Z_{N,−1}. The latter is normal and affine of relative dimension 1 over M_{1} = Spec(R_{N}[j]). In addition, the coarse moduli scheme H′_{N} := M(H′_{N}) of H′_{N} is an open subscheme of H_{N} and hence also of Z_{N,−1}.

Remark 4.4 As a first approximation, we can view the previously considered moduli spaces H_{E/K,N} as the fibres of the morphism p_{N} : H_{N} → M_{1}. Indeed, since each E/K gives rise to a point x_{E} ∈ M_{1} such that κ(x_{E}) = K and j(E) = j(x_{E}), we might expect that the fibre (H_{N})_{x} over x is “essentially” (i.e. up to twists) the curve H_{E/K,N}. While this is true “in general”, it is not always true. For example, at a (geometric) point x where Aut(E_{x}) ≠ ±1, the fibre of p_{N} is an affine curve whose compactification has lower genus than those of the generic case.

By the above corollary we see that H_{N} and Z_{N,−1} are (quasi-) affine normal surfaces over Spec(R_{N}). We next want to identify the surfaces in terms of the modular diagonal quotient surfaces introduced in [KS].

Even though these surfaces can be defined over R_{N} (cf. [Ka7]), we shall explain their construction only over the ring R̃_{N} = R_{N}[ζ_{N}] = ℤ[\frac{1}{2N} , ζ_{N}], where ζ_{N} is a primitive N-th root of unity. (This situation suffices for geometric purposes, but not for arithmetic applications.) Over R̃_{N} we have the smooth affine curve X(N)′_{/R̃_{N}} which represents the moduli problem [Γ(N)]^{can} of [KM], p. 283. It comes equipped with a natural action of the group G_{N} = SL_{2}(ℤ/Nℤ)/{±1}; thus G_{N} × G_{N} acts on the product Y_{N} = X(N)′_{/R̃_{N}} ×R̃_{N} X(N)′_{/R̃_{N}}. For any ε ∈ (ℤ/Nℤ)^{×}, let

Δ_{ε} = {(g, τ_{ε}(g)) : g ∈ G_{N}} ≤ G_{N} × G_{N}

denote the “twisted diagonal subgroup”, i.e. the graph of the automorphism τ_{ε} ∈ Aut(G_{N}) defined by τ_{ε}(g) = σ_{ε}gσ_{ε}^{−1}, where σ_{ε} = \binom{ε 0}{0 1} ∈ GL_{2}(ℤ/Nℤ). Then the quotient

Z_{N,ε} = Y_{N}/Δ_{ε}

is called the modular diagonal quotient surface of type (N, ε).

Proposition 4.5 The modular diagonal quotient surface Z_{N,ε} is the coarse moduli space of the moduli problem Z_{N,ε}; i.e. M(Z_{N,ε}) = Z_{N,ε}, and hence the coarse moduli spaces H_{N} and H′_{N} are open subsets of Z_{N,−1}.

Proof (Sketch). As was mentioned above, the moduli problem [Γ(N)]_{/R̃_{N}}^{can} is represented by E_{N}/X, where X = X(N)′ ; R̃_{N}, Thus, since Z_{N,ε} is affine and relatively representable by Theorem 4.1, it follows that the simultaneous moduli problem P_{N,ε} := [Γ(N)]^{can} × Z_{N,ε} is represented by p^{∗}E/Y_{ε} where p : Y_{ε} → X represents the functor X_{E_{N} /X,N,ε} on \underline{Sch}_{/X}; cf. [KM], p. 109. Thus, by [KM], p. 224, we see that the coarse moduli space of Z_{N,ε} is M(P_{N,ε})/G_{N} because [Γ(N)]_{/R̃_{N}}^{can} is etale with group G_{N}.

Now Y_{ε} ≃ X ×R̃_{N} X because X also represents the functor X_{E_{N} /R̃_{N} ,N,ε} on \underline{Sch}R̃_{N}, and via this identification the group G_{N} is mapped to Δ_{ε} ≤ G_{N} ×G_{N}. Thus M(Z_{N,ε}) = Z_{N,ε}, as claimed. The last assertion follows from Corollary 4.3.

While the above result gives a concrete interpretation of the coarse moduli space Z_{N,−1} and hence of the subspaces H_{n} and H′_{n}, it has the disadvantage that its interpretation as a coarse moduli space has nothing to with covers of curves. As a result, it seems somewhat mysterious how the spaces H_{n} and H′_{n} are embedded in Z_{n,−1}. To remedy this, we now explain how Z_{n,−1} is the coarse moduli space of suitable Hurwitz functor.

As a first step towards this, let A_{N} denote the moduli problem on \underline{Ell} which, for a fixed elliptic curve E/S, is the set

A_{N}(E/S) = A_{E/S,N}(S) = {(J, λ, h)}/ ≃,

of isomorphism classes of tripels (J, λ, h) consisting of a principally polarized abelian surface (J, λ) over S and and injective homomorphism h : E → J of degree N in the sense of [Ka5], §5.1 (p. 24). We then have:

Proposition 4.6 There is a natural isomorphism Ψ′ : A_{N} →^{∼} Z_{N,−1} of moduli problems, and hence Z_{N,−1} is also a coarse moduli scheme of A_{N}.

Proof. For each E/S ∈ ob(\underline{Ell}) we have by Theorem 5.10 of [Ka5] an iso-

morphism Ψ′_{E/S} : (A_{N})_{E/S} = A_{E/S,N} →^{∼} (Z_{N,−1})_{E/S} = X_{E/S,N,−1} of the

corresponding fibre functors, and these fit together to define the desired isomorphism. The last assertion follows from Proposition 4.5.

We next need to identify A_{N} with a Hurwitz moduli problem. This, however, is more problematic. Consider first the moduli problem H_{N}^{st} on \underline{Ell} which, for a given E/S, classifies finite flat covers f : C → E of degree N where C/S is a stable curve of genus 2 (in the sense of [DM]), i.e.

H_{N}^{st}(E/S) = {f : C → E finite, flat, C/S stable of genus 2, deg(f) = N}/≃ .

Next, let H_{N}^{§} be the subfunctor of H_{N}^{st} consisting of the those S-covers f : C → E such that (J_{C_{s}}, θ_{C_{s}}, f_{s}^{∗}) ∈ A_{N}(Spec(κ(s))), for every s ∈ S; here J_{C_{s}}/κ(s) is the Jacobian of the fibre C_{s} and θ_{s} its canonical polarization. Since the Jacobian exists for any stable curve C/S (cf. [BLR], Th. 9.4/1), the rule (f : C → E) ↦ (J_{C}, θ, f^{∗}) defines a morphism of moduli problems

j_{N}^{§} : H_{N}^{§} → A_{N}.

However, j_{N}^{§} fails to be injective since no normalization condition was imposed. To remedy this, we might like to impose (if N = n is odd) the normalization condition (1), but this will not work if the cover f_{s} : C_{s} → E_{s} is reducible (i.e. C_{s} is a union of two elliptic curves) because the hyperelliptic involution σ_{C_{s}} now has 7 (in place of 6) fixed points. Instead, we could require that (in all cases) f^{−1}(0_{E_{s}}) ∩ Fix(σ_{C_{s}}) has at least 3 points, and (as will be shown elsewhere) this does lead to a moduli problem H_{n}^{†} which is isomorphic to A_{n} and which contains H_{n} as a natural open subproblem.

4.2 Hurwitz Moduli Stacks

We now analyze the moduli space H_{n}^{∗} which was introduced in Subsection 1.3. It turns out the corresponding moduli problem H_{n}^{∗} can be studied by a formalism which is very similar to that used for the moduli problem H_{n}. Here, however, we have to replace the stack \underline{Ell}_{/R} of elliptic curves over R by the stack (\underline{M}_{0})_{/R} of curves of genus 0 over R. The latter is defined completely analogous to \underline{Ell}_{/R}, i.e. (\underline{M}_{0})_{/R} is the category fibred in groupoids over \underline{Sch}_{/R} whose objects are relative (smooth, proper) curves X → S of genus 0, and whose morphisms are cartesian diagrams.

We shall now interpret H_{n}^{∗} as a moduli problem on (\underline{M}_{0})_{/R}, i.e. as a contravariant functor H_{n}^{∗} : (\underline{M}_{0})_{/R} → \underline{Sets}; here R is any ring in which n! is invertible.

For this, let X/S ∈ ob((\underline{M}_{0})_{/R}) be a relative genus 0 curve, and consider the set

Cov_{X/S,n}^{∗} := {φ : Y → X : φ is a finite cover of degree n whose fibres φ_{t} satisfy (∗) }/ ≃_{X} .

Then the rule H_{n}^{∗}(X/S) = Cov_{X/S,n}^{∗} defines a moduli problem on (\underline{M}_{0})_{/R}, i.e. a functor H_{n}^{∗} : (\underline{M}_{0})_{/R} → \underline{Sets}.

Similar to the case of moduli problems on \underline{Ell}_{/R}, each moduli problem P on (\underline{M}_{0})_{/R} gives rise to a functor P̃ : \underline{Sch}_{/R} → \underline{Sets} by considering isomorphism classes of P-structures. In particular, for P = H_{n}^{∗} we obtain:

H̃_{n}^{∗}(S) = {(Y → X) ∈ Cov_{X/S,n}^{∗} : X/S is a (relative) curve of genus 0}/∼,

where, as before, the indicated equivalence relation is the weak equivalence of covers. Note that if S = Spec(K) is a field, then the set H̃_{n}^{∗}(S) coincides with the set H̃_{n}^{∗}(K) defined in Remark 1.5(a).

Remark 4.7 (a) Although we don’t need this here, we observe that the moduli problem H_{n}^{∗} ≃ H̃_{n}^{∗} is relatively representable, i.e. for each X/S ∈ ob((\underline{M}_{0})_{/R}), the fibre functor (H_{n}^{∗})_{X/S} : \underline{Sch}_{/S} → \underline{Sets} defined by T ↦ H_{n}^{∗}(X_{T} /T) is representable. Indeed, if X = ℙ_{K}^{1}, where K is a field, then the results of Fried-Völklein and Wewers show that this functor is represented by the scheme H^{in}(S_{n}, C) ⊗ K,as was explained in subsection 1.3. For a general scheme X/S, this follows from the results of Wewers [We].

(b) For later usage we observe here that the concept of a moduli problem on (\underline{M}_{0})_{/R} can be reformulated in terms of moduli stacks over (\underline{M}_{0})_{/R}: the latter are categories p : \underline{C} → (\underline{M}_{0})_{/R} over (\underline{M}_{0})_{/R} which are fibred in groupoids. Indeed, given a moduli problem P : (\underline{M}_{0})_{/R} → \underline{Sets}, we can define its classifying stack \underline{P} by the rule that its fibre category over X/S ∈ ob((\underline{M}_{0})_{/R}) is \underline{P}_{X/S} = P(X/S) (where the morphisms are the identities). Conversely, each moduli stack \underline{C} over (\underline{M}_{0})_{/R} gives rise to a moduli problem P\underline{C} on (\underline{M}_{0})_{/R}.

(c) In the case of the moduli problem H_{n}^{∗}, it is immediate that the associated classifying stack is equivalent (in the sense of categories) to the fibred category \underline{H}_{n}^{∗} whose objects are morphisms are triples Y → X → S with X/S ∈ ob((\underline{M}_{0})_{/R}) and Y/X ∈ H_{n}^{∗}(X/S), and whose morphisms are (extended) cartesian diagrams. Thus, by the discussion of Subsection 1.3, we see that we can identify \underline{H}∗_{n} with the stack H_{0,0,m} of [BR], Theorem 6.22, for a suitable monodromy m = (G, H, ξ). (More precisely, G = S_{n} and H ≃ S_{n−1} is the stabilizer of an element of the set {1, …, n} (on which S_{n} acts), and ξ is the Hurwitz data given by the class C.)

It is clear that the moduli problem H_{n}^{∗} cannot be representable because the group scheme Aut(ℙ^{1}) acts non-trivially. However, we have:

Proposition 4.8 The moduli problem H_{n}^{∗} has a coarse moduli space which is given by

M(H_{n}^{∗}) = H_{n}^{∗} = H^{in}(S_{n}, C)/Aut(ℙ^{1}).

Proof. This follows easily from [BR]. Indeed, via identification \underline{H}_{n}^{∗} = H_{0,0,m} of Remark 4.7(c) it follows from [BR], Théorème 6.22 that \underline{H}_{n}^{∗} ≃ H_{g,S_{n},ξ}/Δ(m) for a suitable integer g. But since here N_{G}(H) = H, we see that Δ(m) = 1, and so \underline{H}_{n}^{∗} ≃ H_{g,S_{n},ξ}. Now by formula (6.17) of [BR] we know that H_{g,S_{n},ξ} ≃ FH_{g,S_{n},ξ}/Aut(ℙ^{1}). Since the results of Fried/Völklein/Wewers[We] show that the Fried-Völklein stack FH_{g,S_{n},ξ} is representable by H^{in}(S_{n}, C), the assertion about the coarse moduli spaces follows from [BR], Théorème 6.3 (3).

4.3 The Main Result

We now want to relate the moduli problem H′_{n} (on \underline{Ell}_{/R}) to the moduli problem H_{n}^{∗} (on (\underline{M}_{0})_{/R}). In order to be able to compare them, we shall pass to their classifying stacks \underline{H}′_{n} and \underline{H}_{n}^{∗} (over \underline{Ell} and over \underline{M}_{0}, respectively), and view both of these as fibred categories over \underline{Sch}_{/R}; cf. Remark 4.7(b). We then have:

Theorem 4.9 The rule (C →^{f} E) ↦ (C/⟨ω_{C}⟩ → E/⟨[−1]⟩) defines a functor q = q_{n} : \underline{H}′_{n} → \underline{H}_{n}^{∗}, and the induced map

M(q) : H′_{n} = M(H′_{n}) → H_{n}^{∗} = M(H_{n}^{∗})

on the coarse moduli schemes is surjective and radical. Thus, if R = K is a field of characteristic 0, then M(q) is an isomorphism and hence H′_{n} ≃ H_{n}^{∗} is an irreducible, normal affine surface.

Proof. Let (C →^{f} E) ∈ ob(\underline{H}′_{n}), where C and E are S-curves. Then, since f is normalized, we have fω_{C} = [−1]f, and so we have an induced morphism φ : Y := C/⟨ω_{C}⟩ → X := E/⟨[−1]⟩. (The quotients exist and are smooth S-curves by Lønsted-Kleiman[LK], Theorem 4.12.) Moreover, if s ∈ S, then Disc(f_{s}) is reduced (by hypothesis), and by Proposition 2.4 we see that φ_{s} is a cover of type (∗), which means that (Y →^{φ} X) ∈ ob(\underline{H}_{n}^{∗}). Since this construction is compatible with base-change (because 2 is invertible in S), we see that this defines the desired functor q.

The functor q induces a natural transformation q̃ : H̃′_{n} → H̃_{n}^{∗} between the associated functors on \underline{Sch}_{/R}. Thus, by the universal property of coarse moduli spaces, there is unique morphism M(q) = M(q̃) : H′_{n} = M(H′_{n}) → H_{n}^{∗} = M(H_{n}^{∗}) such that μ′q̃ = h_{M(q)}μ^{∗}, where μ′ : H_{n}^{∗} → h_{M(H′_{n})} and μ^{∗} : H_{n}^{∗} → h_{M(H_{n}^{∗})} are the natural transformations associated to the coarse moduli spaces. (Recall that Propositions 4.5 and 4.8 show that we have the identifications M(H′_{n}) = H′_{n} and M(H_{n}^{∗}) = H_{n}^{∗}.)

Now the discussion of Subsection 2.2 shows: Claim: If S = Spec(k), where k is an algebraically closed field, then

q̃_{S} : H̃′_{n}(S) → H̃_{n}^{∗}(S)

is a bijection.

Thus, since μ′_{S} : H̃′_{n}(S) →^{∼} h_{M(H′_{n})}(S) = H′_{n}(k) and μ_{S}^{∗} : H̃_{n}^{∗}(S) →^{∼}

H̃_{n}^{∗}(S) = H_{n}^{∗}(k) are bijections, it follows that M(q)_{k} : H′_{n}(k) → H_{n}^{∗}(k) is a bijection. Thus, M(q) is surjective and radical; cf. [EGA], (I,3.6.3) and (I,3.7.1). Moreover, if R = K is a field of characteristic 0, then H′_{n} is normal and irreducible (cf. Proposition 4.5), and so we see that H_{n}^{∗} is also irreducible, and that hence M(q) is a birational map. In addition, M(q) is separated because H′_{n} is quasi-affine and H_{n}^{∗} is affine. Thus, since H_{n}^{∗} is normal (cf. Remark 1.5(b)), it follows from Zariski’s Main Theorem that M(q) is a local isomorphism; cf. [EGA], (Err_{IV}, 30). By [EGA], (I, 4.4.8) we thus have that M(q) is an open immersion, and hence an isomorphism since M(q) is surjective.

Remark 4.10 (a) Even though M(q) is an isomorphism, the functor

q : \underline{H}′_{n} → \underline{H}_{n}^{∗}

is not an isomorphism (of stacks). Indeed, q cannot be an equivalence (of categories) because of the presence of twists of elliptic curves. More precisely, if K ⊃ R is any field for which K^{∗}/(K^{∗})^{2} ≠ 1, then the discussion of Subsection 2.2 shows that q̃_{K} : H̃′_{n}(K) → H̃_{n}^{∗}(K) cannot be injective, and so in particular q cannot be an equivalence of categories.

(b) In his preprint, Fried[Fri] proves the irreducibility of H^{in}(S_{n}, C) by the method of Nielsen classes. This gives a different proof of the irreducibility of H_{n}^{∗}.

4.4 Connection with Humbert surfaces

If H is any Hurwitz scheme which classifies curve covers f : C → C_{1} with fixed genus g = g_{C}, then the forget map (f : C → C_{1}) ↦ C induces a natural morphism

μ_{H} : H → M_{g}

to the moduli space M_{g} which classifies isomorphism classes of genus g curves. The image μ_{H}(H) is thus a subscheme of M_{g} which frequently carries interesting information. For example, if H = H_{d,g} is the Hurwitz space of simple genus g covers of degree d of ℙ^{1}, then the μ_{H}(H_{d,g})’s (d ≥ 2) define cycles on M_{g} which are important for the study of the geometry of M_{g}; cf. [HM], p. 32, 175, for more detail.

Specializing this to our case H = H_{N} which classifies normalized genus 2 covers, we thus obtain a morphism

μ_{N} = μ_{H_{N}} : H_{N} → M_{2}.

It turns out that the image of μ_{N} is essentially a Humbert surface. To make this more precise, recall (cf. [Hum] or [vdG], section IX.2) that for each positive integer D ≡ 0, 1(4), there is an irreducible surface Hum_{D} contained in the space A_{2} of principally polarized abelian surfaces; Hum_{D} is called the Humbert surface with Humbert invariant (or discriminant) D.

Since each point of μ_{N}(H_{N}) corresponds to a genus 2 curve with an elliptic differential, it follows from Humbert[Hum], Théorème 15, ^{3} that μ_{N}(H_{N}) is contained in some Humbert surface Hum_{D} with square invariant D = d^{2}; here we view M_{2} as an open subset of A_{2} via the (birational) map M_{2} → A_{2} (which take a curve to its polarized Jacobian). More precisely:

Proposition 4.11 We have μ_{N}(H_{N}) = Hum_{N^{2}} ∩ M_{2}. More precisely, if τ denotes the involution of Z_{N,−1} induced by the automorphism of X(N) × X(N) which interchanges the two factors, then μ_{N} factors over the quotient H_{N}^{sym} := H_{N}/⟨τ⟩, and the induced map

3In fact, this result was already proven by Biermann in 1883 (cf. [Kr], p. 485), long before the discovery of Humbert surfaces.

μ_{N}^{sym} : H_{N}^{sym} → Hum_{N^{2}} ∩ M_{2}

is the normalization map of the Humbert surface Hum_{N^{2}} ∩ M_{2}.

Proof. The first assertion follows from Theorem 1.9 of [Ka2] (together with the fact that every minimal cover can be normalized; cf. [Ka5], Proposition 2.2). The second assertion follows from Corollary 1.8 of [Ka2] because H_{N} is precisely the (open) part of Z_{N,−1} whose points correspond to genus 2 curve covers; cf. Corollary 4.3, together with the discussion of the next section.

Remark 4.12 If D is not a square, then the Humbert surface H_{D} is birationally isomorphic to the symmetric Hilbert modular surface Hilb_{D}^{sym}; the latter is the quotient of the Hilbert modular surface Hilb_{D} = SL_{2}(𝔒_{D})\(ℌ×ℌ) by the involution τ induced by the involution of ℌ × ℌ which interchanges the two factors; cf. [vdG], Proposition (IX.2.5). Thus, the Corollary 1.8 of [Ka2] may be viewed as an extension of this fact to square discriminants by viewing the modular diagonal quotient surfaces Z_{N,ε} as “degenerate Hilbert modular surfaces”. This is the point of view taken by Hermann, who investigated the geometry of Z_{N,ε} and of Z_{N,ε}^{sym} = Z_{N,ε}/⟨τ⟩ as analogues of Hilbert modular surfaces; cf. [He1], [He2].

This point of view is also adopted by McMullen in his papers. Indeed, he uses a more general definition of Hilbert modular surfaces Hilb_{D} in which also square discriminants D are allowed, and then shows in all cases that Hum_{D} is birationally isomorphic to Hilb_{D}^{sym}; cf. [Mc2], Theorem 4.5.

In the case that D = N^{2}, this result may be stated as follows: if C/\overline{K} is a curve of genus 2 with Jacobian J_{C}, then

⟨C⟩ ∈ Hum_{N^{2}}(\overline{K}) ⇔ End(J_{C}) has real multiplication by 𝔒_{N^{2}};

the latter means that we have an embedding ρ : 𝔒_{N^{2}} := ℤ[x]/(x^{2} − Nx) ↪ End(J_{C}) which is primitive, i.e. ρ(𝔒_{N^{2}})ℚ ∩ End(J_{C}) = ρ(𝔒_{N^{2}}). [Indeed, if ⟨C⟩ ∈ Hum_{N^{2}}(\overline{K}), then by Proposition 4.11 we have a (normalized) subcover f : C → E of degree N to some elliptic curve E, and then the rule x ↦ f^{∗}f_{∗} induces such an embedding. Conversely, if we have such an embedding, then ε = \frac{1}{N} ρ(x) ∈ End^{0}(J_{C}) is a non-trivial idempotent, and one concludes (using Lange[Lan]) that ρ(x) = Nε = f^{∗}f_{∗}, for a suitable minimal cover f : C → E.]

5 The Compactification of H′_{n} and of H_{n}^{∗} and Related Boundary Curves

5.1 Compactifications: an Overview

As the constructions of the previous section show, the moduli spaces H′_{n} and H_{n}^{∗} are not compact. It is thus of interest to construct natural compactifications of these spaces and to investigate whether or not the boundary components have a modular interpretation in terms of covers of curves.

Since H′_{n} was constructed as an open subset of the affine surface Z_{n,−1}, the natural compactification \overline{Z}_{n,−1} of Z_{n,−1} (which was studied in [KS]) also serves as a compactification of H′_{n}. While this compactification has the advantage of being very explicit, its disadvantage is that it does not readily lead to a modular interpretation in terms of covers. However, such an interpretation can be obtained by studying the fibres of the canonical compactification C of the universal cover C → E_{H} over H = H_{E/K,n} for each fixed elliptic curve E/K; cf. Subsection 3.3.

More precisely, we saw in Theorem 3.9 that the study of the fibres of C lead to the following four types of covers f_{x} : C_{x} → E of E:

I. C_{x} is a smooth curve of genus 2 and Diff(f_{x}) is reduced;

II. C_{x} is smooth curve of genus 2 and Diff(f_{x}) = 2P with P ∈ E[2];

III. C_{x} is the union of two elliptic curves meeting at a point;

IV. C_{x} is a singular irreducible curve of arithmetic genus 2.

These cases correspond to the stratification

\overline{X(N)} = H′_{E/K,N} ˙∪ H_{E/K,N}^{∂} ˙∪ S_{1} ˙∪ S_{0}

which was constructed in Section 3. By varying E, we should expect that these induce a similar stratification for \overline{Z}_{N,−1}. This, however, is not quite correct since we also have to allow the case that E degenerates. Thus, we have a fifth case:

V. C → E is a cover of a stable curve E of arithmetic genus 1.

The precise nature of the covers of type V will be discussed in the next section.

On the other hand, the theory of Wewers [We] gives a recipe for an (abstract) compactification of H^{in}(S_{n}, C) in terms of covers of ℙ^{1}; here the boundary components correspond to covers in which the monodromy group, the ramification type and the curve are (suitable) degenerations the generic situation. This, therefore, also gives a compactification of H_{n}^{∗} = H^{in}(S_{n}, C)/Aut(ℙ^{1}) ≃ H′_{n}, and so one might expect that these boundary components match up with those of H′_{n} defined by the first method. This is indeed the case. Thus, in both interpretations the boundary curves of the Hurwitz spaces correspond to interesting degenerations of covers.

5.2 The Boundary Curves of H′_{n}

In the case of the Hurwitz space H′_{n} ⊂ Z_{n,−1} which parameterizes normalized genus 2 covers of elliptic curves, the above overview (§5.1) shows that boundary curves naturally split into four main types. As we shall see, these correspond to the stratification induced by the inclusions

H′_{n} ⊂ H_{n} ⊂ Z_{n,−1} ⊂ Z̄_{n,−1}^{(1)} ⊂ \overline{Z}_{n,−1}.

Here Z̄_{n,−1}^{(1)} = ψ_{1}^{−1}(X(1)) is the inverse image of X(1) ≃ 𝔸^{1} of the morphism

ψ_{1} = pr_{1} ∘ ψ : \overline{Z}_{n,−1} → \overline{X(1)} × \overline{X(1)} → \overline{X(1)} ≃ ℙ^{1},

where, as in [KS], ψ : \overline{Z}_{n,−1} → \overline{X(1)} × \overline{X(1)} is the map induced by the inclusion Δ_{−1}(G_{n}) ≤ G_{n} × G_{n}. Note that the fibres of ψ_{1} are geometrically irreducible (but not necessarily reduced); cf. [KS], p. 353.

We have two tasks: (i) to interpret (if possible) these boundary curves as the (coarse) moduli curves of suitable Hurwitz functors; and (ii) to identify the components of the boundary curves and to study their geometric properties.

Type I: the generic case (the points in H′_{n}): By the modular description of H′_{n} mentioned in subsection 4.1, the points of H′_{n} classify genus 2 covers of type I; i.e. normalized genus 2 covers f : C → E of some elliptic curve E with reduced discriminant; cf. Corollary 4.3. We also know that H′_{n} is an open subset of the irreducible projective normal surface \overline{Z}_{N,−1}.

Type II: The points in ∂_{II} := H_{n} \ H′_{n}.

Since H_{n} is a coarse moduli scheme for the moduli problem H_{n} (cf. Corollary 4.3), we would expect that ∂_{II} is the coarse moduli space for the (closed) subproblem ∂H_{n} of H_{n} which classifies normalized genus 2 covers with nonreduced discriminants. This is almost correct. It is easy to see (using Theorem 3.1 and Proposition 3.4) that ∂H_{n} is relatively representable and closed in H_{n}. Thus, we have a coarse moduli space M(∂H_{n}) (of dimension 1) and a canonical morphism M(∂H_{n}) → M(H_{n}) = H_{n} whose image is ∂_{II}. However, without further work we only know that this map is radical, but not necessarily that it is a closed immersion (or that M(∂H_{n}) is reduced).

Recall from subsection 3.2 that the covers of Type II break up into four subcases, which we shall label here as types II_{1}, …, II_{4}. (Thus, for i = 1, …, 4, type II_{i} corresponds to case i of subsection 3.2.) These give rise to subproblems ∂H_{n}^{(i)} of H_{n}, and the corresponding coarse moduli schemes induce curves ∂_{II_{i}} = ∂H_{n}^{(i)} on H_{n}. One might expect these curves to be irreducible, and to have large genus for large n.

Remark 5.1 The work of [Mc1] and [HL1] (and others) shows that this expectation is correct for the curves ∂_{II_{2}} and ∂_{II_{4}}. More precisely, they show that their (birational) images in M_{2} with respect to μ_{n} : H_{n} → M_{2} (as defined in Subsection 4.4) can be interpreted as Teichmüller curves. This means in particular that for i = 0, 1 there exist subgroups Γ_{i,n} ≤ SL_{2}(ℤ) of finite index and a birational map

Γ_{i,n}\ℌ → μ_{n}(∂H_{n}^{(2+2i)}) ⊂ M_{2}, for i = 0, 1.

Here i is the value of the spin as defined in [Mc1]. As Hubert/Lelièvre show, Γ_{i,n} has index d_{i,n} = \frac{3}{16n}(n − 1 − 2i)sl(n) in SL_{2}(ℤ), at least if n is prime; note that the case i = 0 corresponds to their type A. Furthermore, they show that the genus of both curves is asymptotic to \frac{1}{64}n^{3} as n → ∞; cf. [HL1], Theorem 1.2. (Moreover, one can show, using the results of [Mc1], that their genus is ≥ 3 for n ≥ 11, n prime.) It is interesting to note that in general Γ_{i,n} is not a congruence subgroup; cf. [HL2].

Type III: The points in ∂_{III} := Z_{n,−1} \ H_{n}. By the discussion at the end of Subsection 4.1 we know that Z_{n,−1} is a coarse moduli space for the functor A_{n} and so the points in ∂_{III} correspond to singular (stable) genus 2 curves whose Jacobian is smooth, i.e. to curves C that are the union of two elliptic curves meeting in a single point. In [Ka3], Theorem 2.3, the set ∂_{III} was interpreted in terms of conditions involving the functor Z_{N,−1}, i.e. in terms of triples (E_{1}, E_{2}, ψ) ∈ Z_{N,−1}(K). Roughly speaking, the condition is that ψ should be induced by an isogeny h : E_{1} → E_{2} of a specific type. In the next section we will see in general that the set of ψ’s induced by an isogeny lie on so-called Hecke curves T_{m,k} ⊂ Z_{n,−1}, and that T_{m,k} is birationally equivalent to X_{0}(m), provided that mk^{2} ≡ −1 (mod n); cf. Subsection 6.2 below. Reinterpreting [Ka3], Corollary 2.4 and Remark 2.5 in terms of Hecke curves, we obtain:

Proposition 5.2 If m = s(n − s)/t^{2} and k ≡ ts^{∗} (n), where 1 ≤ s ≤ n − 1, (s, n) = 1, and s^{∗}s ≡ 1 (n) then T_{m,k} is an irreducible component of ∂_{III}. Furthermore, if n = p is prime, then every component of ∂_{III} has this form, and hence

∂_{III}(H_{p}) = ⋃_{s=1}^{\frac{p−1}{2}} ⋃_{t^{2}|s(p−s)} T_{\frac{s(p−s)}{t^{2}} ,ts^{∗}}.

Type IV: The points in ∂_{IV} := Z̄_{n,−1}^{(1)} \ Z_{n,−1}. We first work out the geometric structure of ∂_{IV}. For this we note that since \overline{X(1)} = X(1) ∪ \{P_{∞}\}, we have (\overline{X(1)} × \overline{X(1)}) \ (X(1) × X(1)) = P_{∞} × \overline{X(1)} ∪ \overline{X(1)} × P_{∞} = pr_{1}^{−1}(P_{∞}) ∪ pr_{2}^{−1}(P_{∞}), and so

\overline{Z}_{n,−1} \ Z_{n,−1} = ψ^{−1}(pr_{1}^{−1}(P_{∞}) ∪ pr_{2}^{−1}(P_{∞}))_{red} = ψ_{1}^{−1}(P_{∞})_{red} ∪ ψ_{2}^{−1}(P_{∞})_{red},

where ψ and ψ_{i} are as defined at the beginning of this subsection. By the discussion preceding Proposition 2.5 of [KS] we know that C_{∞,i} := ψ_{i}^{−1}(P_{∞})_{red} is an irreducible curve isomorphic to X_{1}(n). We thus see that

\overline{Z}_{n,−1} \ Z_{n,−1}^{(1)} = ψ_{1}^{−1}(P_{∞})_{red} = C_{∞,1},

and that hence

∂_{IV} = Z_{n,−1}^{(1)} \ Z_{n,−1} = C_{∞,2} \ C_{∞,1}

is an open subset of C_{∞,2} ≃ X_{1}(n).

In order to see how the points of ∂_{IV} give rise to curve covers, we first note that the surface \overline{Z}_{n,−1} has the following modular interpretation (as is shown in [Ka7]): it is the coarse moduli space of the functor \overline{Z}_{n,−1} which classifies triples (E_{1}, E_{2}, ψ) where E_{1} and E_{2} are generalized elliptic curves “of type n” over a scheme S (i.e. the fibres of E_{i}/S are either smooth elliptic curves or Néron polygons of length n) and ψ : E_{1}[n] → E_{2}[n] is an isomorphism of group schemes of determinant −1.

Via this interpretation, the points of ∂_{IV} (\overline{K}) can be identified with triples (E_{1}, E_{2}, ψ) where E_{1}/\overline{K} is an elliptic curve and E_{2}/\overline{K} is the Néron polygon of length n. By an extension of the “basic construction” (cf. section 2.3) we can construct a stable curve C_{ψ} on J_{ψ} = (E_{1} × E_{2})/Graph(ψ) together with morphism f : C_{ψ} → E_{1} (induced by the projection onto the first factor). Note that the Classification Theorem 3.9 shows that C_{ψ} is an irreducible singular curve whose normalization is an elliptic curve and which has a unique singular point.

Type V: The points in ∂_{V} := \overline{Z}_{n,−1} \ Z̄_{n,−1}^{(1)}. By definition, ∂_{V} = ψ_{1}^{−1}(P_{∞})_{red} = C_{∞,1} ≃ X_{1}(n). For purposes of discussing the associated covers, it is useful to write

∂_{V} = ∂_{V}^{∗} ˙∪ ∂_{V I}, where ∂_{V I} = C_{∞,1} ∩ C_{∞,2}.

Via the above modular description (see type IV), the points of ∂_{V}^{∗} (\overline{K}) (respectively, of ∂_{V I}) correspond to triples (E_{1}, E_{2}, ψ) where E_{1}/\overline{K} is the Néron polygon of length n and E_{2}/\overline{K} is an elliptic curve (respectively, E_{2}/\overline{K} is the Néron polygon of length n).

If (E_{1}, E_{2}, ψ) ∈ ∂_{V}^{∗} (\overline{K}), then by the same method as for type IV we obtain an irreducible singular curve C_{ψ} on J_{ψ}. Here, however, the projection onto the first factor only induces a rational map f_{0} : C_{ψ} ⋯ → E_{1}. Nevertheless, if we compose f_{0} with the map c_{n} : E_{1} → Ē_{1} which contracts all components of E_{1} (and hence Ē_{1} is the irreducible Néron polygon of length 1), then f = c_{n} ∘ f_{0} : C_{ψ} → Ē_{1} is a morphism and hence defines a cover of degree n.

If (E_{1}, E_{2}, ψ) ∈ ∂_{V I}, then the situation is more complicated (and has not been fully worked out). Here we need to look at stable curves C_{ψ} (of genus 2) whose Jacobian is totally degenerate; thus, C_{ψ} is the reduction of a suitable Mumford curve (of genus 2). In addition, we should have (as above) a morphism f : C_{ψ} → Ē_{1} of degree n.

Note that ∂_{V I} is a finite set; in fact, one can show that its cardinality is #∂_{V I}(\overline{K}) = \frac{ϕ(n)}{2}, where ϕ is the Euler totient function. It is interesting to observe that this count agrees with the intersection formula (19) of [KS].

5.3 The Boundary Curves of H_{n}^{∗}

According to Wewers[We], p. 65, the space H_{r}^{N}(S_{n}) ⊃ H^{in}(S_{n}, C) has a natural compactification \overline{H}_{r}^{N}(S_{n}) whose boundary components classify “admissible m-covers” (as defined on p. 60 of [We]). As Wewers shows on p. 68ff, these components can be further analyzed in terms of “degeneration structures” of conjugacy classes.

Here we will not directly follow this description; instead we will only use it as a motivation for the ideas leading to the compactification of H^{in}(S_{n}, C) given below. In particular, we shall use the following two key ideas for the construction of the compactification: 1) We need to allow covers of stable curves (which are degenerations of smooth curves) and 2) we need to consider other ramification data (which is obtained by degeneration of the generic data).

In our case, this means that we need to analyze the different types of admissible covers (and their ramification data) that can arise as a degeneration of covers φ : ℙ^{1} → ℙ^{1} of type (∗).

For this, let us first consider the degenerations of (normalized) covers f : C → E. Then we have the following possibilities:

1) E and C are smooth, but the ramification data “degenerates”;

2) E is smooth and C is a stable reducible curve (of arithmetic genus 2);

3) E is smooth and C is a stable singular irreducible curve;

4) E is a stable singular curve of arithmetic genus 1.

Note that the above cases correspond to the boundary components of the compactification of H′_{n} of Subsection 5.2; indeed, the cases 1) to 4) correspond to the boundary components ∂_{II} to ∂_{V}, respectively.

Each degenerate cover f : C → E gives rise to a cover φ : \overline{C} := C/⟨ω_{C}⟩ → ℙ^{1} because f is still quasi-normalized. (More precisely, C and E come equipped with involutions ω_{C} and [−1]_{E}, respectively, such that [−1]_{E}f = fω_{C}.)

Now one has to analyze the ramification structure of the cover φ. We shall restrict ourselves to the cases 1) - 3). So the base E is an elliptic curve, and the branch points P_{1}, ..., P_{4} remain distinct (because they are also the branch points of π_{E} : E → ℙ^{1}). Thus, in these cases P_{5} will move into one of P_{1}, ..., P_{4}, and because of symmetry, it is enough to look at the case that P_{5} moves to P_{4} or to P_{3}.

To see how the associated tuple σ changes if we coalesce two branch points P, P′ of the cover φ, we choose a homotopy basis of ℙ^{1} minus the branch points in the following way: Take a small disc D containing P and P′, but none of the other branch points, and choose the loops around P and P′ such that they agree outside of D and their product is homotopic (in ℙ^{1} minus the branch points) to a loop that winds once around D and agrees with the other two loops outside of D. If (σ_{1}, ..., σ_{r}) is the tuple associated with φ relative to this homotopy basis, and we coalesce the last two branch points, then the resulting cover is associated with the tuple (σ_{1}, ...σ_{r−2}, σ_{r−1}σ_{r}). In other words, coalescing the last two branch points means replacing the last two entries of the tuple by their product.

Since we are interested in the cases that P_{5} moves either to P_{3} or P_{4} we put τ = σ_{3}σ_{5} and ρ = σ_{4}σ_{5}. Define H := ⟨σ_{1}, σ_{2}, σ_{3}⟩ = ⟨σ_{1}, σ_{2}, σ_{3}, ρ⟩ and F = ⟨σ_{1}, σ_{2}, σ_{4}⟩ = ⟨σ_{1}, σ_{2}, τ⟩.

Case 1: C is smooth. The ramification types occurring in this case were already considered above and are listed in the table of Proposition 3.6.

Case 2: C is reducible. A very interesting case for arithmetical investigations is that C is reducible ([FK]). It is the union of two isogenous elliptic curves Ê_{1} and Ê_{2} linked at one point.

The cover f : C → E of degree n induces two finite covers f_{i} = f_{|Ê_{i}} : Ê_{i} → E of degree n_{i} with n_{1} + n_{2} = n. We can assume that n_{1} is even and n_{2} is odd.

This case is classified in the world of Galois representations on torsion points of elliptic curves in [Ka4]: the isomorphism α on points of order n is induced by restriction of this isogeny and satisfies an additional condition.

Here we give a classification in terms of degeneration of ramification types. One possible ramification type of τ is (4)^{1}(1)^{1}(2)^{(n−3)/2}. In this case

F = N⟨σ_{1}⟩ where N is abelian of order n and σ_{1} acts by inversion on N. F has exactly two orbits of length n_{1} and n_{2} with n_{1} + n_{2} = n.

One possible ramification type of ρ is (1)^{5}(2)^{(n−5)/2}. Then H is not a dihedral group, has two orbits of length n_{1} and n_{2} and necessarily 4 | n_{2}.

Case 3: C is singular and irreducible. Here C is obtained from a genus one curve Ê by identifying two points. The induced cover f̃: Ê → E is unramified by the Riemann-Hurwitz formula. Hence it is an isogeny of degree n for suitable choice of the zero points of E and Ê. The two points of Ê that coincide in C are then n-division points. This situation is wellunderstood, and the corresponding boundary curve of the diagonal surface is a curve which is an open subset of X_{1}(n), as we saw above in the discussion of type IV.

The characterization in terms of covers of ℙ^{1} is given as follows: P_{5} coalesces with P_{4}, and the ramification type of ρ is (1)^{1}(2)^{(n−1)/2}. The monodromy group H = D_{2n} is the dihedral group naturally embedded into S_{n}.

By using group theory and Riemann’s existence theorem, Völklein[V2] proves the following.

Proposition 5.3 (Völklein) The above list covers all possibilities for the cycle types of ρ and τ. All these cases do actually occur, for any n, n_{1}, n_{2} satisfying the given conditions.

6 Rational Points Related to Isogenies

As we have seen in Section 4, rational points on the Hurwitz space H_{n}^{∗} correspond to rational points on H′_{n}. Because of the large rigidity number it may be preferable to use the modular interpretation of such points as corresponding to pairs of elliptic curves with isomorphic level-n-structures and then to exploit arithmetical properties of Z_{n,−1} rather than to use group theoretical methods. At the same time this will shed some light to the conjectures in Subsection 1.2. On the other hand, the interpretation of points by covers will give information about the questions asked in Subsection 1.1 (see Section 7).

6.1 Construction of Points by Isogenies

By the above-mentioned modular interpretation we know that if E and E′ are elliptic curves over K and if ψ : E[n] → E′[n] is a G_{K}-isomorphism which is an anti-isometry with respect to the Weil pairings, then the triple (E, E′, ψ) determines a K-rational point on Z_{n,−1}. By imposing further restrictions, one can ensure that this point lies on the open subset H′_{n} ⊂ Z_{n,−1}.

One way to construct such G_{K}-isomorphisms is by considering K-isoge-nies of elliptic curves. Thus, we assume in the following that there exists a cyclic K-isogeny

η : E → E′.

Its degree will be denoted by d = deg(η). To avoid trivial cases we always assume that d > 1 and that η is separable.

Let α_{n} denote the restriction of η to E[n]. We then get G_{K}-isomorphisms z · α_{n} from E[n] to E′[n] for all n prime to d and z ∈ ℤ prime to n.

Of course, there may be other G_{K}-isomorphisms between E[n] and E′[n]. We call the triples (E, E′, z · α_{n}) “generic” because of the following evident observation.

Lemma 6.1 Assume that the centralizer of G_{K} in Aut(E_{n}) is ℤ · id_{E[n]} and that n is prime to d. Then every G_{K}-isomorphism between E[n] and E′[n] is of the form z · α_{n} with z ∈ ℤ prime to d. Moreover, if E has no complex multiplication, then there is a number c (depending on E and K) such that this property holds for every n with (n, c) = 1.

Proposition 6.2 For all n prime to d and all z ∈ ℤ prime to n the abelian variety J_{z,n} := (E × E′)/Graph(z · α_{n}) is isomorphic to E × E′.

Proof (cf. [DiFr]). The isogeny

Φ : E × E′ ⟶ E × E′

given by the matrix ( \begin{matrix}n ∘ id_{E} & 0 \\ −z · η & id_{E′} \end{matrix} ) has kernel Graph(z · α_{n}).

In order to get points on H_{n} via the “basic construction” we need two additional properties:

Firstly, the graph of ψ = z · α_{n} has to be isotropic with respect to the Weil pairing and secondly, one has to verify that the resulting curve C_{ψ} is irreducible.

The first condition means that

deg(z · η) = z^{2} · d ≡ −1 mod n.

The second condition was analyzed in [Ka3], Theorem 2.3. In the special case n = p is prime, this criterion states that C_{ψ} is irreducible if and only if there is no isogeny θ : E → E′ of degree k(p − k) with 1 ≤ k ≤ p − 1 such that θ_{|E[n]} = kψ (= kzα_{n}); cf. [Ka3], Theorem 3.

We state the result in the simplest case (cf. [Fr1]).

Proposition 6.3 Assume that d and n are distinct odd primes. In addition, assume that E has no complex multiplication.^{4} Then there is an element z ∈ ℤ such that z · α_{n} induces a covering C → E of degree n if and only if n is split into two non-principal prime ideals in ℤ[\sqrt{−d}].

Remark 6.4 (a) There is a similar (but more complicated) criterion in the case of a general d; cf. [Fr1], Proposition 3.1.

(b) If d is odd and if E′ ≄ E, then there always exists a cover C → E of degree 2 with J_{C} ≃ E × E′; cf. [DiFr]. In particular, E × E′ is the Jacobian of a curve of genus 2.

(c) On the other hand, if d = 1, 2, 4, 6, 10, 12, 18, 22, 28, 30, 42, 58, 60, 70, 78, 102, 130, 190, 210, 330 or 462 and E does not have CM, then there is no (irreducible) curve C/\overline{K} such that J_{C} ≃ E × E′; cf. [Ka8]. In fact, the above list of values of d are conjectured to be all the values for which no such curve exists. As is explained in [Ka8], this would follow from an old Conjecture of Gauss on the structure of class groups of imaginary quadratic fields (which known to be true under the Generalized Riemann Hypothesis).

The following result may be viewed as a converse to Proposition 6.2.

Proposition 6.5 Let C/K be a curve with Jacobian J_{C} ≃ E × E′. If Hom_{K}(E, E′) = Hom\overline{K}(E, E′), then there is a cyclic isogeny η : E → E′, and integers n ≥ 2 and z with deg(η)z^{2} ≡ −1(mod n) such that C ≃ C_{ψ}, where ψ = zη_{|E[n]}. Thus, there is a minimal cover f : C → E of degree n. such that f_{∗} ∘ α = pr_{E}, for a suitable isomorphism α : E × E′ → J_{C}.

Proof (cf. [Ka8]). Fix an isomorphism α : A := E × E′ →^{∼} J_{C} and let λ′ = α̂λ_{C}α : A →^{∼} Â be the principal polarization induced by λ_{C} : J_{C} →^{∼} Ĵ_{C}. The given hypothesis implies that the Néron-Severi group NS(A ⊗ \overline{K}) of A ⊗ \overline{K} is K-rational, i.e. NS(A ⊗ \overline{K}) = NS(A), and so there exists a divisor D ∈ Div(A) which defines the polarization λ′, i.e. we have λ′ = ϕ_{D}. By Riemann-Roch we can choose D to be effective, and then we have that C ⊗ \overline{K} ≃ D × \overline{K}. By a Galois descent argument similar to that for Torelli’s theorem (cf. Milne[Mi], p. 203) one shows that in fact C ≃ D. Thus, we have an embedding j : C → J_{C} such that j(C) = α(D).

Now every D ∈ Div(A) has the form D ≡ aθ_{1} + bθ_{2} + Γ_{h}, where a, b ∈ ℤ and h ∈ Hom(E, E′); here θ_{i} = pr_{i}^{∗}(0_{E_{i}}) (where E_{1} = E, E_{2} = E′), and Γ_{h} denotes the graph of h ∈ Hom(E, E′). Since D defines a principal polarization (so D is ample and D^{2} = 2), it follows that there are positive integers n and r such that b = n − 1, a = r − deg(h) and nr − deg(h) = 1. Thus, deg(h) ≠ 0 for else D ≡ θ_{1} + θ_{2}, which is impossible since D is is irreducible. Thus, h = kη, for some cyclic isogeny η : E → E′ and k ≠ 0. We thus have

4For the case that E has CM see [Fr1].

(20) D ≡ (r − k^{2}d)θ_{1} + (n − 1)θ_{2} + Γ_{kη}, where d = deg(η), rn − k^{2}d = 1.

Let ψ = −kη_{|E[n]} : E[n] → E′[n]. Since d(−k)^{2} ≡ −1 (mod n), we see that ψ is an anti-isometry. Let Φ : A → A be as defined in the proof of Proposition 6.2 (with z = −k). We thus obtain an isomorphism β : J_{ψ} →^{∼} A such that βπ_{ψ} = Φ. Moreover, one checks that Φ^{∗}(D) ≡ n^{2}(θ_{1} + θ_{2}), and so C_{ψ} ≡ β^{∗}D ≡ β^{∗}α^{∗}(j(C)).

Now the basic construction gives a normalized cover f_{ψ} : C_{ψ} → E of degree n such that (f_{ψ})_{∗}∘π_{ψ} = [n]_{E} ∘pr_{E}. Put γ = (f_{ψ})_{∗}∘β^{−1}∘α^{−1} : J_{C} → E, and let f = γ ∘ j : C → E. Then f_{∗} = γ and deg(f) = deg(f_{ψ}) = n because β^{∗}α^{∗}j(C) ≡ C_{ψ}.

From the definition of Φ we see that pr_{E} ∘ Φ = [n] ∘ pr_{E} = (f_{ψ})_{∗} ∘ π_{ψ} = f_{∗} ∘ α ∘ β ∘ π_{ψ} = f_{∗} ∘ α ∘ Φ, and hence f_{∗} ∘ α = pr_{E} because Φ is an isogeny.

Corollary 6.6 In the above situation put d = deg(η) and r = \frac{dz^{2}+1}{n}. Then for any pair of integers x, y ∈ ℤ with (dx, y) = 1, there is a minimal elliptic subcover f_{x,y} : C → E such that

(f_{x,y})_{∗} ∘ α = ypr_{E} + xη̂pr_{E′} and deg(f_{x,y}) = drx^{2} − 2dzxy + ny^{2}, (21)

and hence there exist infinitely many minimal subcovers C → E. Furthermore, if E does not have complex multiplication, then every minimal cover f : C → E is of this form (up to translates), and hence (d, deg(f)) = 1.

Proof. Choose integers a, c ∈ ℤ such that ay−cdx = 1, so g := \binom{a x}{cd y} ∈ Γ_{0}(d). Put α_{g} = \binom{a −xη̂}{−cη y} ∈ End(A), where A = E × E′. Note that α ∈ Aut(A) because \binom{y xη̂}{cη x} is the inverse of α_{g}. Now g^{t}\binom{dr dk}{dk n}g = \binom{dr_{g} dk_{g}}{dz_{g} n_{g}}, for some integers r_{g}, n_{g}, k_{g} ∈ ℤ with n_{g}r_{g} − k_{g}^{2}d = 1 and n_{g}, r_{g} > 0. Moreover, n_{g} = drx^{2} + 2dkxy + ny^{2} = drx^{2} − 2dzxy + ny^{2}. Define D_{g} ∈ Div(A) by (20), using (r_{g}, n_{g}, k_{g}) in place of (r, n, k). If D = α^{∗}(j(C)) is as in Proposition 6.5, then we have α_{g}^{∗}(D) ≡ D_{g} (cf. [Ka8]), so αα_{g}(D_{g}) ≡ j(C). By (the proof of) Proposition 6.5 we thus have a minimal cover f_{g} : C → E of degree n_{g} such that (f_{g})_{∗} ∘ α ∘ α_{g} = pr_{E}, and so (f_{g})_{∗} ∘ α = pr_{E} ∘ α_{g}^{−1} = pr_{E} ∘ \binom{y xη̂}{cη x} = ypr_{E} + xη̂pr_{E′}. Thus, f_{x,y} := f_{g} satisfies (21).

Now suppose that E has no complex multiplication. Then End(E) = ℤ1_{E} and Hom(E′, E) = ℤη̂, so Hom(A, E) = ℤpr_{E} + ℤη̂pr_{E′}. Let f : C → E be a minimal cover. Then h := f_{∗} ∘ α ∈ Hom(A, E), so ∃x, y ∈ ℤ such that h = ypr_{E} + xη̂pr_{E′}. We have (x, y) = 1 for otherwise h factors over the isogeny [g]_{E}, where g = (x, y), and then f cannot be minimal. Moreover, we have q := (d, y) = 1 for otherwise we can factor η as η = η_{2}η_{1}, where ker(η_{1}) = ker(η)[q], and then h factors over the isogeny η̂_{1} of degree q because h := η̂_{1}(\frac{y}{q}η_{1}pr_{E} + xη̂_{2}pr_{E′}). Thus, (dx, y) = 1, and so f is the same as f_{x,y} up to a translation because f_{∗} = (f_{x,y})_{∗}. Note that the formula (21) for the degree shows that (deg(f), d) = 1 because (n, d) = 1 (as dz^{2} ≡ −1 (mod n)).

6.2 The Universal Construction

Let j be transcendental over the prime field K_{0} of K and let F_{d} = K_{0}(j, j_{d}), where j_{d} is the j-invariant of an elliptic curve E_{j,d} = E_{j}/Ker(η_{d}) obtained by applying a cyclic isogeny η_{d} of degree d to E_{j}, the elliptic curve defined over K_{0}(j) (let’s say with Hasse invariant -1/2). Hence F_{d} is the function field of X_{0}(d)/K_{0}, the modular curve which parameterizes elliptic curves with cyclic isogenies of degree d. For any n and z such that dz^{2} ≡ −1 (mod n) we thus obtain an F_{0}-rational point P_{d,z} = (E_{j}, E_{j,d}, (zη_{d})_{|E[n]}) ∈ Z_{n,−1}(F_{0}) or, equivalently, a rational map τ_{d,z} : X_{0}(d) → Z_{n,−1} (which is defined over K_{0}).

The above construction has the following natural modular interpretation. Let X_{0}(d) denote the moduli functor which classifies triples (E, E′, η), where η : E → E′ is a cyclic isogeny of degree d; thus, as was mentioned above, X_{0}(m) is coarsely represented by the modular curve X_{0}(d). Then the rule (E, E′, η) ↦ (E, E′, zη_{|E[n]}) defines a morphism of functors and hence of moduli schemes

τ_{d,z} = τ_{d,z}^{(n)} : X_{0}(d) → Z_{n,−1} and τ_{d,z} = τ_{d,k}^{(n)} : X_{0}(d) → Z_{n,−1},

Thus, the image T_{d,z} = T_{d,z}^{(n)} := τ_{d,k}^{(n)}(X_{0}(d)) is a curve lying on Z_{n,−1} with generic point P_{d,z} ∈ Z_{n−1}(F_{d}). We call T_{d,z} a Hecke curve because it is induced by a certain Hecke correspondence on X(n); cf. [KS], p. 364 or [Ka7]. Note that

T_{d,z} = T_{d′,z′} ⇔ d = d′ and z ≡ ±z′ (mod n),

as is easy to see. (In fact, if T_{d,k} ≠ T_{d′,k′}, then these curves can intersect only at CM-points, i.e. at points (E, E′, ψ) ∈ Z_{n,−1}(\overline{K}), where E and E′ are elliptic curves with complex multiplication (or are supersingular).) Note also that τ_{d,z} is easily seen to be a birational equivalence (so T_{d,z} is birationally equivalent to X_{0}(d)), but in general τ_{d,z} cannot be an isomorphism because T_{d,z} has singularities (at certain CM-points) whereas X_{0}(d) does not.

We now want to find conditions to ensure that P_{d,z} lies on H_{n} and on H′_{n} or, equivalently, that the curve T_{d,z} meets H_{n} and H′_{n}. Note that there are values of d and z such that this is not the case, as we saw above in Proposition 5.2.

For this, assume that ℚ(\sqrt{−d}) does not have class number 1 and that d is prime. Then by Chebotartev there exist infinitely many prime ideals which are not principal, and so by Proposition 6.3 there are infinitely many prime numbers n such that there is a curve C_{j,n} of genus 2 defined over F_{d} = K_{0}(j, j_{d}) covering E_{j} of degree n. Hence we constructed a non-constant point on H_{n}(F_{d}).

However, it seems more difficult to decide whether or not such a point lies in H′_{n}. Here we prove:

Proposition 6.7 Let d and n be odd integers such that dz^{2} ≡ −1 mod n for some z ∈ ℤ. If T_{d,z} ∩H_{n} ≠ ∅, then either T_{d,z} ∩H′_{n} ≠ ∅ or T_{d,z} is a component of ∂_{II_{3}}(H′_{n}).

Proof. By the basic construction, the hypothesis means that (z · η_{d})_{|E_{j}[n]} induces a cover f_{n} of E = E_{j} by an absolutely irreducible curve C of genus 2 over F_{d} = K_{0}(j, j_{d}).

Suppose that T_{d,z} ∩ H′_{n} = ∅. Then the ramification type of f_{n} is one of the cases (1)–(4) listed in Proposition 3.6. Note that if we are in case (3), then we are done, for then T_{d,z} is a component of ∂_{II_{3}}(H′_{n}).

In cases (1) and (2) we see that E_{j} has an F_{d}-rational point of order 2 because the discriminant divisor of f_{n} is rational over F_{d}. This, however, contradicts the fact E_{j} has no such point over F_{d} (because otherwise F_{2d} ⊂ F_{d}, which is impossible).

Finally, suppose that we are in case (4). Since the different divisor Diff(f_{n}) = 2W′_{k} is rational, it follows that the Weierstraß point W′_{k} is rational. Let f′_{n} : C → E′ = E_{j,d} be the complementary cover. Since f_{n}^{−1}(0_{E}) ∩ (f′_{n})^{−1}(0_{E}) is a subgroup of odd order, we see that f′_{n}(W′_{k}) = P′, for some P′ ∈ E′[2] \ {0_{E′}}. Thus, E′ has an F_{d}-rational point of order 2, and hence so does E_{j} (because d = deg(η) is odd). Thus, we obtain the same contradiction.

Remark 6.8 In Corollary 7.8 below we shall show that for a given d with (d, 30) = 1, there are only finitely many n’s such that T_{d,z_{n}}^{(n)} is a component of ∂_{II_{3}}(H′_{n}). This is in marked contrast to the components of ∂_{III}(H_{n}): if there is one (prime) n such that T_{d,z}^{(n)} is a component of ∂_{III}(H_{n}), then there are infinitely many primes n_{i} such that T_{d,z_{i}}^{(n_{i})} is a component of ∂_{III}(H_{n_{i}}). Indeed, if d is prime, then this follows from Proposition 6.3 and Chebotarev, and the general case is similar.

Example 6.9 Take d = 13. Then X_{0}(d) has genus 0 and ℚ(\sqrt{−d}) has class number 2, so for infinitely many n’s we get a rational curve in H_{n} which is defined over K_{0}. Moreover, by Corollary 7.8 below we see that infinitely many of these also lie in H′_{n} (and hence also in H_{n}^{∗}).

6.3 The Answer to Q4 is Yes

Recall Question Q4 which asks in our context: For given n_{0} ∈ N and finitely generated field K, are there curves C of genus 2 defined over K such that there are projective absolutely irreducible Galois covers of C of degree ≥ n_{0}?

By results of [Fr1] and [Ki] we know already that the answer is positive for finite fields K. If K is infinite we can do even better.

By Example 6.9 we find a prime n ≥ n_{0} and a K-cover φ : ℙ^{1} → ℙ^{1} of degree n which is ramified in points P_{1}, …, P_{5} (say) of order ≤ 2 with monodromy group S_{n}.

Now choose P_{6} ∈ ℙ^{1}(K) different from P_{1}, …, P_{5} and let π : C_{φ} → ℙ^{1} be a genus 2 cover of degree 2 which is ramified at P_{1}, …, P_{5}, P_{6}.

Then the normalization of the fibre product of φ : ℙ^{1} → ℙ^{1} and π : C_{φ} → ℙ^{1} over ℙ^{1} is unramified over C_{φ} (Abhyankar’s Lemma) and its Galois closure C̃ has Galois group S_{n}. Since P_{6} is unramified in φ and ramified in P_{6} it follows that C̃ is regular.

So we get

Proposition 6.10 For all K and all n_{0} there is a curve C of genus 2 defined over K which has a regular unramified Galois cover of degree > n_{0}.

Moreover, if K is not a finite field, there are infinitely many such curves C which are not a twist of a curve defined over a finite field.

6.4 Finiteness Conjectures

We now discuss the Diophantine background of the conjectures presented in Subsection 1.2.

In view of (the reverse of) the basic construction, Conjecture 3 predicts that for given elliptic curve E and for large enough natural numbers N, the twisted modular curves H_{E/K,N} have only “obvious” K-rational points coming from isogenies. This is a conjecture of the type of Fermat’s Last Theorem: We have an infinite family of curves of growing genus depending on a fixed “parameter” (here the fixed ramification points P_{1}, …, P_{4}) and we look at the union of all solutions over K of these curves. So it is not surprising that a proof of Conjecture 3 requires not only Faltings’ theorem about the finiteness of rational solutions on each of the curves but also additional arithmetical information about these solutions. As was mentioned in Subsection 1.2, the height conjecture for elliptic curves (see [Fr1] delivers such an information.

If one looks at small N (less or equal to 5) the related modular curves are of genus 0 and so one can explain why there are so many examples for non-isogenous elliptic curves with the isomorphic torsion structures. But by experiments one finds easily many examples for N = 7 (and not so easily) examples for N = 11 for K = ℚ. The examples become very rare for N = 13 and just recently Cremona has found examples for N = 17. As far as we know this is the record for the ground field ℚ. These phenomena cannot be explained by properties of modular curves, and in fact, a question of Mazur concerning these examples was one of the main motivations for the investigations described in this paper (and in [Ka7]).

In Mazur’s question, as in Conjectures 1 and 2, the elliptic curve E/K is no longer fixed, and so this leads to the study of pairs (E, E′) of elliptic curves with G_{K}-isomorphic N-torsion structures. Any such isomorphism ψ : E[N] → E′[N] has a “determinant” ε ∈ (ℤ/Nℤ)^{×}, and the triple (E, E′, ψ) determines a unique K-rational point on the modular diagonal quotient surface Z_{N,ε} (which is a natural generalization of the surface Z_{N,−1} which was considered above).

By the work of Hermann, Kani and Schanz [KS], one knows that for N ≤ 10, the (desingularization of the) compactification \overline{Z}_{N,ε} of Z_{N,ε} is either a rational surfaces, a K3-surface or an elliptic surface. Thus, we can find infinitely many curves of genus ≤ 1 lying on Z_{N,ε}, and each of these will have infinitely many points which are rational over suitable a finite extension of K. But for N ≥ 13 (or for N = 11, 12 and for some ε’s such as ε = −1), it turns out that these surfaces are of general type, so one expects that there are only finitely many such curves on these surfaces.

Indeed, Lang’s conjecture predicts that for surfaces S/K of general type almost all of the K-rational points lie on the curves of genus ≤ 1 on S, and that there are only finitely many such curves (even over \overline{K}). This conjecture plays the role of Faltings’ theorem (Mordell conjecture) for curves of genus ≥ 2.

Even if we believe this conjecture, the Conjectures 1 and 2 would not be immediate consequences. As in the case of families of curves, we would need to have additional information to find all curves of low genus on the surfaces Z_{n,ε}.

In Subsection 6.2 we saw that the Hecke curves T_{m,k} (for m small) give rise to such curves on Z_{N,−1}, and the same construction yields curves on the surfaces Z_{N,ε} whenever mk^{2} ≡ ε (mod N). One might hope that every curve is of this form:

Conjecture 4 (Kani) For prime numbers n ≥ 23, all curves on ℤ_{n,ε} of genus ≤ 1 are Hecke curves.

One might expect that Lang’s Conjecture and Conjecture 4 imply Conjecture 2. This is indeed the case, but is not completely obvious, due to the fact that different (non-isomorphic) triples (E_{i}, E′_{i}, ψ_{i}) give rise to the same K-rational point on Z_{N,ε}. However, if j(E_{i}) ≠ 0, 1728, then (as shown in [Ka7]) this can happen if and only if the points are simultaneous twists of each other, i.e. if there is a (quadratic) character χ on G_{K} such that (E_{2}, E′_{2}, ψ) ≃ ((E_{1})_{χ}, (E′_{1})_{χ}, ψ_{χ}) where (E_{1})_{χ} and (E′_{1})_{χ} denote the χ-twists of E_{1} and E′_{1}, respectively and ψ_{χ}(E_{1})_{χ}[N] → (E′_{1})_{χ}[N] is the isomorphism induced by ψ. (In particular, we see that if K isn’t finite, then there are infinitely many non-isomorphic triples which give rise to the same point on Z_{N,ε}.) We thus obtain:

Proposition 6.11 Lang’s Conjecture and Conjecture 4 imply Conjecture 2.

7 Towers

We now want to discuss the questions concerning the existence of towers of unramified regular Galois covers of curves of genus 2. Like in the proof of Proposition 6.10 we want to use Abhyankar’s Lemma to construct composites of covers of a given curve C ramified at the same places to get unramified covers. To get an infinite tower by this strategy, we have to find infinitely many covers f_{n} : C → E with the same discriminant divisor. Hence, we have to analyze ramification points (i.e. the discriminant) of covers resulting from the universal construction in Subsection 6.2.

7.1 Ramification points

We continue to assume that E and E′ are isogenous curves with a K-rational isogeny η of minimal degree d. From now on we shall assume in addition that this degree is odd and that the j-invariant of E is not contained in an algebraic extension of the prime field K_{0} of K. Thus, E does not have complex multiplication and we have Hom_{K}(E, E′) = Hom\overline{K}(E, E′) = ℤη.

Let C be a curve over defined over K with J_{C} ≃ E × E′. Then by Proposition 6.5 we know that C has a subcover f : C → E which is induced by an isogeny; in fact, C has infinitely many such subcovers.

Let I = I(C, E, E′) denote the set of integers n > 1 for which there exists a minimal subcover

f_{n} : C → E

of degree n which is induced by an isogeny. (Fix one for each n ∈ I.) Let Δ_{n} = P_{1}^{n} + P_{2}^{n} denote the discriminant divisor (branch locus) of f_{n} and let Diff(f_{n}) = Q_{1}^{n} + Q_{2}^{n} its different divisor (ramification locus). Then by Proposition 6.7 we know that either P_{1}^{n} ≠ P_{2}^{n} or that P_{1}^{n} = P_{2}^{n} = 0_{E} and Q_{1}^{n} ≠ Q_{2}^{n} (Case (3)). We denote the set of n’s for which the first case holds by I_{1} and the rest by I_{0}.

We remark that if P_{1}^{n} ≠ 0_{E} is K-rational then Q_{1}^{n} is K-rational.

Lemma 7.1 If the set of points \{P_{j}^{n} ∈ E(\overline{K}) : n ∈ I_{1}\} is finite, then the set \{Q_{j}^{n} ∈ C(\overline{K}) : n ∈ I_{1}\} is finite.

Proof. Assume that {P_{j}^{n} : n ∈ I_{1}} is finite. Thus all points in this set are K′-rational, for a finite extension K′/K, and hence {Q_{j}^{n} : n ∈ I_{1}} is finite. Here we use the result of Faltings/Grauert/Samuel that the set of K′- rational points on C is finite. Note that C is not isotrivial over K_{0} because J_{C} ≃ E × E′ isn’t isotrivial (as j(E) ∉ \overline{K}_{0}).

In the sequel we shall use a criterion of [DiFr] about equality of ramification divisors. This is based on the following criterion for ramification points.

Lemma 7.2 Let ι : C → J_{C} be an embedding, and let

c : C → E

be a minimal cover which maps Q to 0_{E}. Then c is unramified in Q if and only if ker(c_{∗}) intersects transversally with ι(C) in ι(Q).

Proof. The subscheme ι(c^{−1}(O_{E})) of ι(C) has degree deg c, contains ι(Q) and is equal to ker(c_{∗}) ∩ ι(C). Since Q isn’t a ramification point if and only if Q occurs with multiplicity 1 in this scheme, the lemma follows.

Since translations on E are étale we can apply the above criterion to arbitrary minimal covers. In [DiFr] the following result is deduced from this:

Proposition 7.3 ([DiFr]) Assume that

c_{i} : C → E_{i}, for i = 1, 2

are two minimal covers such that ker(c_{1∗}) ∩ ker(c_{2∗}) a finite group scheme. Then the ramification locus of c_{1} is different (and hence disjoint) from the ramification locus of c_{2} if and only if ker(c_{1∗}) ∩ ker(c_{2∗}) is a reduced group scheme.

Corollary 7.4 Let K be a field of characteristic 0, and let c_{1} and c_{2} be minimal covers from C to E with c_{1} ≠ ±c_{2}. Then the ramification loci of c_{1} and c_{2} are disjoint.

Thus, if {f_{n}; n ∈ I_{1}} is an infinite set of minimal covers from C to E, then the set of ramification loci on C is infinite.

The case of positive characteristic behaves totally different.

Proposition 7.5 Assume that char(K)= p > 0. Then the ramification points of all minimal covers from C to E lie in a finite set of order ≤ p^{dim_{ℤ} Hom_{K}(J_{C},E)}.

Proof. Let c_{1} and c_{2} be two minimal covers with c_{1∗} ≡ c_{2∗} modulo p · Hom_{K}(J_{C}, E). We claim that then Diff(c_{1}) = Diff(c_{2}). Indeed, if ker(c_{1∗}) = ker(c_{2∗}), then this is clear since then c_{2} = g ∘ c_{1} for some automorphism g, so assume that ker(c_{1∗}) ≠ ker(c_{2∗}). Now the group schemes ker(c_{i∗})[p] (of rank p^{2}) are equal as closed subschemes of J_{C} and hence are contained in the finite group scheme ker(c_{1∗}) ∩ ker(c_{2∗}). Thus, the latter is not étale hence Diff(c_{1}) = Diff(c_{2}) by Proposition 7.3.

Note that the above two propositions apply to arbitrary genus 2 curve covers C → E. We now return to the situation of the curve C as at the beginning of this subsection and prove:

Proposition 7.6 If (d, 15) = 1, then the set I_{0} is finite.

Proof. Suppose first that char(K) = p > 0. If I_{0} were infinite, then by Proposition 7.5 there exists an infinite subset I′_{0} ⊂ I_{0} such that D := Diff(f_{n}) = Diff(f_{m}) for all n, m ∈ I′_{0}. Note that since C, η and the f_{n}’s are all defined over F_{d} = 𝔽_{p}(j_{E}, j_{E,d}), we can and will assume henceforth that K = F_{d}.

Write D = Q_{1}+Q_{2}, and put D′ = Q_{1}−Q_{2}. Note that D′ is not necessarily rational over K = F_{d}, but that σ(D′) = ±D′, for all σ ∈ G_{K} (because D is K-rational). Then cl(D′) ∈ ker(f_{n∗}) ∩ ker(f_{m∗}) for all n, m ∈ I′_{0} and hence cl(D′) has finite order. Thus, H := ⟨cl(D′)⟩ is a finite, K-rational subgroup on ker(f_{n∗}) ≃ E′, and so H ≤ ker(η^{t}) because all finite etale K-rational subgroup schemes of E′ have this property. Let q|r := #H be a prime. Since q|d, we have that H[q] = ker(η^{t})[q], and so \frac{ϕ(q)}{2}|[K(H[q]) : K]. But by construction, [K(H[q]) : K]|2, so ϕ(q)|4, i.e. q ≤ 5. But since (d, 30) = 1, it follows that H = 1. This is impossible because Q_{1} ≠ Q_{2}, and so we conclude that I_{0} must be finite.

Now suppose that char(K) = 0, so K_{0} = ℚ. As above we can assume that K = F_{d} = ℚ(j_{E}, j_{E,d}). Choose a prime p ∤ 2d. Since X_{0}(d) has good reduction at p (Igusa), we can reduce K to K_{p} = 𝔽_{p}(j_{p}, j_{p,d}) (where j_{p} is transcendental over 𝔽_{p}). Then E, E′ reduce to elliptic curves E_{p}, E′_{p}/K_{p}, and η reduces to a cyclic isogeny between E_{p} and E′_{p}. Thus, the G_{K}-isomorphisms ψ_{n} : E[n] → E′[n] reduce to G_{K_{p}}-isomorphisms ψ_{n,p} : E_{p}[n] → E′_{p}[n], and hence each cover f_{n} : C → E reduces to a cover f_{n,p} : C_{p} → E_{p} of the same degree. Note that C_{p} is again an irreducible curve because if not, there exists an isogeny diamond configuration (f, H_{1}, H_{2}) for ψ_{n,p} (in the sense of [Ka3]), and this could be lifted to one of ψ_{n} because Hom(E, E′) → Hom(E_{p}, E′_{p}) is an isomorphism.

Thus, I(C, E, E′) ⊂ I(C_{p}, E_{p}, E′_{p}), and hence we also have I_{0}(C, E, E′) ⊂ I_{0}(C_{p}, E_{p}, E′_{p}) since the discriminants specialize properly. Now by the first part of the proof we know that I_{0}(C_{p}, E_{p}, E′_{p}) and hence so is I_{0}(C, E, E′).

Combining the last proposition with Corollary 7.4 and Lemma 7.1 yields Corollary 7.7 If char(K) = 0, then the set \{Q_{j}^{n} ∈ C(\overline{K}) : n ∈ I\} and the set \{P_{j}^{n} ∈ E(\overline{K}) : n ∈ I\} are infinite, provided that (d, 30) = 1.

We can now finally prove the following fact which was mentioned in Remark 6.8.

Corollary 7.8 If (d, 30) = 1, then there are only finitely many n’s such that T_{d,z_{n}}^{(n)} is a component of ∂_{II}(H′_{n}), for a suitable z_{n} ∈ ℤ.

Proof. Let Ĩ denote the set of integers n ≥ 2 such that T_{d,z_{n}}^{(n)} is a component of ∂_{II}(H′_{n}). Then for each n there exists an irreducible curve C_{n} and a minimal cover f_{n} : C_{n} → E defined over K = F_{d} = K_{0}(X_{0}(d)) with J_{C_{n}} ≃ E × E′, where E, E′ are as at the beginning of Subsection 6.2. (In particular, E and E′ satisfy the conditions of this subsection.)

By the proof of Proposition 6.5 we know that every C_{n} is isomorphic to an effective divisor D_{n} on A = E × E′ which defines a principal polarization of A. Since there are only finitely many such (up to automorphisms), we conclude that there are finitely many curves C_{n_{1}}, …, C_{n_{t}} such that every curve C_{n} is K-isomorphic to one of these.

By Proposition 6.7 we know that Ĩ ⊂ ∪_{i}I_{0}(C_{n_{i}}, E, E′). Since the latter is a finite set by Proposition 7.6, we see that Ĩ is finite as well.

The above discussion shows that our strategy for building infinite unramified towers breaks down over fields of characteristic 0 because by Corollary 7.7 the ramification and branch loci f_{n} : C → E are not contained in finite sets. Nevertheless, it turns out that the different divisors of these covers cannot be too “big” in the sense that they lie in a set of bounded height. Note that this gives another proof of the finiteness result of Proposition 7.5.

Proposition 7.9 The divisors {Diff(f_{n}) : n ∈ I} have bounded height over K = F_{d}.

Proof. Let F′_{d} be the function field of X := X(d)/K_{0}, and let C/X be the minimal model of C ⊗ F′_{d}/F′_{d}. We will prove that there is an effective divisor B_{0} on C such that for all n ∈ I we have

(ω_{C/X}^{0}.\overline{Diff(f_{n})}) ≤ (ω_{C/X}^{0})^{2} − \frac{1}{12}sl(d) − B_{0}^{2} + 4(B_{0}.L), (22)

where \overline{D} denote the closure of the effective divisor D = Diff(f_{n}) in C, and L is any ample invertible sheaf on C/X.

To see that this implies the statement of the proposition, recall (cf. Lang[La], p. 58) that if K = k(T) is a function field of a normal projective curve T/k, then the height of a point P ∈ ℙ_{K}^{n}(K) is given by h_{K}(P) = deg_{T} (s_{P}^{∗} (L)), where s_{P} : T → ℙ_{T}^{n} is the section defined by P and L = O(1) is the usual very ample line bundle on ℙ_{T}^{n}. We can extend h_{K} to a function on ℙ_{K}^{n}(\overline{K}) by setting h_{K}(P′) := \frac{1}{[K′:K]}h_{K′}, for P′ ∈ ℙ_{K′}^{n}, where K′/K is a finite extension. Thus, if V/K is any projective variety and φ : V ↪ ℙ_{K}^{n} is a projective embedding, then h_{K,φ} := h_{K} ∘ φ is the height function on V (\overline{K}) determined by φ. A basic result is that, up to a bounded function, h_{K,φ} depends only on D = φ^{∗}(O_{P_{K}^{n}}(1)); cf. [La], p. 85. Note that if V/T is a projective model of V/K, and if φ extends to morphism φ̃ : V → ℙ_{T}^{n}, then with D = φ̃^{∗}O(1) we have

h_{K}, φ(P) = \frac{1}{deg(φ(P))} deg_{T_{P}} (s_{P}^{∗} D) = \frac{1}{deg(φ(P))}(\overline{P} .D)_{V}, for all P ∈ V (\overline{K});

here T_{P} is the normalization of T in the extenion κ(P) of K, s_{P} : T_{P} → V the induced T -morphism, and \overline{P} = s_{P} (T_{P}) is the closure of P in V (if we view P as a closed point on V).

If V = C is a curve, then the usual properties of height functions ([La], p. 94) show that if S ⊂ C(\overline{K}) has bounded height with respect to one height function h_{K,φ_{0}}, then it has bounded height with respect to every height function h_{K,φ}.

We now specialize the discussion to C/K as in the proposition (except that F_{d} has been replaced by F′_{d}). Since C ∼ E × E and E/k(X) has semistable reduction, the same is true for C; i.e. C/X is a semi-stable (cf. proof of Step 2 of Theorem 3.9). Now although ω_{C/X}^{0} isn’t necessarily relatively ample over T, we do have a sufficient large multiple (ω_{C|X}^{0})^{⊗m} defines a T -morphism φ_{m} : C → ℙ_{T}^{N} for some N = N_{m}. (Indeed, if C_{s}/X denotes the associated stable model with contraction morphism c : C → C_{s}, then ω_{C_{s}/X}^{0} is relatively ample (cf. [DM]) and so the assertion follows since ω_{C/X}^{0} ≃ c^{∗}ω_{C_{s}/X}^{0}.) Thus, by the above formula we see that (a multiple of) the left hand side of (22) can be interpreted as a height function, provided we restrict to those f_{n}’s for which Diff(f_{n}) is irreducible. (The reducible ones can be dealt with in a similar manner, but they are anyways finite in number by Faltings/Grauert.)

Thus, the assertion of the proposition follows once we have proved (22). For this, let q : E → X denote the minimal model of E/K. Since E/X is semi-stable, we have (cf. [DR], p. 175)

(23) ω_{E/X}^{0} ≃ q^{∗}𝔄, where 𝔄 = q_{∗}ω_{E/X}^{0} ∈ Pic(X).

Note that by (6) we know that h := deg(𝔄) = h_{E/X} = \frac{1}{24}sl(d).

Now consider f_{n} : C → E. By Proposition 8.1 below there exists an effective divisor B_{n} ≥ 0 consisting entirely of components of the (reducible) fibres of the structure map p : C → X such that

ω_{C/X}^{0} ∼ p^{∗}𝔄 + \overline{D}_{n} + B_{n}, (24)

where D_{n} = Diff(f_{n}). Here we have used the fact that (f_{n}^{∗}(ω_{E/X}^{0}))_{K} ≃ p^{∗}𝔄 which follows from (23) because p_{K} = q_{K} ∘ f_{n}.

Fix one n_{0} ∈ I and write D_{0} = D_{n_{0}} and B_{0} = B_{n_{0}}. Then from (24) (for n and n_{0}) we obtain B_{n} − B_{0} ∼ \overline{D}_{0} − D_{n}. (25)

Now for any component Γ of a fibre we have 0 ≤ (Γ.\overline{D}_{n}) ≤ (p^{∗}(p(Γ)).\overline{D}_{n}) = 2, and so |(Γ.(D_{0} − D_{n}))| ≤ 4. Thus, since D_{0} =: ∑ b_{Γ}Γ ≥ 0, we obtain from (25) that |(B_{0}.(B_{n} − B_{0}))| = ∑ b_{Γ}|(Γ.(B_{n} − B_{0}))| ≤ 4 ∑ b_{Γ} ≤ 4(B_{0}.L), the latter since (Γ.L) ≥ 1 for all Γ because L is ample. Thus

−4(B_{0}.L) + B_{0}^{2} ≤ (B_{0}.B_{n}) ≤ B_{0}^{2} + 4(B_{0}.L). (26)

Next, since \overline{D}_{0} and B_{n} are effective and without common components we see that \overline{D}_{0}.B_{n} ≥ 0. Thus, by (24) we obtain

(ω.B_{n}) = (p^{∗}𝔄 + \overline{D}_{0} + B_{0}).B_{n} = \overline{D}_{0}.B_{n} + B_{0}.B_{n} ≥ B_{0}.B_{n},

where ω := ω_{C/X}^{0}. From this, together with (24) and (26) we obtain

(ω.\overline{D}_{n}) = ω.(ω − p^{∗}𝔄 − B_{n}) ≤ ω^{2} − 2h − B_{0}.B_{n} ≤ ω^{2} − 2h − B_{0}^{2} + 4(B_{0}.L);

here we have also used the fact (\overline{D}_{n}.p^{∗}𝔄) = deg(D_{n}) deg(𝔄) = 2h. Since h = \frac{1}{24}sl(d), this proves (22).

7.2 Towers of unramified extensions

In this subsection we discuss the question whether we can use different covers of C to E to compose them to towers of regular unramified Galois covers. First of all we will restrict ourselves to fields K of positive characteristic p, and we can assume that K is of transcendence degree 1 over K_{0}. Then we know by Proposition 7.5 that there are infinitely many covers

f_{i} : C → E

with the same different divisor. In [DiFr] one finds examples of curves C which admit infinitely many subcovers f_{i} : C → E with the same discriminant divisor. We now want analyze the condition that (certain) genus 2 curve C admit subcovers with this property.

Recall the situation: We assume that we have an isogeny η : E → E′ of degree d and that J_{C} is isomorphic to E × E′. Moreover, we shall impose here the restriction that d > 5 is prime. Then we have that every elliptic curve E′′ which is separably isogenous to E over K = F_{d} = 𝔽_{p}(j_{E}, j_{E′}) is isomorphic to E or to E′, and hence does not have any non-trivial torsion points in any quadratic extension of K. In particular, the ramification points of the f_{i}’s on C are mapped to non-torsion points on E (or to 0_{E}).

In the following we denote by I the set of integers n such that there is a minimal cover f_{n} : C → E of degree n with Disc(f_{n}) reduced. For each n, fix such a cover f_{n}. Note that it follows from Corollary 6.6 and Proposition 7.6 that the set I is infinite. We also assume that the covers are compatible in the sense that there is an embedding ι : C → J_{C} such that (f_{n})_{∗} ∘ ι = f_{n}, for all n.

As was mentioned above, it follows from Proposition 7.5 that there is an infinite subset I′ of I such that the covers f_{n} with n ∈ I′ all have the same ramification points on C. We now want to find conditions which ensure that they also have the same branch point on E.

Lemma 7.10 Assume that Q is a ramification point on C for f_{n_{1}} and f_{n_{2}} with n_{1} ≠ n_{2} and that f_{n_{1}}(Q) = f_{n_{2}}(Q). Then there is a unique elliptic subgroup E_{0} on J_{C} which contains ι(Q).

Proof. Let h = f_{n_{1},∗} − f_{n_{2},∗} : J_{C} → E. Since n_{1} ≠ n_{2}, its kernel is a group scheme of dimension 1 and so the connected component of ker(h) is an elliptic curve E_{0} defined over K. If h_{0} : J_{C} → \overline{E}_{0} := J_{C}/E_{0} denotes the quotient map, then h = h_{1}h_{0}, for some isogeny h_{1} : \overline{E}_{0} → E.

By hypothesis we have ι(Q) ∈ ker(h), so h_{0}(ι(Q)) ∈ ker(h_{1}) has finite order. Since \overline{E}_{0} is an elliptic curve isogenous to E, it has no non-torsion points in quadratic extensions (by our hypothesis on K), and thus we have h_{0}(ι(Q)) = 0. This means that ι(Q) ∈ E_{0}, so ι(Q) lies on the elliptic subgroup E_{0}. Note that E_{0} is the only elliptic subgroup containing ι(Q), for if E′_{0} ≠ E_{0} were another, then ι(Q) is in the finite subgroup scheme E_{0} ∩ E′_{0}, which contradicts the fact that ι(Q) has infinite order.

Lemma 7.11 If E_{1} ≤ A := E × E′ is an elliptic subgroup, then there is another elliptic subgroup E_{2} on A such that (E_{1}.E_{2}) = 1. In particular, if i_{E_{k}} : E_{k} → A denote the inclusion maps, them the map α_{E_{1},E_{2}} = i_{E_{1}} + i_{E_{2}} : E_{1} × E_{2} →^{∼} A is an isomorphism.

Proof. Write θ_{1} = pr_{E}^{∗}(0_{E}), θ_{2} = pr_{E′}^{∗}(0_{E′}) and Γ^{∗} = −dθ_{1} − θ_{2} + Γ_{−η}. Then θ_{1}, θ_{2}, Γ^{∗} is a basis of NS(A) (cf. [Ka8]), so E_{1} ≡ m_{1}θ_{1} + m_{2}θ_{2} + m_{3}Γ^{∗}, for some m_{i} ∈ ℤ. Now since (m_{1}θ_{1} + m_{2}θ_{2} + m_{3}Γ^{∗})^{2} = 2(m_{1}m_{2} − m_{3}^{2}d), it follows that m_{1}m_{2} = m_{3}^{2}d as E_{1}^{2} = 0. Moreover, since E_{1} is primitive (cf. [Ka2], Theorem 2.8) we have that (m_{1}, m_{2}, m_{3}) = 1. Since d is square free, it follows that (m_{1}, m_{2}) = 1. Thus d = d_{1}d_{2} with (d_{1}, d_{2}) = 1 and d_{i}|m_{i}, and similarly m_{3}^{2} = (n_{1}n_{2})^{2} with (n_{1}, n_{2}) = 1 and n_{i}|m_{i}. Thus m_{i} = n_{i}^{2}d_{i}, for i = 1, 2, and (n_{1}d_{1}, n_{2}d_{d}) = 1, and hence there exist k_{1}, k_{2} ∈ ℤ such that k_{1}m_{1}d_{1} − k_{2}m_{2}d_{2} = 1. Put D = k_{2}^{2}d_{2}θ_{1} + k_{1}^{2}d_{1}θ_{2} + k_{1}k_{2}Γ^{∗}. Then D^{2} = 0, and so there is an elliptic subgroup E_{2} with D ≡ mE_{2} for some m ∈ ℤ (cf. [Ka2], Prop. 2.3). But D is primitive since (k_{2}^{2}d_{2}, k_{1}^{2}d_{1}) = 1, so m = ±1. Moreover, since D.θ_{1} = k_{1}^{2}d_{1} > 0, we must have m = 1, i.e. E_{2} ≡ D. Now (E_{1}.E_{2}) = \frac{1}{2}(E_{1} + E_{2})^{2} = (n_{1}^{2}d_{1} + k_{2}^{2}d_{2})(n_{2}^{2}d_{2} + k_{1}^{2}d_{1}) − (n_{1}n_{2} + k_{1}k_{2})^{2}d = (n_{1}k_{1}d_{1} − n_{2}k_{2}d_{2})^{2} = 1^{2} = 1, and so E_{2} has the desired properties. The second assertion is an immediate consequence of the first.

Corollary 7.12 If E_{0} ≤ E × E′ is an elliptic subgroup, then either pr_{E} ∘ α ∘ i_{E_{0}} = 0, or pr_{E′} ∘ α ∘ i_{E_{0}} = 0, for some automorphism α ∈ Aut(E × E′).

Proof. By Lemma 7.11 there is an elliptic subgroup E′_{0} ≤ A and an isomor-

phism β : A_{0} := E_{0} × E′_{0} →^{∼} A such that β ∘ i_{E_{0}×0}^{A_{0}} = i_{E_{0}}^{A}.

Suppose first that E_{0} ≃ E. Then E′_{0} ≃ A/E_{0} ≃ A/E ≃ E′, so we have

isomorphisms α_{0} : E_{0} →^{∼} E and α_{1} : E′_{0} →^{∼} E′. Thus, α = β ∘ (α_{0}^{−1} × α_{1}^{−1}) ∈

Aut(A) satisfies α∘i_{E_{0}} = i_{E×0} ∘α_{0}. Thus, pr_{E′} ∘α∘i_{E_{0}} = pr_{E′} ∘i_{E×0} ∘α_{0} = 0.

Next, suppose E_{0} ≄ E. Then (since d is prime) E_{0} ≃ E′, and one has by a similar argument as above that E′_{0} ≃ E, so we have isomorphisms

α_{0} : E_{0} →^{∼} E′ and α_{1} : E′_{0} →^{∼} E. If τ : E × E′ →^{∼} E′ × E denotes the natural

map, then α = β ∘ (α_{0}^{−1} × α_{1}^{−1}) ∘ τ ∈ Aut(A) satisfies α ∘ i_{E_{0}} = i_{0×E′} ∘ α_{0}, and so pr_{E} ∘ α ∘ i_{E_{0}} = pr_{E} ∘ i_{0×E′} ∘ α_{0} = 0.

Proposition 7.13 Let d > 5 be a prime and let E/K be an elliptic curve defined over K = F_{d} with j-invariant j = j_{E} transcendental over 𝔽_{p} ⊂ F_{d}. Let η : E → E′ be an isogeny of degree d to some curve E′/K, and let C/K be a genus 2 curve such that J_{C} ≃ E × E′. Then the following conditions are equivalent.

(i) There exist infinitely many minimal covers f_{n} : C → E with the same branch locus.

(ii) There exist two minimal covers f_{n_{k}} : C → E of different degrees with the same ramification locus and the same branch locus.

(iii) There is a minimal subcover f : C → E such that its ramification locus Q lies on some elliptic subgroup E_{0} ≤ J_{C}.

Proof. (i) ⇒ (ii): Clear by by Proposition 7.5.

(ii) ⇒ (iii): Lemma 7.10.

(iii) ⇒ (i): Fix an isomorphism β : J_{C} →^{∼} E × E′. Applying Corollary 6.6 to β(E_{0}) ≤ E × E′, we have two possibilities: Case 1: pr_{E} ∘ α ∘ i_{β(E_{0})} = 0, for some α ∈ Aut(E × E′). Then as in the proof of Corollary 6.6 there are integers x_{0}, y_{0} ∈ ℤ with (dx_{0}, y_{0}) = 1 such that f_{∗}∘β^{−1}∘α^{−1} = y_{0}pr_{E} +x_{0}η̂pr_{E′}. Take y ∈ ℤ such that y ≡ y_{0} (mod p) and (y, x_{0}d) = 1, and put f_{y} := (ypr_{E}+x_{0}η̂pr_{E′})∘α∘β_{|ι(C)}. By (the proof of) Corollary 6.6 we know that f_{y} : C → E is minimal. Moreover, since (f_{y})_{∗} = f_{∗} + (y − y_{0})pr_{E} ∘ α ∘ β ≡ f_{∗} (mod pEnd(J_{C})), the proof of Proposition 7.5 shows that f and f_{y} have the same ramification point Q. Moreover, we have (f_{∗} − (f_{y})_{∗})(Q) = (y − y_{0})pr_{E}(α(β(Q))) = 0, and so f and f_{y} have the same branch point, and hence (i) holds in this case. Case 2: pr_{E′} ∘ α ∘ i_{β(E_{0})} = 0, for some α ∈ Aut(E × E′). Again by Corollary 6.6 there are integers x_{0}, y_{0} ∈ ℤ with (dx_{0}, y_{0}) = 1 such that f_{∗} ∘ β^{−1} ∘ α^{−1} = y_{0}pr_{E} + x_{0}η̂pr_{E′}. Take x ∈ ℤ such that x ≡ x_{0} (mod p) and (y_{0}, x) = 1, and put f_{x} := (y_{0}pr_{E} + xη̂pr_{E′}) ∘ α ∘ β_{|ι(C)}. By (the proof of) Corollary 6.6 we know that f_{x} : C → E is minimal. Here (f_{x})_{∗} = f_{∗} + (x − x_{0})η̂ ∘ pr_{E′}α ∘ β ≡ f_{∗} (mod pEnd(J_{C})), so as before f and f_{x} have the same ramification point Q. Moreover, we have (f_{∗} − (f_{y})_{∗})(Q) = (x − x_{0})η̂(pr_{E′}(α(β(Q)))) = 0, and so f and f_{x} have the same branch point, and hence (i) holds here as well.

In [DiFr] it is shown that there always exists a cover f : C → E satisfying the condition (iii) of Proposition 7.13, the degree of f being 2. In addition, there is an explicit description of an infinite family of minimal covers f_{n} : C → E which have the same ramification and branch points.

Now we are nearly done. To end we have to overcome a technical difficulty. The monodromy groups occurring for the covers are the full symmetric groups S_{n_{i}}. These are only “nearly simple” groups and so when we take composites of covers (with normalization) it could happen that there occur constant field extensions with 2-elementary Galois groups. Hence we have to generalize the discussion and replace the ground field K by an appropriate affine scheme Spec(S) with function field K such that Pic(S)/(Pic(S)^{2} is finite, and then show that there are infinitely many minimal covers with the same branch divisor over S.

This implies that there is a finite extension K_{1} of K such that the curve C × K_{1} has an infinite tower of K_{1}-rational regular unramified Galois extensions with Galois group being a product of alternating groups A_{n}. Hence we get

Theorem 7.14 Let K be a field of finite type of odd positive characteristic. Let E be an elliptic curve over K which has a K-rational isogeny η of prime degree d > 5. (For instance, take E belonging to a K-rational point on X_{0}(13).)

Then there is a curve C of genus 2 over K with J_{C} ≃ E × η(E) and a finite extension K_{1} of K such that C has an infinite unramified regular Galois pro-cover defined over K_{1}.

If K is not a finite field, then the curve C can be chosen such that it is not a twist of constant curves, i.e. such that it is not isotrivial.

Note that in the above theorem we can take for the curve C/K a twist of the curve discussed in Proposition 7.13 and constructed in [DiFr].

Corollary 7.15 Let K be a finitely generated field over F_{p}. There is an algebraic subset of positive dimension in the moduli space of curves of genus 2 over K such that Question Q3 has a positive answer for all curves in this subset.

Remark 7.16 It is easy to see that in the infinite towers constructed in Theorem 7.14 over the non-constant curve C, there is no point of C which splits totally. So we get no answer to Question Q2 by this construction.

Question: Can one find a curve C/F_{d} which satisfies the condition (iii) of Proposition 7.13 which is not isomorphic to the one used in [DiFr]? (For a given d, this curve is uniquely characterized by the property that J_{C} ≃ E×E′ and that it has a subcover f : C → E of degree 2.)

8 Appendix: The Riemann-Hurwitz Relation for Relative Curves

Let k be a field and let T/k be a smooth projective irreducible curve with function field K = κ(T). Let X/T be a relative curve, i.e. X/k is a smooth projective irreducible surface with a surjective morphism p_{X} : X → T. We denote by X_{K} = X ⊗ K the generic fibre of p_{X} (which is a smooth projective curve over K).

Now suppose Y/T is another relative curve and that we have a nonconstant, separable morphism

f_{K} : X_{K} = X ⊗ K → Y_{K} = Y ⊗ K

of their generic fibres. Then the usual Riemann-Hurwitz relation states that we have an isomorphism

ω_{X_{K}/K} ≃ f_{K}^{∗}(ω_{Y_{K}/K}) ⊗ L(D_{f_{K}}), (27)

where D_{f_{K}} ≥ 0 is an effective divisor (the different divisor of f_{K}). We now want to extend this relation to similar one involving the relative dualizing sheaves of X/T and Y/T, which are given by

ω_{X/T}^{0} = ω_{X/k} ⊗ p_{X}^{∗}ω_{T/k}^{−1} and ω_{Y/T}^{0} = ω_{Y/k} ⊗ p_{Y}^{∗} ω_{T/k}^{−1},

where ω_{X/k} = ∧^{2}Ω_{X/k}^{1} and ω_{Y/k} = ∧^{2}Ω_{Y/k}^{1}; cf. Kleiman[Kl].

In the case that f_{K} extends to a T -morphism f : X → Y, such a formula was established in the Appendix of [Ka6]. Here we want to show that a similar formula holds without this assumption.

For this, note first that the map f_{K} naturally extends to a morphism f : U → Y, where U ⊂ X is an open set such that X \ U has codimension 2, i.e. X \ U consists of finitely many closed points. Thus, every divisor of U extends uniquely to a divisor on X, and hence every invertible sheaf L on U extends uniquely to an invertible sheaf L_{X} on X.

Proposition 8.1 Let \overline{D} denote the schematic closure in X of the different divisor D_{f_{K}} of f_{K}. Then there exists an effective divisor B = ∑ b_{Γ}Γ ≥ 0 consisting entirely of fibre components of p_{X} such that if we write D_{f} = \overline{D}+B then we have ω_{X/T}^{0} ≃ (f^{∗}ω_{Y/T}^{0} )_{X} ⊗ L(D_{f}). (28)

Proof. First note that since f^{∗}Ω_{Y/k}^{1} and Ω_{U/k}^{1} are locally free sheaves of rank 2 on U, we have the exact sequence

0 → f^{∗}Ω_{Y/k}^{1} → Ω_{U/k}^{1} → Ω_{U/Y}^{1} → 0.

Let D(U/Y) = F^{0}(Ω_{U/Y}^{1} ) denote the Kähler different of f which is by definition (cf. Kunz[Ku], p. 159) the 0-th Fitting ideal sheaf of the sheaf of differentials Ω_{U/Y}^{1} . From the above exact sequence it follows that

f^{∗}ω_{Y/k} = D(U/Y)ω_{U/k}

(viewed as subsheaves of ω_{U/k} = ∧^{2}Ω_{U/k}^{1}), and hence D(U/Y) is invertible. Thus, if D_{f} denotes the associated effective divisor defined by the invertible ideal sheaf D(U/Y), then we have

ω_{U/k} ≃ f^{∗}ω_{Y/k} ⊗ L(D_{f}),

from which equation (28) follows immediately.

Finally we show that the divisor D_{f} has the form asserted in the proposition. Indeed, since D(U/Y)_{|X_{K}} = D(X_{K}/Y_{K}) by general properties of the different (cf. [Ku], p. 160, property a)), we see that D_{f} = \overline{D} + B, where B is an effective divisor consisting entirely of fibre components.

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