Non-constant Curves of Genus 2 with Infinite Pro-Galois Covers

Claus Diem and Gerhard Frey

November 14, 2006

Abstract For every odd prime number p, we give examples of non-constant smooth families of curves of genus 2 over fields of characteristic p which have pro-Galois (pro-étale) covers of infinite degree with geometrically connected fibers. The Jacobians of the curves are isomorphic to products of elliptic curves.

Key words: Covers of curves, fundamental groups, families of curves, curves of genus 2, curves with split Jacobian. MSC2000: 14H30, 14H45, 14H10.

1 Introduction and Main Result

In the following, unless stated otherwise, a curve over a field is assumed to be smooth, proper and geometrically connected. This work is motivated by the following general problem.

Consider all curves of a fixed genus g over fields of a certain type (e.g. algebraically closed, local, finitely generated). Among these curves, does there exist a curve C over a field K which allows an infinite tower of nontrivial unramified covers ⋯ ⟶ C_{i} ⟶ ⋯ ⟶ C_{0} = C such that for all i, C_{i} is also a (geometrically connected) curve over K and C_{i} ⟶ C is Galois?

We concentrate on this problem for curves of genus 2 over finitely generated fields. Some examples of curves of genus 2 over finite fields allowing such a tower of covers are known, and we address the question whether there also exist non-constant curves of genus 2 (over function fields) with such a tower. We show with explicit examples that this is the case.

Example 1.1 For every odd prime p, the (smooth, proper) curve of genus 2 given by the (affine) equation

Y^{2} = (t − 1) · (1 − t^{p−1}) · (X^{2} − 1) · (X^{2} − t^{p−1}) · (X^{2} − 1 − t − ⋯ − t^{p−1}) over the global function field K = 𝔽_{p^{2}}(t)[\sqrt{t}, \sqrt{t − 1}, …, \sqrt{t − (p − 1)}] allows such a tower of covers.

1.1 Further Background Information

We will use the following terminology. A curve cover over K is a surjective morphism of curves over K. A projective limit of a projective system of curves (where the morphisms are curve covers) over K is called a pro-curve over K. A pro-curve cover of a curve C over K is a surjective morphism π : D ⟶ C where D is a pro-curve.

Definition 1.2 A curve cover is Galois if it is unramified and the corresponding extension of function fields is Galois. A pro-curve cover is pro-Galois if it is a projective limit of a projective system of Galois covers.

For brevity, we speak of pro-Galois curve covers instead of pro-Galois procurve covers.

With this terminology, the general problem posed above can be reformulated as follows: Are there curves of a fixed genus g over fields of certain types which allow pro-Galois curve covers of infinite degree?

As an example consider the case that K is an algebraically closed field. Then a curve C over K allows a pro-Galois curve cover of infinite degree and only if the genus of C is ≥ 1. The situation is very different over finitely generated fields.

No curve of genus 1 over a finitely generated field K has a pro-Galois curve cover of infinite degree. There even exists a universal bound n = n(K) such that all Galois curve covers of a genus 1 curve with a rational point over K have degree ≤ n. Indeed, by the Theorem of Mazur-Kamienny-Merel (see [16]) and an induction argument on the absolute transcendence degree of K, there exists a number n = n(K) such that all elliptic curves over K have at most n K-rational torsion points – from this the assertion follows; see [7, Section 2] for details.

If K is finitely generated and C is a curve over K corresponding to the generic point of the moduli scheme of curves of a certain genus, the authors expect that C also does not have a pro-curve cover of infinite degree. But for special curves the situation changes. For every natural number g ≥ 3 and every field K containing the 4-th roots of unity, explicit examples of curves over K of genus g having a pro-Galois curve cover of infinite degree can be given. If K is not finite over its prime field the curves can be chosen to be non-constant and even non-isotrivial (i.e. for every extension of the ground field, they stay non-constant); see [7, 6].

For finite fields, examples of curves of genus 2 with a pro-Galois curve cover of infinite degree are known (see [11, §3, Examples 2,3], [7]), but to the knowledge of the authors, no examples of non-constant curves of genus 2 with a pro-Galois curve cover have been known so far.

1.2 The Main Result

In this subsection we shall formulate the main result of this paper. We thereby freely use some results of [1] which we recall at the beginning of Section 3. We begin with some notation.

Let p be an odd prime, and let N be an odd natural number. Let S be a smooth variety over a finite field of characteristic p. We assume that there is an S-isogeny

τ : ℰ ⟶ ℰ′

between two elliptic curves over S having degree N. The kernel of the multiplication by 2 of ℰ is denoted by ℰ[2], the image of the zero-section by [0_{ℰ}]. We denote by ℰ[2]^{#} the S-scheme ℰ[2] − [0_{ℰ}]. For the curve ℰ′ we use similar definitions.

Notation 1.3 Let 𝒫 := ℰ/⟨−1⟩, 𝒫′ := ℰ′/⟨−1⟩ with projections

ρ : ℰ ⟶ 𝒫 and ρ′ : ℰ′ ⟶ 𝒫′.

Remark 1.4 The quotients 𝒫 and 𝒫′ are ℙ^{1}-bundles over S which are isomorphic to ℙ_{S}^{1} if ℰ[2] is S-isomorphic to the group scheme (ℤ/2ℤ)^{2}. Without weakening the results of the paper very much the reader may assume this.

There exists a unique S-isomorphism

γ : 𝒫 ˜ ⟶𝒫′

such that

ρ′ ∘ τ|_{ℰ[2]^{#}} = γ ∘ ρ|_{ℰ[2]^{#}} .

Assumption 1.5 Let the following two equivalent conditions be satisfied.

• For no geometric point s of S, there exists an isomorphism α : ℰ_{s} ⟶ ℰ′_{s} such that α|_{ℰ_{s}[2]} = τ_{s}|_{ℰ_{s}[2]}.

• γ(ρ([0_{ℰ}])) and ρ′([0_{ℰ′}]) are disjoint.

Notation 1.6 Let 𝒞 be the normalization of the integral scheme ℰ ×_{𝒫′} ℰ′ (where the product is with respect to γ ∘ ρ and ρ′).

We thus have the commutative diagram

\xymatrix{ & 𝒞 \ar[dl] \ar[dr] \\ ℰ \ar[d]_{ρ} &  & ℰ′ \ar[d]^{ρ′} \\ 𝒫 \ar[rr]^{γ} &  & 𝒫′ .}

Now 𝒞 is a curve of genus 2 over S and the morphisms 𝒞 ⟶ ℰ, 𝒞 ⟶ ℰ are degree 2 covers. (For the terminology concerning relative curves, see Subsection 1.6.) It is the curve 𝒞 which is the basic object of this article.

Our main result is the following theorem.

Theorem 1 We use the notations from above, and we assume that Assumption 1.5 is satisfied. So 𝒞 is a curve of genus 2 over S which is defined as in Notation 1.6.

Then there exists a connected Galois cover T ⟶ S with Galois group a (finite) elementary abelian 2-group such that the curve 𝒞_{T} over T has a pro-Galois curve cover whose Galois group 𝒢 fits into an exact sequence

1 ⟶ ∏_{i=1}^{∞} A_{n_{i}} ⟶ 𝒢 ⟶ (ℤ/2ℤ)^{r} ⟶ 1

with r ≤ 4, n_{i} are pairwise distinct natural numbers and A_{n_{i}} is the alternating group on n_{i} elements.

Remark 1.7 It is easy to see that the Jacobian variety J_{𝒞} of 𝒞 is isogenous to ℰ × ℰ′. We shall prove the stronger result that J_{𝒞} is in fact isomorphic to ℰ × ℰ′; see Proposition 3.5 as well as Subsection 5.1 for additional information.

1.3 Outline of the Proof

The proof of Theorem 1 is organized as follows:

In Section 2, we show some basic results on ramification loci on varying covers of a fixed elliptic curve by a fixed genus 2 curve over an algebraically closed field. In Section 3, we fix an isogeny τ : ℰ ⟶ ℰ′ as in Theorem 1 (but over a more general base scheme). Using results from [1], we first show that 𝒞 is a curve of genus 2 whose Jacobian is isomorphic to ℰ × ℰ′. We then show with the results of Section 2 that there are infinitely many covers from 𝒞 to ℰ with the same ramification and branch loci and pairwise distinct degrees. In Section 4, we show how our main result follows from the existence of infinitely many minimal genus 2 covers of a given elliptic curve with the same branch loci (under some conditions).

Theorem 1 follows by combining the final result of Section 2, Proposition 3.10, with the main result of Section 4, Proposition 4.1.

1.4 Derivation of Examples

Theorem 1 can be used to obtain Example 1.1. Let p be an odd prime, let S := 𝔸_{𝔽_{p}}^{1} − {0, 1, …, p − 1} and let us denote the coordinate on 𝔸_{𝔽_{p}}^{1} by t. Let ℰ be the genus 1 curve over S given by

Y^{2}Z − X(X − Z)(X − tZ) = 0.

We want to fix a section of ℰ over S in order to turn ℰ into an elliptic curve. For this, let ℰ_{a} be the S-scheme given by Y^{2} − X(X − 1)(X − t) = 0 and let a : S ⟶ ℰ_{a} be section given by X ↦ t, Y ↦ 0. We have a natural inclusion ι : ℰ_{a} ↪ ℰ which is compatible with the projection to S, and ι ∘ a is a section of ℰ over S. We take this section as the zero-section.

Similarly, let ℰ′ be the elliptic curve over S given by

Y^{2}Z − X(X − Z)(X − t^{p}Z) = 0

with zero-section determined by X ↦ t^{p}, Y ↦ 0, and let τ : ℰ ⟶ ℰ′ be the Frobenius endomorphism given by

X ↦ X^{p}, Y ↦ Y^{p}, Z ↦ Z^{p}.

We note that the quotients ℰ ⟶ 𝒫, ℰ′ ⟶ 𝒫′ can be identified with the usual projections to ℙ_{S}^{1} (to the “X-coordinate”), and under these identifications, γ : 𝒫 ⟶ 𝒫′, becomes the identity. We also note that we have chosen S in such a way that the assumptions of Theorem 1 are satisfied. The curve 𝒞 defined in Notation 1.6 is the normalization of ℰ ×_{ℙ_{S}^{1}} ℰ′. The field K in the introduction is obtained as the function field of the “maximal” connected Galois cover T ⟶ S which has an elementary abelian 2-group as Galois group, that is, it is the maximal Galois extension of 𝔽_{p}(t) which is unramified outside the places corresponding to 0, 1, …, p − 1 and the place ∞ and has an elementary abelian 2-group as Galois group. (The explicit description of the field can for example be obtained via Kummer theory.) The curve in the introduction is the restriction of the curve 𝒞_{T} to the generic fiber of T, that is, to Spec(K).

Further examples can be obtained as follows:

Let p be an odd prime and N ≥ 3 and assume that p and N are coprime. It is well-known that there exists a fine moduli scheme for elliptic curves with cyclic subgroups of order N and level-4-structure (of fixed determinant ζ_{4}) over schemes over 𝔽_{p}(ζ_{4}). (We use that p is odd.) Let us denote this scheme by Y_{0}(N; 4)_{𝔽_{p}(ζ_{4})} – it is a smooth affine curve over 𝔽_{p}(ζ_{4}).

Let τ : ℰ ⟶ ℰ′ be the universal isogeny over Y_{0}(N; 4)_{𝔽_{p}(ζ_{4})}. There exists a uniquely determined open subscheme S of Y_{0}(N; 4)_{𝔽_{p}(ζ_{4})} such that the assumptions of Theorem 1 are satisfied. Obviously, S is non-empty (as otherwise the universal elliptic curve over X_{0}(N; 4)_{𝔽_{p}(ζ_{4})} would have complex multiplication).

If we now apply Definition 1.6 to the restriction of the universal isogeny τ : ℰ ⟶ ℰ′ to S, we obtain a curve 𝒞 over S. We can now apply Theorem 1 to this curve.

1.5 Open Problems

Problem 1.8 It is an open problem whether there exists a curve of genus 2 over a finitely generated field of characteristic 0 with a pro-Galois curve cover of infinite degree. The existence of such a curve would of course be implied by the existence of a genus 2 curve 𝒞 with a pro-Galois curve cover of infinite degree over an open part of Spec(𝒪_{K}), where 𝒪_{K} is the principal order in a number field. In Subsection 5.2 we discuss difficulties occurring when one tries to adapt the proof of Theorem 1 to the “mixed-characteristic case”.

There is a sharpening of the condition on a curve over a field having a pro-Galois curve cover of infinite degree.

Let C be a curve over a field K and let P ∈ C(K). Following ideas of [11], the K-rational geometric fundamental group is introduced in [7]. Let us denote this group by π_{1}^{geo}(C, P). The group π_{1}^{geo}(C, P) classifies pro-étale (curve) covers c : D ⟶ C such that P splits completely. If deg(c) is finite, this means that the topological point P has exactly deg(c) preimages under c. Thus the group π_{1}^{geo}(C, P) is infinite if and only if a pro-étale (or pro-Galois) (curve) cover of D ⟶ C of infinite degree exists such that P splits completely.

For every prime p, there exists a curve over a finite field K of characteristic p with infinite K-rational geometric fundamental group. There also exists a non-isotrivial curve over a function field in one variable over a finite field of given prime characteristic p with infinite K-rational fundamental group; this follows from [6, Theorem 4.3].

Moreover, for every natural number g ≥ 3 and every field K containing the 4-th roots of unity of characteristic congruent 3 modulo 4, explicit examples of curves over K of genus g having an infinite K-rational fundamental group can be given. If K is not finite over its prime field the curves can be chosen to be non-isotrivial; see [6, Theorem 1.1].

Problem 1.9 It is an open problem whether any curve over a finitely generated field K of characteristic 0 with infinite K-rational geometric fundamental group exists,^{1} and the case g = 2 remains open in any characteristic.

1.6 Terminology and Facts

The usage of “Galois” follows [8, Exposé V]. This means in particular: If Y is a locally noetherian, integral, normal scheme, X is an integral scheme and f : X ⟶ Y is a finite surjective morphism, then f is a Galois cover if and only if it is unramified and the corresponding extension of function fields is Galois; see [8, Exposé I, Corollaire 9.11] and [8, Exposé V, §8].

Let S be a connected locally noetherian scheme. A (relative) curve of genus g over S is a smooth, proper S-scheme all of whose geometric fibers are connected non-singular curves. We denote curves over S by 𝒞, 𝒟, ℰ, . . .. If S is the spectrum of a field, we also use the notation C, D, E, . . ..

By the first corollary to the Theorem in [20, Chapter II, §5], the Euler-Poincaré characteristic of the geometric fibers of a relative curve is constant, thus the genus of the fibers is also constant; we define the genus of a relative curve to be the genus of any of its fibers.

We will often use the fact that a curve over a regular scheme is regular; see [8, Exposé II, Proposition 3.1].

A curve cover of a curve 𝒞 over S is a finite and flat S-morphism π : 𝒟 ⟶ 𝒞, where 𝒟 is also a curve over S.

Note that an S-morphism π : 𝒟 ⟶ 𝒞 between two curves over S which induces curve covers on the fibers is automatically finite and flat, that is, it is a curve cover. (The morphism is fiberwise flat ([10, Proposition 9.7]) and thus flat by [9, IV (11.3.11)], it is finite by [10, Corollary 4.8] and [9, IV (8.11.1)].)

A pro-curve over S is a projective limit of a projective system of curves with respect to morphisms which are curve covers over S.

A pro-curve cover of a curve 𝒞 over S is a projective limit of a projective system of curve covers of 𝒞. A pro-Galois pro-curve cover is abbreviated as pro-Galois curve cover, and its automorphism group is called Galois group.

We shall identify effective Cartier divisors on a locally noetherian integral scheme X with the locally principal closed subschemes of X; cf. [10, Remark 6.17.1], [9, 21.2.12]. (This means in particular that Cartier divisors on an integral, regular schemes are identified with closed subschemes of codimension 1.) If additionally X is an S-scheme, then under this identification, relative effective Cartier divisors on X over S correspond to locally principal closed subschemes of X which are flat over S; see [18, Definition 3.4].

1The proof of [7, Theorem 4.22] and thus also the proof of the “Result” at the end of the introduction of [7] are incorrect: As stated in [6, Remark 5.4], the condition char(K) ≡ 3 mod 4 has to be inserted.

Let S be a scheme. If S is integral, we denote its function field by κ(S). If X is an S-scheme and T ⟶ S is a morphism, we denote the pull-back of X to T by X_{T}. If α : X ⟶ Y is a morphism of S-schemes, we denote its pull-back via T by α_{T} : X_{T} ⟶ Y_{T}. Note that if X is a curve over S and T is connected and locally noetherian, X_{T} is a curve over T.

If 𝒜 is an abelian scheme over S, the dual abelian scheme 𝒜̂ exists by [3, Theorem 1.9]. If φ : 𝒜 ⟶ ℬ is a homomorphism of abelian schemes over S, we denote the dual homomorphism by φ̂: ℬ̂ ⟶ 𝒜̂.

If 𝒜 and ℬ are two abelian schemes over S, we denote the product over S by 𝒜 × ℬ. If 𝒜_{1}, …, 𝒜_{n} and ℬ_{1}, …, ℬ_{m} are abelian schemes over S, we have a canonical isomorphism between the group of homomorphisms from ∏_{j} 𝒜_{j} to ∏_{i} ℬ_{i} and the group of matrices (a_{i,j})_{i=1...,m,j=1,...,n}, where a_{i,j} ∈ Hom_{S}(𝒜_{j}, ℬ_{i}). The composition of homomorphisms (if applicable) corresponds to the multiplication of the matrices. If we use matrices, we identify ℤ with its image in any endomorphism ring.

If 𝒞 is a curve over S, we denote its Jacobian by J_{𝒞}. The dual of the Jacobian is denoted by J_{𝒞} and the canonical principal polarization on J_{𝒞} by λ_{𝒞} : J_{𝒞} ⟶ Ĵ_{𝒞}; cf. [19, Proposition 6.9]. If π : 𝒟 ⟶ 𝒞 is a curve cover over S, we have canonical homomorphisms π^{∗} : J_{𝒞} ⟶ J_{𝒟} and π_{∗} := λ_{𝒞}^{−1} π̂^{∗} λ_{𝒟} : J_{𝒟} ⟶ J_{𝒞}.

Let ℰ be an elliptic curve over S, 𝒞 a genus 2 curve over S, π : 𝒞 ⟶ ℰ a curve cover. We call π minimal if it does not factor through a non-trivial isogeny ℰ_{1} ⟶ ℰ. Note that this is equivalent to ker(π_{∗} : J_{𝒞} ⟶ ℰ) being an elliptic curve, and it is equivalent to π^{∗} : ℰ ⟶ J_{𝒞} being a closed immersion; see the beginning of Section 2 of [12] and Point 7) in Section 7 of [12].

Let π : X ⟶ Y be a finite morphism of locally noetherian schemes. Then the image π(x) ∈ Y of a ramification point x ∈ X is called branch point. The set of ramification points as well as the set of branch points are closed in X and Y respectively (the set of ramification points is closed because it the support of Ω_{X/Y}, and the set of branch points is then the image of a closed set under a finite morphism, thus also closed). The corresponding schemes with the reduced induced scheme structures are called ramification locus and branch locus respectively.

We say that a field extension K|k is regular if K ⊗_{k} \overline{k} is a domain (\overline{k} = algebraic closure of k) (cf.[14, VIII, §4]). This notion should not be confused with the notion of a regular scheme.

Acknowledgments

The authors would like to thank E. Kani and E. Viehweg for fruitful discussions on questions related to this work. They thank the anonymous referee for carefully reading the manuscript and for various helpful suggestions.

2 The Ramification Loci of Covers of Genus 2 of an Elliptic Curve

Let \overline{κ} be an algebraically closed field. When we speak of the “intersection” of two curves on a smooth surface, we mean the scheme-theoretic intersection. We will use some easy results from intersection theory which we recall in an appendix to this section.

Let C be a curve of genus 2 over \overline{κ}.

Proposition 2.1 Let P be a closed point of C, let c : C ⟶ E be a cover which maps P to 0_{E}, let ι : C ⟶ J_{C} be the canonical immersion which maps P to 0_{E}. Then the following assertions are equivalent.

• c is unramified in P.

• ι(C) and ker(c_{∗}) intersect transversely in ι(P) (inside J_{C}).

Proof. Let c^{−1}(0_{E}) = C_{0_{E}} = C ×_{E} 0_{E} be the fiber of c at 0_{E}. Note that c = c_{∗} ∘ ι. Consider the following Cartesian diagram.

\xymatrix{c^{−1}(0_{E}) = C ×_{E} 0_{E} \ar@{^{(}->}[r] \ar[d] & C \ar@{^{(}->}[d]^{ι} \\ ker(c_{∗}) \ar[d] \ar@{^{(}->}[r] & J_{C} \ar[d]^{c_{∗}} \\ 0_{E} \ar@{^{(}->}[r] & E}

We know that c^{−1}(0_{E}) is a 0-dimensional closed subscheme on C, and its support contains P. Both statements of the proposition are equivalent to the multiplicity of c^{−1}(0_{E}) at P being 1. □

Proposition 2.2 Let c_{1} : C ⟶ E_{1}, c_{2} : C ⟶ E_{2} be minimal covers such that there does not exist an isomorphism of curves σ : E_{1} ˜ ⟶E_{2} with σ ∘ c_{1} = c_{2}. Then ker(c_{1∗}) and ker(c_{2∗}) are abelian subvarieties of J_{C} which intersect in a finite group scheme. There is the following alternative:

• Either the intersection of ker(c_{1∗}) and ker(c_{2∗}) is a reduced (étale) finite group scheme and the ramification loci of c_{1} and c_{2} are disjoint.

• or the intersection of ker(c_{1∗}) and ker(c_{2∗}) is a non-reduced finite group scheme and the ramification loci of c_{1} and c_{2} are equal.

Note that if char(\overline{κ}) = 0 the first alternative holds.

Proof. The fact that the abelian subvarieties ker(c_{1∗}) and ker(c_{2∗}) of J_{C} intersect in a finite group scheme follows because by assumption there does not exist a σ̃ ∈ Iso(E_{1}, E_{2}) with σ̃ ∘ c_{1}_{∗} = c_{2}_{∗}.

Note that ker(c_{1∗}) and ker(c_{2∗}) intersect transversely at 0 if and only if the intersection of the two elliptic curves is a reduced (étale) finite group scheme; see also Lemma 2.6).

Let P be a closed point of C. By composing the c_{i} with suitable translations on E_{i}, we can assume that c_{i}(P) = 0_{E_{i}} (i = 1, 2). We can now apply the previous proposition. By the fact that “non-transversal intersection at P ” is an equivalence relation (see Lemma 2.7), we conclude:

If ker(c_{1∗}) ∩ ker(c_{2∗}) is a non-reduced finite group scheme, P is a ramification point of c_{1} if and only if it is a ramification point of c_{2}.

If on the other hand ker(c_{1∗}) ∩ ker(c_{2∗}) is a reduced finite group scheme, P cannot be a ramification point of both c_{1} and c_{2}. □

Proposition 2.3 Assume that char(\overline{κ}) = p > 0. Let c_{1} : C ⟶ E and c_{2} : C ⟶ E be minimal covers. Let \overline{c}_{1}_{∗}, \overline{c}_{2}_{∗} ∈ Hom\overline{κ}(J_{C}, E) ⊗_{ℤ} 𝔽_{p} be the induced elements, and assume that \overline{c}_{1}_{∗} = \overline{c}_{2}_{∗}. Then the ramification loci of c_{1} and c_{2} are equal.

Proof. Under the assumption of the proposition, ker(c_{1}_{∗})[p] is equal to ker(c_{2∗})[p] (as closed subschemes of J_{C}). In particular, ker(c_{1}_{∗}) = ker(c_{2}_{∗}) or ker(c_{1}_{∗}) ∩ ker(c_{2}_{∗}) is non-reduced. The result follows with the last proposition. □

Appendix to Section 2: Facts about Intersection Theory

Let \overline{κ} be an algebraically closed field, let X be a smooth surface over \overline{κ}, and let P ∈ X be a closed point. Let Y be a curve in X. As always we assume that Y is smooth. For the following arguments it would suffice that X and Y are smooth at P. Note that X is locally factorial, so if Y ↪ X is a 1-dimensional closed subscheme of X containing P with canonical closed immersion ι_{Y} : Y ↪ X, the kernel of ι_{Y}^{#} : 𝒪_{X,P} ⟶ 𝒪_{Y,P} is generated by a single element f ∈ 𝒪_{X,P} (unique up to a unit). We call such an f a local equation of Y at P. Note that f ∈ 𝔪_{X,P}.

Definition 2.4 Let Y_{1} and Y_{2} be two curves on X such that P ∈ Y_{1}, P ∈ Y_{2}. Then Y_{1} and Y_{2} intersect transversely at P if the local equations of Y_{1} and Y_{2} at P generate 𝔪_{X,P}.

Lemma 2.5 Let Y be a curve in X, let P be a closed point on Y. Then the local equation of Y at P does not lie in 𝔪_{X,P}^{2} .

Proof. This follows easily from the fact that X is regular and 2-dimensional whereas Y is regular and 1-dimensional. □

For the following lemma, note that the surjection 𝔪_{X,P} ⟶ 𝔪_{Y_{i},P} induces by dualization injections of (Zariski) tangent spaces T_{Y_{i},P} ↪ T_{X,P}.

Lemma 2.6 Let Y_{i} ⊂ X (i = 1, 2) be two curves in X meeting in P. Let Y_{1} ∩ Y_{2} := Y_{1} ×_{X} Y_{2} be the scheme-theoretic intersection of Y_{1} and Y_{2}. The following statements are equivalent:

a) Y_{1} and Y_{2} intersect transversely at P.

b) Viewed as elements in 𝔪_{X,P} /𝔪_{X,P}^{2} , the local equations of Y_{1} and Y_{2} at P are linearly independent.

c) The canonical homomorphism 𝔪_{X,P} /𝔪_{X,P}^{2} ⟶ 𝔪_{Y_{1},P} /𝔪_{Y_{1},P}^{2} ×𝔪_{Y_{2},P} /𝔪_{Y_{2},P}^{2} is an isomorphism.

d) The canonical homomorphism T_{Y_{1},P} ×T_{Y_{2},P} ⟶ T_{X,P} is an isomorphism.

e) There exists a neighborhood U of P such that (Y_{1} ∩ Y_{2})|_{U} is equal to the closed subscheme given by the closed immersion Spec (\overline{κ}) ↪ U at P.

Proof. a) and b) are equivalent by Nakayama’s Lemma. The equivalence of b) and c) is easy, and d) is a “dual formulation” of c). The local ring of Y_{1} ∩ Y_{2} at P is isomorphic to 𝒪_{X,P} /(f_{1}, f_{2}). This implies that a) holds if and only if the local ring of Y_{1} ∩ Y_{2} at P is isomorphic to Spec(\overline{κ}). This in turn is equivalent to e). □

We define a relation on the set of all curves Y lying in X and meeting P as follows: Y_{1} is equivalent to Y_{2} if and only if Y_{1} and Y_{2} do not intersect transversely at P.

Lemma 2.7 The relation just defined is an equivalence relation.

Proof. We only have to prove the transitivity. This follows from point b) in the above equivalences with Lemma 2.5. □

3 Minimal Covers with Prescribed Branch Loci

In this section, we first show how results of [1] imply the statements which were implicitly used in Subsection 1.2. Then we use the results of the previous section to derive a preliminary result on our way to prove Theorem 1.

Let us start off as in Subsection 1.2: Let p be an odd prime number, and let N be an odd natural number. Let S be a locally noetherian, integral, regular scheme of characteristic p. Let τ : ℰ ⟶ ℰ′ be an isogeny of elliptic curves over S of degree N.

Also as in Subsection 1.2, let ℰ[2]^{#} := ℰ[2]−[0_{ℰ}], where [0_{ℰ}] is the Cartier divisor associated to the zero-section 0_{ℰ} of ℰ (similar definitions for ℰ′), let 𝒫 := E/⟨−1⟩, 𝒫′ := E′/⟨−1⟩ with projections ρ : ℰ ⟶ 𝒫 and ρ′ : ℰ′ ⟶ 𝒫′.

Let ψ := τ|_{ℰ[2]} : ℰ[2] ⟶ ℰ′[2].

It is well-known that 𝒫 is isomorphic to the ℙ^{1}-bundle ℙ(q_{∗}(ℒ(0_{ℰ})), where q : ℰ ⟶ S is the structure morphism (see for example the discussion above Theorem 2 in [1]), and a similar statement holds for 𝒫′. Note that ℰ[2]^{#} ⟶ S and ℰ′[2]^{#} ⟶ S are étale covers of degree 3 and ℰ[2]^{#} ⟶ 𝒫 as well as ℰ′[2]^{#} ⟶ 𝒫′ are closed immersions. By [1, Propsition B.4], there exists a unique S-isomorphism γ : 𝒫 ⟶ 𝒫′ with γ∘ρ|_{ℰ[2]^{#}} = ρ′∘ψ. (The fact that 𝒫 is isomorphic to ℙ_{S}^{1} if ℰ[2] ≃ (ℤ/2ℤ)^{2} also follows from [1, Propsition B.4].)

Lemma 3.1 The conditions

• for no geometric point s of S, there exists an isomorphism α : ℰ_{s} ⟶ ℰ′_{s} such that α|_{ℰ_{s}[2]^{#}} = ψ

• γ ∘ ρ([0_{ℰ}]) and ρ′([0_{ℰ′}]) are disjoint

are equivalent.

Proof. It is enough to prove the lemma in the case that S is the spectrum of an algebraically closed field field. In this situation one can use [1, Lemma A.1]. □

For the rest of this section, we assume that the conditions of Lemma 3.1 are satisfied.

Lemma 3.2 ℰ ×_{𝒫′} ℰ′ is integral.

As in Notation 1.3, let 𝒞 be the normalization of ℰ ×_{𝒫′} ℰ′, and let

π : 𝒞 ⟶ ℰ, π′ : 𝒞 ⟶ ℰ′

be the canonical projections.

Proposition 3.3 𝒞 is a curve of genus 2 over S and the maps π, π′ are degree 2 covers.

For Lemma 3.2 and Proposition 3.3, see [1, Proposition 4.4].

In the notation of [1] ([1, Definition 2.7, Proposition 2.15]), (𝒞, π, π′) is a “normalized symmetric pair” corresponding to (ℰ, ℰ′, ψ). This means that • ker(π_{∗}) = Im((π′)^{∗}) and ker(π^{∗}) = Im(π′_{∗}),

• π_{∗}W_{𝒞} = ℰ[2]^{#} and π′_{∗}W_{𝒞} = ℰ′[2]^{#},

• π^{∗}|_{ℰ[2]} = (π′)^{∗} ∘ ψ.

Here, W_{𝒞} is the Weierstraß divisor of 𝒞 and ℰ[2]^{#} := ℰ[2]−[0_{ℰ}], where [0_{ℰ}] is the Cartier divisor associated to the zero-section 0_{ℰ} of ℰ (similar definition for ℰ′[2]^{#}).

Remark 3.4 Even without the assumption that S is regular, one can prove that a genus 2 curve 𝒞 over S and two degree 2 covers π, π′ with the above conditions exist. Moreover, the triple (𝒞, π, π′) is up to unique isomorphism (defined in an obvious way) uniquely determined by the three conditions. On the other hand, if the two equivalent conditions in Lemma 3.1 are not satisfied, no normalized symmetric pair corresponding to (ℰ, ℰ, ψ) exists. For more information on these issues, we refer the reader to [1].

Let δ : ℰ × ℰ′ ⟶ J_{𝒞} be the homomorphism given by the matrix ( π^{∗} (π′)^{∗} ). (We identify ℰ and ℰ′ with their dual abelian varieties.)

By [1, Proposition 2.14], the kernel of δ is Graph(−ψ) = Graph(ψ), and the pull-back of the canonical principal polarization of J_{𝒞} via π is twice the canonical product polarization of ℰ × ℰ′.

Proposition 3.5 The Jacobian J_{𝒞} is isomorphic to ℰ × ℰ′.

Proof. As said above, J_{𝒞} is isomorphic to (ℰ × ℰ′)/ Graph(−ψ). The latter is in turn isomorphic to ℰ × ℰ′. In fact, the isogeny

Φ : ℰ × ℰ′ ⟶ ℰ × ℰ′

given by the matrix ( \begin{matrix}2 & 0 \\ τ & 1 \end{matrix} ) has kernel Graph(−ψ), thus it induces an

isomorphism (ℰ × ℰ′)/ Graph(−ψ) ˜ ⟶ℰ × ℰ′. □

We use the group structure on ℰ and obtain for all a, b ∈ ℤ a morphism aπ + bτ̂π′ : 𝒞 ⟶ ℰ. We have a bilinear form

β : ℤ^{2} × ℤ^{2} ⟶ End(ℰ), ((a, b), (c, d)) ↦ (aπ + bτ̂π′)_{∗} (cπ + dτ̂π′)^{∗}

with β((a, b), (a, b)) = deg(aπ + bτ̂π′). As by definition π_{∗} ∘ (π′)^{∗} = 0 and π′_{∗} ∘ π^{∗} = 0, we have β((a, b), (c, d)) = 2ac + 2Nbd ∈ ℤ. This implies that

deg(aπ + bτ̂π′) = 2a^{2} + 2Nb^{2}.  (1)

Let us fix the following notation: For (a, b) ∈ ℤ^{2}, we denote the ramification locus of aπ + bτ̂π′ by V_{(a,b)} and the branch locus by Δ_{(a,b)}. Further, we set V := V_{(1,0)} and Δ := Δ_{(1,0)}. The following lemma is [1, Proposition 3.11].

Lemma 3.6 π′|_{V} is the zero-element in the abelian group ℰ′(V).

Proposition 3.7 If p|b, then V_{(1,b)} = V and Δ_{(1,b)} = Δ.

Proof. Let \overline{κ} be any algebraically closed field, and let s be any point of S(\overline{k}). By Proposition 2.3, π_{s} and (π + bτ̂π′)_{s} : 𝒞_{s} ⟶ ℰ_{s} have the same ramification locus.

Now let x be a \overline{κ}-valued point of 𝒞 lying over s. Then the support of x is a ramification point of π if and only if it is a ramification point of π_{s}: both conditions are equivalent to the inclusion of \overline{κ} into the local ring of x in the geometric fiber 𝒞_{π(x)} being not surjective. A corresponding statement holds for π + bτ̂π′.

This implies that π and π + bτ̂π′ : 𝒞 ⟶ ℰ have the same ramification locus, that is, the underlying sets of V_{(1,b)} and V are equal. As both schemes are endowed with the reduced induced scheme structure, we have that V_{(1,b)} = V.

By Lemma 3.6, it follows that

(π + bτ̂π′)|_{V} = π|_{V} + bτ̂π′|_{V} = π|_{V} ,

In particular, V is mapped under π+bτ̂π′ to Δ. This implies that Δ_{(1,b)} = Δ. □

Proposition 3.8 If 2|b, then π + bτ̂π′ is minimal of degree 2 + 2Nb^{2}.

Proof. The statement on the degree follows from (1).

To show that π +bτ̂π′ is minimal, that is, that it does not factor through a non-trivial isogeny ℰ̃ ⟶ ℰ, it suffices to show that the homomorphism

π_{∗} + bτ̂π′_{∗} : J_{𝒞} ⟶ ℰ

does not factor through a non-trivial isogeny ℰ̃ ⟶ ℰ.

So assume that π_{∗} + bτ̂ factors through the isogeny ℰ̃ ⟶ ℰ. Then

(π_{∗} + bτ̂π′_{∗}) ∘ π^{∗} : ℰ ⟶ ℰ

also factors through ℰ̃ ⟶ ℰ. Now (π_{∗} + bτ̂π′_{∗}) ∘ π^{∗} = 2 · id_{ℰ}. This implies that the degree of the isogeny ℰ̃ ⟶ ℰ divides 4.

To rule out that the degree is 2 or 4, we consider the commutative diagram

\xymatrix{ℰ × ℰ′ \ar[d]_{δ} \ar[dr]^{Φ} \ar[dd]_{[2]} \\ J_{𝒞} ≃ (ℰ × ℰ′)/ Graph(−ψ) \ar[r]^{∼} \ar[d]_{δ̂λ_{𝒞}} \ar[dd]_{π_{∗}+bτ̂π′_{∗}} & ℰ × ℰ′ , \ar[dl]^{Ψ} \\ ℰ × ℰ′ \ar[d] \\ ℰ}

where Φ is given as in Proposition 3.5, Ψ : ℰ × ℰ′ ⟶ ℰ × ℰ′ is given by

the matrix ( \begin{matrix}1 & 0 \\ −τ & 2 \end{matrix} ), and the last vertical arrow is given by the matrix

( 1 bτ̂ ). The homomorphism

δ̂λ_{𝒞} : J_{𝒞} ⟶ ℰ × ℰ′

is given by the matrix ( \begin{matrix}π_{∗} \\ π′_{∗} \end{matrix} ).

Under the horizontal isomorphism in the diagram, π_{∗} + bτ̂π′_{∗} is given by the matrix ( 1 − bN 2bτ̂ ). Let ι : ℰ ⟶ ℰ × ℰ′ ≃ J_{𝒞} be the inclusion of the first summand. Then (π_{∗} + bτ̂π′_{∗}) ∘ ι = (1 − bN) · id_{ℰ}. As by assumption 1 − bN is odd, the degree of ℰ̃ ⟶ ℰ cannot be divisible by 2. It is thus an isomorphism.

We conclude that π + bτ̂π′ : 𝒞 ⟶ ℰ is minimal. □

The last two propositions imply:

Proposition 3.9 The covers π_{i} := π+2ipτ̂π′ : 𝒞 ⟶ ℰ (i ∈ ℕ) are minimal of degree 2 + 8N(ip)^{2} and have the same ramification loci and the same branch loci.

We summarize the results stated in Lemma 3.1, Lemma 3.2 and in Propositions 3.3, 3.5 and 3.9 in the following proposition.

Proposition 3.10 Let p be an odd prime, let N be odd. Let S be a locally noetherian, integral, regular scheme of characteristic p, let τ : ℰ ⟶ ℰ′ be a cyclic isogeny of degree N over S.

Then there exists a unique S-isomorphism γ : 𝒫 ˜ ⟶𝒫′ such that ρ′ ∘ τ|_{ℰ[2]^{#}} = γ ∘ρ|_{ℰ[2]^{#}} . Assume that the following two equivalent conditions are satisfied.

• For no geometric point s of S, there exists an isomorphism α : ℰ_{s} ⟶ ℰ′_{s} such that α|_{ℰ_{s}[2]} = τ_{s}|_{ℰ_{s}[2]}.

• γ(ρ([0_{ℰ}])) and ρ′([0_{ℰ′}]) are disjoint.

Let 𝒞 be the normalization of the integral scheme ℰ×_{𝒫′} ℰ′ (where the product is with respect to γ∘ρ and ρ′). Then 𝒞 is a curve of genus 2 with J_{𝒞} ≃ ℰ×ℰ′, the canonical morphisms π : 𝒞 ⟶ ℰ and π′ : 𝒞 ⟶ ℰ′ are degree 2 covers and there exists a sequence (π_{i})_{i∈ℕ_{0}} of minimal covers π_{i} : 𝒞 ⟶ ℰ with pairwise distinct degrees and π_{0} = π such that the ramification loci as well as the branch loci of the π_{i} are all equal.

4 Pro-Galois Curve Covers of Infinite Degree

The goal of this section is to prove the following proposition which together with Proposition 3.10 implies Theorem 1.

Proposition 4.1 Let S be an integral, regular scheme of finite type over ℤ[1/2] or more generally a locally noetherian, integral, regular scheme over ℤ[1/2] such that π_{1}(S)/π_{1}(S)^{2} is finite. (Here π_{1}(S) denotes the fundamental group of S with respect to some base point.)

Let ℰ be an elliptic curve over S. Assume that there exists a sequence (π_{i})_{i∈ℕ_{0}} of minimal covers π_{i} : 𝒞_{i} ⟶ ℰ (where the 𝒞_{i} are curves of genus 2 over S) with pairwise distinct degrees and deg(π_{0}) = 2 as well as deg(π_{i}) ≥ 5 for i ≥ 1 such that the branch loci of the π_{i} are all equal.

Let 𝒞 := 𝒞_{0}. Then there exists a connected Galois cover T ⟶ S with Galois group a (finite) elementary abelian 2-group such that the curve 𝒞_{T} over T has a pro-Galois curve cover whose Galois group 𝒢 fits into an exact sequence

1 ⟶ ∏_{i=1}^{∞} A_{n_{i}} ⟶ 𝒢 ⟶ (ℤ/2ℤ)^{r} ⟶ 1

for some r ≤ 4, where n_{i} := deg(π_{i}) and A_{n_{i}} is the alternating group on n_{i} elements.

The rest of this section is devoted to a proof of this proposition.

We first show that any integral, regular scheme S of finite type over ℤ[1/2] the assumption that π(S)/π(S)^{2} is finite is satisfied. We thereby give some background information on this condition as well.

Let S be any scheme over ℤ[1/2]. Then we have an exact sequence

0 ⟶ Γ(S, 𝒪_{S})^{∗}/Γ(S, 𝒪_{S})^{∗2} ⟶ Hom(π_{1}(S), ℤ/2ℤ) ⟶ Pic(S)[2] ⟶ 0 ;

see [17, Proposition 4.11] together with [17, Corollary 4.7]. We thus see that π_{1}(S)/π_{1}(S)^{2} is finite if and only if both Pic(S)[2] and Γ(S, 𝒪_{S})^{∗}/Γ(S, 𝒪_{S})^{∗2} are finite.

If now S is an integral, normal scheme of finite type over ℤ, both groups Pic(S)[2] and Γ(S, 𝒪_{S})^{∗}/Γ(S, 𝒪_{S})^{∗2} are in fact finite, and hence so is π_{1}(S)/π_{1}(S)^{2}; see [13, Theorems 7.4 and 7.5].

Now let the assumptions of the proposition be satisfied. Let Δ be the branch locus of each of the π_{i}. As all residue characteristics are distinct from 2 and deg(π_{0}) = 2, for any s ∈ S (π_{0})_{s} is generically étale, in particular, (π_{0})_{s} is finite, and all branch points in codimension 1 on ℰ lie in the generic fiber over S.

More precisely, we have by [1, Lemma 3.13]:

Lemma 4.2 The canonical morphism Δ ⟶ S is an étale cover of degree 2.

Note that the curve ℰ over S gives rise to an elliptic curve over κ(S). In particular, the field extension κ(ℰ)|κ(S) is regular. Let L_{i}|κ(ℰ) be the Galois closure of the extension of function fields π_{i}^{#} : κ(ℰ) ↪ κ(𝒞_{i}).

Let us fix some compositum L_{i}\overline{κ(S)} of L_{i} and \overline{κ(S)} over κ(S).

Lemma 4.3 The Galois group of L_{i}\overline{κ(S)}|κ(ℰ)\overline{κ(S)} is isomorphic to the symmetric group S_{n_{i}}. In particular, the Galois group of L_{i}|κ(ℰ) is isomorphic to S_{n_{i}}, and L_{i}|κ(S) is regular.

Proof. As the extension of function fields κ(𝒞_{i})\overline{κ(S)}|κ(ℰ)\overline{κ(S)} over \overline{κ(S)} has 2 branched places, the conorm of each of these places has the form 𝔓_{1}^{2}𝔓_{2} ⋯ 𝔓_{n_{i}−1}, where the 𝔓_{j} are pairwise distinct places. It follows that the Galois group of L_{i}\overline{κ(S)}|κ(ℰ)\overline{κ(S)} (seen as permutation group on n_{i} elements) contains a transposition.

Moreover, as π_{i} : 𝒞_{i} ⟶ ℰ is minimal, κ(𝒞_{i})|κ(ℰ) has no proper intermediate fields, and the same is true for κ(𝒞_{i})\overline{κ(S)}|κ(ℰ)\overline{κ(S)}.

Hence the Galois group of L_{i}\overline{κ(S)}|κ(ℰ)\overline{κ(S)} is a primitive transitive subgroup of S_{n_{i}} with a transposition, and hence it is equal to S_{n_{i}}; see [23, Theorem 13.3]. □

Let L_{i}^{0} := L_{i}^{A_{n_{i}}}. Let 𝒞̃_{i} be the normalization of ℰ in L_{i}, and let 𝒞̃_{i}^{0} be the normalization of ℰ in L_{i}^{0}.

Let us fix inclusions of κ(𝒞̃_{i})|κ(ℰ) into some algebraic closure of κ(ℰ). For t ∈ ℕ, let 𝒟_{t} be the normalization of ℰ in the compositum of κ(𝒞̃_{0}), …, κ(𝒞̃_{t}) over κ(ℰ), and let 𝒟_{t}^{0} be the normalization of ℰ in the compositum of κ(𝒞̃_{0}^{0}), …, κ(𝒞̃_{t}^{0}) over κ(ℰ) (both composita with respect to the inclusions into the fixed algebraic closure of κ(ℰ)). The extensions κ(𝒟_{t})|κ(ℰ) and κ(𝒟_{t}^{0})|κ(ℰ) are Galois, and κ(𝒟_{t}^{0})|κ(ℰ) has as Galois group an elementary abelian 2-group.

[figure: 𝒟_{t}; 𝒞̃_{0}; 𝒞̃_{1}; ⋯; ⋯; 𝒞̃_{t}; 𝒟_{t}^{0}; 𝒞̃_{0}^{0}; 𝒞̃_{1}^{0}; ⋯; 𝒞̃_{t}^{0}; 𝒞 = 𝒞_{0}; π_{0}; 𝒞_{1}; π_{1}; ⋯; ⋯; 𝒞_{t}; π_{t}; ℰ]

Our goal is now to prove the following proposition.

Proposition 4.4 For all t ∈ ℕ, we have:

a) The morphisms 𝒟_{t} ⟶ 𝒞 and 𝒟_{t}^{0} ⟶ 𝒞 are Galois covers.

b) The Galois group of 𝒟_{t}^{0} ⟶ 𝒞 is an elementary abelian 2-group.

c) The restrictions Gal(κ(𝒟_{t})|κ(𝒟_{t}^{0}))) ⟶ Gal(κ(𝒞̃_{i})|κ(𝒞̃_{i}^{0})) induce an isomorphism

Gal(𝒟_{t} → 𝒟_{t}^{0}) ≃ Gal(κ(𝒟_{t})|κ(𝒟_{t}^{0})) ˜ ⟶ ∏_{i=1}^{t} Gal(κ(𝒞̃_{i})|κ(𝒞̃_{i}^{0})) ≈ ∏_{i=1}^{t} A_{n_{i}}.

d) Let F_{t} be the algebraic closure of κ(S) in κ(𝒟_{t}^{0}), let S_{t}^{0} be the normalization of S in F_{t}. Then S_{t}^{0} ⟶ S is a Galois cover, and 𝒟_{t}^{0} as well as 𝒟_{t} are in a canonical way curves over S_{t}^{0}. The induced morphism 𝒟_{t}^{0} ⟶ 𝒞_{S_{t}^{0}} is a Galois curve cover over S_{t}^{0} with Galois group isomorphic to (ℤ/2ℤ)^{r} for some r ≤ 4. We have an exact sequence of Galois groups

1 ⟶ Gal(𝒟_{t}^{0} → 𝒞_{S_{t}^{0}}) ⟶ Gal(𝒟_{t}^{0} → 𝒞) ⟶ Gal(S_{t}^{0} → S) ⟶ 1.

In particular, Gal(S_{t}^{0} → S) is an elementary abelian 2-group.

Let us for the moment assume that we have proved Proposition 4.4. Then it is not difficult to derive Proposition 4.1:

For any t ∈ ℕ, the cover 𝒟_{t}^{0} ⟶ 𝒞 is a composite of the cover 𝒟_{t}^{0} ⟶ 𝒞_{S_{t}^{0}} and the cover 𝒞_{S_{t}^{0}} ⟶ 𝒞. By the last item of Proposition 4.4 and the assumption that π_{1}(S)/π_{1}(S)^{2} is finite, there exists some t_{0} ∈ ℕ such that for t > t_{0}, the canonical morphism 𝒟_{t}^{0} ⟶ 𝒟_{t_{0}}^{0} is an isomorphism.

Let 𝒟^{0} := 𝒟_{t_{0}}^{0} = \underleftarrow{lim} 𝒟_{t}^{0}, let T := S_{t_{0}}^{0}. Then for all t ≥ t_{0}, 𝒟_{t} is a curve over T, and so is 𝒞_{T}. Let 𝒟 := \underleftarrow{lim} 𝒟_{t}. Then 𝒟 is a pro-Galois curve cover of 𝒞_{T} of infinite degree. Moreover, its Galois group is an extension of a group of the form (ℤ/2ℤ)^{r} with r ≤ 4 by ∏_{i=1}^{∞} Gal(κ(𝒞_{i})|κ(𝒞_{i}^{0})) ≈ ∏_{i=1}^{∞} A_{n_{i}}. This implies Proposition 4.1.

∏_{i=1}^{∞} A_{n_{i}}

𝒟

[figure]

pro-Galois ; curve cover ; over T

\begin{matrix}(ℤ/2ℤ)^{r} \\ with r ≤ 4 \end{matrix}

𝒟^{0}

[figure]

𝒞_{T}

[figure] [figure]

[figure]

𝒞

[figure]

[figure]

T

[figure]

Galois with

elementary abelian

2-group S

Now we give the proof of Proposition 4.4. It is divided into several lemmata.

Proof of Assertion a)

Let us recall Abhyankar’s Lemma.

Lemma 4.5 (Abhyankar’s Lemma) Let K be a field, L|K, M|K finite separable extensions of K, N = LM a compositum of M and L over K. Let v be a discrete valuation of N, v_{M}, v_{L}, v_{K} the restrictions of v to M, L, K respectively. Assume that the extensions v_{M}|v_{K} and v_{L}|v_{K} are tame and that e(v_{M}|v_{K})|e(v_{L}|v_{K}). Then v|v_{L} is unramified.

For a proof see [22, Proposition III.8.9] (the assumptions in [22, Proposition III.8.9] that M|K be an extension of function fields in one variable and v a valuation of function fields is not necessary).

Lemma 4.6 The morphisms 𝒞̃_{i} ⟶ ℰ and 𝒟_{t} ⟶ ℰ are finite, the branched points in codimension 1 are the generic points of Δ, and the corresponding ramification indices are 2.

Proof. The first statement follows from the following general fact: If one normalizes an integral noetherian scheme in a finite separable extension of its function field, the canonical morphism is finite; cf. [15, Chapter 4, Proposition 1.25]. By Abhyankar’s Lemma, the ramified points in codimension 1 are the generic points of Δ and the ramification indices divide 2. As the corresponding extensions of function fields are Galois, the ramification indices for points above the branched points in codimension 1 are 2. □

Lemma 4.7 The morphisms 𝒟_{t} ⟶ 𝒞̃_{0} = 𝒞 are étale covers.

Proof. Let t ∈ ℕ be fixed.

By Lemma 4.6, the branch loci of 𝒟_{t} ⟶ ℰ and 𝒞 ⟶ ℰ are equal, and the ramification indices of the branched points in codimension 1 are dividing 2. As L_{0}|κ(ℰ) has degree 2, this implies that for all points x in codimension 1 of ℰ, all ramification indices of 𝒟_{t} ⟶ ℰ at x divide all ramification indices of 𝒞̃_{0} ⟶ ℰ at x.

By Abhyankar’s Lemma, 𝒟_{t} ⟶ 𝒞 is unramified at all points of codimension 1. As S is regular and 𝒞 is smooth over S, 𝒞 is regular (in particular normal).

By “purity of the branch locus” ([8, Exposé X, 3.1]), 𝒟_{t} ⟶ 𝒞 is unramified everywhere, and because 𝒞 is normal, it is étale; see [8, Exposé I, Corollaire 9.11]. □

Lemma 4.8 The morphisms 𝒟_{t} ⟶ 𝒞, 𝒟_{t} ⟶ 𝒟_{t}^{0} and 𝒟_{t}^{0} ⟶ 𝒞 are Galois covers.

Proof. By the last lemma, we already know that the morphisms 𝒟_{t} ⟶ 𝒞 are étale covers. Moreover, the corresponding extensions of functions fields are Galois. As 𝒞 (respectively 𝒟_{t}^{0}) is normal, this implies that the covers 𝒟_{t} ⟶ 𝒞 and 𝒟_{t} ⟶ 𝒟_{t}^{0} are Galois. This implies with [8, Corollaire 3.4] that 𝒟_{t}^{0} ⟶ 𝒞 is étale. It is Galois because the corresponding extension of function fields is Galois (and 𝒞 is normal). □

Proof of Assertions b) and c)

We need a group theoretical lemma.

Lemma 4.9 Let G_{1}, …, G_{t} be groups such that for any i, j with i ≠ j, there does not exist any non-trivial simple group which is a quotient group of both G_{i} and G_{j}. Let G be a group with surjective homomorphisms p_{i} : G ⟶ G_{i}. Then the induced homomorphism p : G ⟶ ∏_{i=1}^{t} G_{i} is surjective.

Proof. The proof can be done by induction on t. For t = 2, the proof is as follows.

Let N := ⟨ker(p_{1}) ∪ ker(p_{2})⟩ – this is a normal subgroup of G. We have canonical surjective homomorphisms G_{i} ≃ G/ ker(p_{i}) ⟶ G/N. By assumption, G/N is trivial, that is, G = N.

This implies that p_{1}|_{ker(p_{2})} : ker(p_{2}) ⟶ G_{1} and p_{2}|_{ker(p_{1})} : ker(p_{1}) ⟶ G_{2} are surjective. This in turn implies that the image of p contains G_{1} ×{1} and {1} × G_{2}, thus p is surjective. □

Now we can proceed with the proof of Proposition 4.4.

By construction, κ(𝒟_{t}^{0})|κ(ℰ) is Galois with Galois group an elementary abelian 2-group. It follows that the Galois group of 𝒟_{t}^{0} ⟶ 𝒞 is an elementary abelian 2-group. This proves Assertion b).

The covers 𝒟_{t} ⟶ 𝒟_{t}^{0} are more interesting:

Lemma 4.10 The restrictions Gal(κ(𝒟_{t})|κ(𝒟_{t}^{0}))) ⟶ Gal(κ(𝒞̃_{i})|κ(𝒞̃_{i}^{0})) are surjective and induce an isomorphism

Gal(𝒟_{t} → 𝒟_{t}^{0}) ≃ Gal(κ(𝒟_{t})|κ(𝒟_{t}^{0})) ˜ ⟶ ∏_{i=1}^{t} Gal(κ(𝒞̃_{i})|κ(𝒞̃_{i}^{0})) ≈ ∏_{i=1}^{t} A_{n_{i}}.

Proof. The group Gal(κ(𝒞̃_{0})|κ(𝒞̃_{0}^{0})) is trivial, so let i ≥ 1. The group Gal(κ(𝒞̃_{i})|κ(𝒞̃_{i}^{0})) is isomorphic to A_{n_{i}}, and the group Gal(κ(𝒟_{i}^{0})|κ(𝒞̃_{i}^{0})) is an elementary abelian 2-group. As n_{i} ≥ 5 (because i ≥ 1), the group A_{n_{i}} is simple (see [21, Theorem 3.15]) and in particular has no non-trivial elementary abelian 2-group as quotient. By Lemma 4.9, the extensions κ(𝒟_{t}^{0}) and κ(𝒞̃_{i}) are linearly disjoint over κ(𝒞̃_{i}^{0}) (inside κ(𝒟_{t})) (this will also be used in the proof of Lemma 4.12). In particular, the restriction map Gal(κ(𝒟_{t}^{0}) κ(𝒞̃_{i})|κ(𝒟_{t}^{0})) ⟶ Gal(κ(𝒞̃_{i})|κ(𝒞̃_{i}^{0})) is an isomorphism.

Again by Lemma 4.9, the induced homomorphism Gal(κ(𝒟_{t})|κ(𝒟_{t}^{0})) ⟶ ∏_{i=1}^{t} Gal(κ(𝒞̃_{i})|κ(𝒞̃_{i}^{0})) ≈ ∏_{i=1}^{t} A_{n_{i}} is surjective. It is obvious that it is injective. □

Proof of Assertion d)

Again we first need a group theoretical lemma.

Lemma 4.11 Let G_{1}, …, G_{t} be finite groups such that any i, j with i ≠ j, there is no simple group which occurs as a composition factor of both G_{i} and G_{j}. Let G be a group with a homomorphism φ : ∏_{i=1}^{t} G_{i} ⟶ G such that for i = 1, …, t, the restriction of φ to G_{i} (regarded as a subgroup of ∏_{i=1}^{t} G_{i}) has kernel N_{i}. Then φ has kernel ∏_{i=1}^{t} N_{i} (regarded as a subgroup of ∏_{i=1}^{t} G_{i}).

Proof. Recall that if G is a finite group and N ◁ G is a normal subgroup, then composition series of both N and G/N in a canonical way give rise to a composition series of G. In particular, the set of composition factors of G is the union of the sets of composition factors of N and G/N.

The assumption and this remark imply that for any j = 2, …, t, there is no simple group which occurs in the composition series of both ∏_{i=i}^{j−1} G_{i} and G_{j}. Because of this, the general case follows by induction from the case for t = 2.

In this case, the proof is as follows.

Obviously, N_{1} × N_{2} in contained in the kernel of φ. The group φ(G_{1} × {1}) ∩ φ({1} × G_{2}) is normal in both φ(G_{1} × {1}) and φ({1} × G_{2}).

However by assumption and the remark at the beginning of the proof, the groups φ(G_{1} × {1}) and φ({1} × G_{2}) cannot contain a non-trivial common normal subgroup. This implies that the group φ(G_{1} × {1}) ∩ φ({1} × G_{2}) is trivial.

Now if φ(g_{1}, g_{2}) = 1, then φ(g_{1}, 1) = φ(1, g_{2}^{−1}) ∈ φ(G_{1} × {1}) ∩ φ({1} × G_{2}). Thus this has to be 1. By the definition of N_{1} and N_{2}, g_{1} ∈ N_{1}, g_{2} ∈ N_{2}, that is, (g_{1}, g_{2}) ∈ N_{1} × N_{2}. □

Lemma 4.12 Let F_{t} be the algebraic closure of κ(S) in κ(𝒟_{t}^{0}). Then κ(𝒟_{t})|F_{t} is regular.

Proof. As we have seen in the proof of Lemma 4.10, the fields κ(𝒟_{t}^{0}) and κ(𝒞̃_{i}) are linearly disjoint extensions of κ(𝒞̃_{i}^{0}). Let F_{t,i} be the algebraic closure of F_{t} in κ(𝒟_{t}^{0})κ(𝒞̃_{i}). If we apply Lemma 4.11 to the restriction map Gal(κ(𝒟_{t}^{0})κ(𝒞̃_{i})|κ(𝒞̃_{i}^{0})) ⟶ Gal(F_{t,i}κ(𝒞̃_{i}^{0})|κ(𝒞̃_{i}^{0})), we obtain that F_{t,i}κ(𝒞̃_{i}^{0}) = F_{t}κ(𝒞̃_{i}^{0}). As κ(𝒞̃_{i}^{0})|κ(S) is regular, we obtain F_{t,i} = F_{t}, that is, κ(𝒟_{t}^{0})κ(𝒞̃_{i})|F_{t} is regular.

By the structure of Gal(κ(𝒟_{t})|κ(𝒟_{t}^{0})) and again Lemma 4.11, this implies that κ(𝒟_{t})|F_{t} is regular. □

Lemma 4.13 Let q_{t} : 𝒟_{t}^{0} ⟶ S be the structure morphism, and let S_{t}^{0} be the normalization of S in F_{t}. Then S_{t}^{0} is equal to Spec(q_{t}_{∗}(𝒪_{𝒟_{t}^{0}} )), and the canonical morphism S_{t}^{0} ⟶ S is an étale cover.

Proof. As 𝒟_{t}^{0} is smooth and proper over S, for all s ∈ S, the ring of global sections of the fiber (𝒟_{t}^{0})_{s} is a finite separable algebra over κ(s). This implies with [9, III (7.8.7)] that Spec(q_{t}_{∗}(𝒪_{𝒟_{t}^{0}} )) ⟶ S is an étale cover. Again by [9, III (7.8.7)] applied to the generic point of S, one sees that the total ring of fractions of Spec(q_{t}_{∗}(𝒪_{𝒟_{t}^{0}} )) is F_{t}. This implies that Spec(q_{t}_{∗}(𝒪_{𝒟_{t}^{0}} )) is the normalization of S in F_{t}, that is, it is S_{t}^{0}; see [8, Corollaire 10.2]. □

We can consider 𝒟_{t}^{0} as an S_{t}^{0}-scheme (Stein factorization); let r_{t} : 𝒟_{t}^{0} ⟶ S_{t}^{0} be the structure morphism. Then 𝒪_{S_{t}^{0}} = r_{t}_{∗}(𝒪_{𝒟_{t}^{0}} ), and 𝒟_{t}^{0} has connected and non-empty geometric fibers over S_{t}^{0}; see [9, III (4.3.1), (4.3.4)].

By the universal property of the product, 𝒟_{t}^{0} ⟶ 𝒞 factors through 𝒞_{S_{t}^{0}} ⟶ 𝒞.

Lemma 4.14 𝒟_{t}^{0} is a curve over S_{t}^{0} and 𝒟_{t}^{0} ⟶ 𝒞_{S_{t}^{0}} is an étale curve cover.

Proof. The morphism 𝒟_{t}^{0} ⟶ 𝒞_{S_{t}^{0}} is an étale cover, because 𝒟_{t}^{0} ⟶ 𝒞 and 𝒞_{S_{t}^{0}} ⟶ 𝒞 are étale covers. This implies that 𝒟_{t}^{0} is smooth and proper over S_{0}^{t}. It is a curve because it has connected and 1-dimensional geometric fibers. □

As S_{t}^{0} ⟶ S is an étale cover and F_{t}|κ(S) is Galois with Galois group a quotient of that of κ(𝒞_{t}^{0})|κ(𝒞) (as one easily sees) and S is normal, S_{t}^{0} ⟶ S is a Galois cover with Galois group an elementary abelian 2-group. The Galois group is canonically isomorphic to that of 𝒞_{S_{t}^{0}} ⟶ 𝒞 because basechange does not change the Galois group.

By Galois theory the cover 𝒟_{t}^{0} ⟶ 𝒞_{S_{t}^{0}} is Galois, and we have the exact sequence

1 ⟶ Gal(𝒟_{t}^{0} → 𝒞_{S_{t}^{0}}) ⟶ Gal(𝒟_{t}^{0} → 𝒞) ⟶ Gal(𝒞_{S_{t}^{0}} → 𝒞) ⟶ 1

with Gal(S_{t}^{0} → S) ≃ Gal(𝒞_{S_{t}^{0}} → 𝒞), where all groups are elementary abelian 2-groups. By pull-back to a geometric fiber, we see that Gal(𝒟_{t}^{0} → 𝒞_{S_{t}^{0}}) is a group of the form (ℤ/2ℤ)^{r} with r ≤ 2 · g_{𝒞} = 4.

Now let S_{t} be analogously defined to S_{t}^{0} for 𝒟_{t}. Then 𝒟_{t} has connected geometric fibers over S_{t} and S_{t} is étale over S_{t}^{0}. By Lemma 4.12 and [9, III (7.8.7)], the generic fiber of 𝒟_{t} over S_{t}^{0} is geometrically connected. On the other hand, again by [9, III (7.8.7)] the number of connected components in each geometric fiber of 𝒟_{t} over S_{t}^{0} is equal to the degree of S_{t} over S_{t}^{0}. This implies that S_{t} ˜ ⟶S_{t}^{0}. The fact that 𝒟_{t} ⟶ 𝒟_{t}^{0} is an étale cover (Lemma 4.8) implies that 𝒟_{t} is a curve over S_{t}^{0}. This completes the proof of Proposition 4.4.

5 Complementary Results

5.1 Abelian Surfaces Isogenous and Isomorphic to a Product of Elliptic Curves

Recall that if 𝒞 ⟶ ℰ is a minimal cover of degree n over some scheme S, the Jacobian of 𝒞 is as principally polarized abelian variety isomorphic to ((ℰ × ℰ′)/Graph(−ψ), λ) where ℰ′ is an elliptic curve over S, ψ : ℰ[n] ⟶ ℰ′[n] is an isomorphism which is anti-isometric with respect to the Weil pairing and λ is the (principal) polarization whose pull-back to ℰ × ℰ′ is n times the product polarization; see [12].

An important special case is that ψ is induced by an isogeny τ : ℰ ⟶ ℰ′ (necessarily of degree coprime to n). In this case (ℰ × ℰ′)/Graph(−ψ) is in fact always isomorphic to ℰ × ℰ′ as can be seen by the following easy generalization of the proof of Proposition 3.5.

Let n is some natural number, let ℰ, ℰ′ two elliptic curves over a scheme S, let τ : ℰ ⟶ ℰ′ be any isogeny of degree N coprime to n. (In Proposition 3.5 we treated the case that n is 2 and N is odd.) As in Section 3, let ψ := τ|_{ℰ[n]}.

Then the isogeny Φ : ℰ × ℰ′ ⟶ ℰ × ℰ′ given by the matrix ( \begin{matrix}n & 0 \\ τ & 1 \end{matrix} )

has kernel Graph(−ψ), thus it induces in isomorphism

(ℰ × ℰ′)/ Graph(−ψ) ˜ ⟶ℰ × ℰ′.  (2)

Assume now that ψ is anti-isometric with respect to the Weil pairing, i.e. that N is congruent to −1 modulo n. Then (ℰ×ℰ′)/Graph(−ψ) has a principal polarization λ whose pull-back to ℰ × ℰ′ is n id_{ℰ×ℰ′} ; see [12, Proposition 5.7]. (As in Section 3, we identify elliptic curves (and products of elliptic curves) with their duals, so that the canonical (product) polarizations become the identity.)

Under isomorphism (2), the polarization λ corresponds to the principal polarization λ̃ on ℰ × ℰ′ whose pull-back with Φ is also n id_{ℰ×ℰ′} , i.e. we have Φ̂ λ̃ Φ = n id_{ℰ×ℰ′} . It follows that λ̃ = n Φ̂^{−1} Φ^{−1}, and consequently λ̃ is given by the matrix

\frac{1}{n} ( \begin{matrix}1 & −τ̂ \\ 0 & n \end{matrix} ) ( \begin{matrix}1 & 0 \\ −τ & n \end{matrix} ) = ( \begin{matrix}\frac{1+N}{n} & −τ̂ \\ −τ & n \end{matrix} ) .  (3)

In particular, any principally polarized abelian surface considered in [4, Proposition 3.1] is isomorphic to a product of elliptic curves with a polarization given by (3).

5.2 The Number Field Case

It is interesting to note that the assumptions of Proposition 4.1 cannot be satisfied for S equal to on open part of Spec(𝒪_{K}[1/2]), where 𝒪_{K} is the principal order in a number field K. Indeed, this is contradicted by the following proposition.

Proposition 5.1 Let K be a number field, let S an open part of Spec(𝒪_{K}[1/2]), and let ℰ be an elliptic curve over S. Then there does not exist a sequence (π_{i})_{i∈ℕ_{0}} of minimal covers π_{i} : 𝒞_{i} ⟶ ℰ (where the 𝒞_{i} are curves of genus 2) with pairwise distinct degrees and the same branch locus which is étale of degree 2 over S.

Proof. By Faltings’ proof of the Shafarevich Conjecture ([2]), there exist only finitely many isomorphism classes of curves of genus 2 over S. It thus suffices to prove the proposition under the assumption that the 𝒞_{i} are equal to each other. Now the conclusion is implied by the following proposition.

□

Proposition 5.2 Let K be a number field, E an elliptic curve and C a curve of genus 2 over K. Then there does not exist a sequence (π_{i})_{i∈ℕ_{0}} of minimal covers π_{i} : C ⟶ E with pairwise distinct degrees and the same branch locus which has degree 2 over K.

Proof. Assume there exists a sequence (π_{i})_{i∈ℕ_{0}} of minimal covers with pairwise distinct degrees. Let Δ be the common branch locus, let V_{i} the ramification locus of π_{i}. By the assumption and the Hurwitz genus formula both Δ and V_{i} have degree 2, thus the canonical maps V_{i} ⟶ Δ are isomorphisms. As by Proposition 2.2 the V_{i} are pairwise disjoint, C has infinitely many Δ-valued points. This contradicts that by Faltings’ proof of the Mordell Conjecture ([2]), any curve of genus ≥ 2 over a number field has only finitely many rational points. □

One might ask whether there exists an elliptic curve E over a number field K and a sequence (π_{i})_{i∈ℕ} of minimal covers π_{i} : C_{i} ⟶ E (where the C_{i} are curves of genus 2 over K) with pairwise distinct degrees and equal branch loci. With similar (but easier) arguments as in Section 4, the existence of such a sequence would lead to a curve of genus 2 over the maximal elementary abelian 2-extension of K which has a pro-Galois curve cover of infinite degree.

There is the following conjecture closely related to the height conjecture for elliptic curves and so to the ABC conjecture.

Conjecture Let E be an elliptic curve over a number field K. Then there is a number n_{0}(K, E) such that for all elliptic curves E′ over K with E[n] isomorphic to E′[n] for some n > n_{0}(K, E) it follows that E and E′ are isogenous (over K).

This conjecture is equivalent to Conjecture 5 in [5] in the special case that the base field is a number field. (We use that the Faltings height over a number field is effective and that given two elliptic curves E and E′ over a number field such that for infinitely many natural numbers n, E[n] is isomorphic to E′[n], E and E′ are isogenous.)

Proposition 5.3 Let K be a number field, E an elliptic curve over K. Then under the assumption of the above conjecture, there are only finitely many isomorphism classes of curves of genus 2 occurring as minimal covers of E with covering degree > n_{0}(K, E).

Proof. Let C ⟶ E be a genus 2 cover with covering degree n > n_{0}(K, E), J_{C} ≃ (E × E′)/ Graph(−ψ), say. Then we have the isomorphism ψ : E[n] ≃ E′[n], thus by the conjecture E is isogenous to E′. This means that J_{C} is isogenous to E × E. By Faltings’ results, there are only finitely many abelian surfaces isogenous to E × E, and any of these finitely many abelian surfaces has up to isomorphism only finitely many principal polarizations. The result now follows by Torelli’s Theorem. □

Together with Proposition 5.2, this proposition implies:

Proposition 5.4 Let K be a number field, E an elliptic curve over K. Then under the assumption of the above conjecture, there does not exist a sequence (π_{i})_{i∈ℕ} of minimal covers π_{i} : C_{i} ⟶ E (where the C_{i} are curves of genus 2) with pairwise distinct degrees and the same branch locus which has degree 2 over K.

References

[1] C. Diem. Families of elliptic curves with genus 2 covers of degree 2. Collect. Math., 57:1–25, 2006.

[2] G. Faltings. Endlichkeitssätze für abelsche Varietten über Zahlkrpern. Invent. Math., pages 349–366, 1983.

[3] G. Faltings and C.-L. Chai. Degeneration of Abelian Varieties. Springer-Verlag, Berlin, 1980.

[4] G. Frey. On elliptic curves with isomorphic torsion structures and corresponding curves of genus 2. In Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), pages 79–98. Internat. Press, Cambridge, MA, 1995.

[5] G. Frey. On Ternary Equations of Fermat Type and Relations with Elliptic Curves. In G. Cornell, J Silverman, and G. Stevens, editors, Modular Forms and Fermat’s Last Theorem, pages 527–548. Springer-Verlag, 1997.

[6] G. Frey and E. Kani. Projective p-adic representations of the k-rational geometric fundamental group. Arch. Math., 77(32-46), 2001.

[7] G. Frey, E. Kani, and H. Völklein. Curves with infinite K-rational geometric fundamental group. In Aspects of Galois theory (Gainesville, FL, 1996), volume 256 of London Math. Soc. Lecture Note Ser., pages 85–118. Cambridge Univ. Press, Cambridge, 1999.

[8] A. Grothendieck. Revêtements étales et groupe fondamental (SGA I), volume 1960/61 of Séminaire de Géométrie Algébrique. Institut des Hautes Études Scientifiques, Paris.

[9] A. Grothendieck with J. Dieudonné. Eléments de Géométrie Algébrique (I-IV). ch. I: Springer-Verlag, Berlin, 1971; ch. II-IV: Publ. Math. Inst. Hautes Etud. Sci. 8,11, 17, 20,24, 28, 32, 1961-68.

[10] R. Hartshorne. Algebraic Geometry. Springer, New York, 1977.

[11] Y. Ihara. On unramified extensions of function fields over finite fields. In Galois Groups and their representations, volume 2 of Adv. Study Pure Math., pages 89–97. 1983.

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

[13] S. Lang. Fundamentals of Diophantine Geometry. Springer-Verlag, 1983.

[14] S. Lang. Algebra (Third Edition). Addison-Wesley, 1993.

[15] Q. Liu. Algebraic Geometry and Arithmetic Curves. Oxford University Press, Oxford, 2001.

[16] L. Merel. Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math., 124:437–449, 1996.

[17] J. Milne. Etale Cohomology. Princeton University Press, 1980.

[18] J. Milne. Jacobian varieties. In G. Cornell and J. Silverman, editors, Arithmetic Geometry, pages 167–212. Springer-Verlag, 1986.

[19] D. Mumford. Geometric Invariant Theory. Springer-Verlag, Berlin, 1965.

[20] D. Mumford. Abelian Varieties. Tata Institute for Fundamental Research, 1970.

[21] J. Rotman. An introduction to the theory of groups. Wm. C. Brown, [22] H. Stichtenoth. Algebraic Function Fields and Codes. Springer-Verlag,

Berlin, 1993.

[23] H. Wielandt. Finite Permutation Groups. Academic Press, New York, 1964.

Claus Diem. Universität Leipzig, Fakultät für Mathematik und Informatik, Johannisgasse 26, 04109 Leipzig, Germany. email: diem@math.uni-leipzig.de

Gerhard Frey. Universität Duisburg-Essen, Institut für Experimentelle Mathematik, Ellernstr. 29, 45326 Essen, Germany. email: frey@iem.uni-due.de