THREE POINT COVERS WITH BAD REDUCTION

STEFAN WEWERS

Abstract. We study Galois covers of the projective line branched at three points with bad reduction to characteristic p, under the condition that p strictly divides the order of the Galois group. As an application of our results, we prove that the field of moduli of such a cover is at most tamely ramified at p.

Introduction

Ramified primes in the field of moduli. Let f : Y → X = ℙ_{ℂ}^{1} be a three point cover, i.e. a finite cover of the Riemann sphere branched at 0, 1 and ∞. The ‘obvious direction’ of Belyi’s Theorem states that f can be defined over a number field. Therefore, the absolute Galois group of ℚ acts on the set of isomorphism classes of all three point covers, and we can associate to f the number field K such that Gal(ℚ̄/K) is the stabilizer of the isomorphism class of f. The number field K is called the field of moduli of f. Very little is known about the correspondence between three point covers and their associated field of moduli. One of the most general things that we know about K is given by the following theorem of Beckmann [Bec89]. Let G be the monodromy group of f. Then the extension K/ℚ is unramified outside the set of primes diving the order of G. This result is related to the fact that f has good reduction at each prime ideal of K dividing such primes p.

A recent theorem of Raynaud [Ray99] gives a partial converse to Beckmann’s Theorem. Suppose that the prime p strictly divides the order of G, i.e. that p^{2} does not divide |G|. Let 𝔭 be a prime ideal of K dividing p and suppose that the ramification index e(𝔭/p) is strictly smaller than the number of conjugacy classes in G of elements of order p. Then f has potentially good reduction at 𝔭.

The proof of this theorem depends on an analysis of the stable reduction of f : Y → X at 𝔭. If f does not have potentially good reduction at 𝔭 then this analysis yields a certain lower bound for the ramification index e(𝔭/p). The theorem follows from this bound. We remark that this sort of result is not restricted to three point covers, but extends, under certain conditions, to covers of more general curves X. The essential condition here is that p strictly divides the order of G.

In [Wew] the author has continued Raynaud’s study of the stable reduction, and the present paper is a further continuation of this work. Our results yield an upper bound for the ramification index of p in the field of moduli of certain three point covers. For instance, we prove the following theorem.

2000 Mathematics Subject Classification. Primary 14H30, 11G20.

Key words and phrases. Three point cover, stable reduction, field of moduli.

Theorem 1. Let f : Y → ℙ^{1} be a three point cover, with monodromy group G and field of moduli K. Let p be a prime which strictly divides the order of G. Then p is at most tamely ramified in the extension K/ℚ.

In fact we obtain a formula for the ramification index e(𝔭/p) in terms of the stable reduction of f : Y → X at 𝔭. This formula shows that e(𝔭/p) is prime to p and gives an upper bound.

In contrast to the theorem of Raynaud mentioned above, our results are very particular to three point covers. They depend in an essential way on the fact that three point covers are ‘rigid’ objects, i.e. do not admit any nontrivial deformation.

Stable reduction. Let R be a complete discrete valuation ring, with residue field k of characteristic p > 0 and fraction field K of characteristic zero. Let f_{K} : Y_{K} → X_{K} := ℙ_{K}^{1} be a three point cover defined over K. We assume that f_{K} is Galois, with Galois group G. In order to prove results such as Theorem 1, this is no restriction. After replacing the field K by a finite extension, we may assume that Y_{K} has a semistable model Y_{R} over R such that the ramification points of f_{K} specialize to pairwise distinct smooth points on the special fiber. Among all semistable models with this property, we let Y_{R} be the minimal one. The action of G extends to Y_{R}; let X_{R} := Y_{R}/G denote the quotient. This is a semistable model of ℙ_{K}^{1} over R. Therefore, the special fiber X̄ := X_{R} ⊗_{R} k is a tree of curves of genus zero. The map f_{R} : Y_{R} → X_{R} is called the stable model of the Galois cover f_{K}. Its restriction to the special fiber f̄: Ȳ → X̄ is called the stable reduction of f_{K}.

We say that the cover f_{K} : Y_{K} → X_{K} has good reduction if the curves X_{R} and Y_{R} are smooth over R. If this is the case, then the stable reduction f̄: Ȳ → X̄ = ℙ_{k}^{1} is a three point Galois cover in characteristic p. If f_{K} does not have good reduction we say that it has bad reduction. Our first main result is a structure theorem for the stable reduction f̄: Ȳ → X̄ in the case of bad reduction.

Theorem 2. Suppose that p strictly divides the order of G and that the cover f_{K} : Y_{K} → X_{K} has bad reduction. Then the following holds (compare with Figure 1).

(1) The curve X̄ consists of one central component X̄_{0}, canonically isomorphic to ℙ_{k}^{1}, and r ≥ 1 tails X̄_{1}, …, X̄_{r}, each of which intersects X̄_{0} in one point.

(2) Let Ȳ_{0} be a irreducible component of Ȳ lying above the central component X̄_{0}. Then the map Ȳ_{0} → X̄_{0} induced from f̄ is the composition of a purely inseparable map Ȳ_{0} → Z̄_{0} of degree p and a Galois cover Z̄_{0} → X̄_{0}.

(3) For j = 1, …, r, let Ȳ_{j} be an irreducible component of Ȳ lying over the tail X̄_{j}. Then the map Ȳ_{j} → X̄_{j} induced by f̄ is a Galois cover, ramified at most at two points. More precisely, Ȳ_{j} → X̄_{j} is wildly ramified at the point where X̄_{j} intersects X̄_{0} and is tamely ramified at any point which is the specialization of one of the K-points 0, 1 or ∞.

Part (1) is the essential statement and depends on the assumption that f_{K} is a three point cover. Part (2) and (3) follow from (1), using the results of [Ray99] and the assumption that p strictly divides the order of G.

Under the additional assumption that all the ramification indices of f_{K} are prime to p, Theorem 2 (1) can be deduced from the results of [Wew], via Raynaud’s construction of the auxiliary cover (see the introduction of [Wew]). The proof of the general case given in the present paper is based on a generalization of the

[figure: 0; 1; ∞; ⋯; ⋯; ⋮; ⋮; ⋮; Ȳ; X̄; X̄_{0} = ℙ_{k}^{1}]

Figure 1. The stable reduction of a three point cover

methods of [Wew]. This generalization avoids the assumption on the ramification indices and the use of the auxiliary cover.

Lifting of special G-maps. The stable reduction of a three point Galois cover f_{K} : Y_{K} → X_{K} := ℙ_{K}^{1} is, by definition, a finite map f̄: Ȳ → X̄ between semistable curves over the residue field k, together with an action of a finite group G on Ȳ which commutes with f̄. In the case of bad reduction, the curves X̄ and Ȳ are both singular, and the map f̄ is inseparable over some of the irreducible components of X̄. This suggests the following question. Given a map f̄: Ȳ → X̄ of the sort we have just described, does it occur as the stable reduction of a three point Galois cover f_{K} : Y_{K} → X_{K} in characteristic zero? If this is the case then we say that f_{K} : Y_{K} → X_{K} is a lift of f̄: Ȳ → X̄.

Theorem 2 gives a list of necessary conditions on f̄ for the existence of a lift (under the condition that p strictly divides the order of the group G). There is one extra condition on f̄ which we have omitted from the statement of Theorem 2, but which plays a central role in this paper. This additional condition is the existence of a certain differential form ω_{0} on the curve Z̄_{0} which is ‘compatible’ with f̄ (see Theorem 2 (2) for the definition of Z̄_{0}). For instance, ω_{0} is logarithmic, i.e. of the form du/u, and it is an eigenvector under the action of the Galois group of the cover Z̄_{0} → X̄_{0}. Moreover, the divisor of zeroes and poles of ω_{0} is related to and determined by the map f̄ and the action of G on Ȳ. The existence and the basic properties of ω_{0} follow from work of Green–Matignon [GM99] and Henrio [Hen00]. In some sense, ω_{0} carries infinitesimal information about the action of G on Ȳ which ‘got lost’ by reduction to characteristic p. Following [Wew], we call (Z̄_{0}, ω_{0}) a special deformation datum. (Actually, in [Wew] we assume that ω_{0} has no poles. This is true if and only if all the ramification indices of the three point cover f_{K} : Y_{K} → X_{K} are prime to p.)

In the present paper we define the notion of a special G-map. This is a map f̄: Ȳ → X̄ between semistable curves in characteristic p together with an action of a finite group G on Ȳ satisfying certain conditions. Essentially, these conditions say that f̄ satisfies the conclusion of Theorem 2 and that there exists a special deformation datum (Z̄_{0}, ω) which is ‘compatibel’ with f̄: Ȳ → X̄ and the action of G on Ȳ. Our second main result can be stated as follows. Theorem 3. Let f̄ : Ȳ → X̄ be a special G-map, defined over an algebraically closed field k of characteristic p > 0. Let K_{0} denote the fraction field of the ring of Witt vectors over k. Then the following holds.

(1) There exists a lift of f̄, i.e. a three point Galois cover f_{K} : Y_{K} → X_{K} whose stable reduction is isomorphic to f̄: Ȳ → X̄.

(2) Each lift of f̄: Ȳ → X̄ can be defined over a finite extension K/K_{0} which is at most tamely ramified.

We actually prove more, namely we determine the set of isomorphism classes of all lifts of f̄, together with the action of Gal(K̄_{0}/K_{0}) on this set, in terms of certain numerical invariants attached to f̄. From this more precise result one can deduce an upper bound for the degree of the minimal extension K/K_{0} over which a given lift of f̄ can be defined. In any case, Theorem 1 follows from Theorem 2 and Theorem 3 by a straightforward argument.

Part (1) of Theorem 3, i.e. the mere existence of a lift, follows already from the results of [Wew] (assuming that ω_{0} has no poles). Part (2) is more difficult and needs new methods. The key step in the proof of (2) is the study of the deformation theory of a certain curve with group scheme action associated to the special deformation datum (Z̄_{0}, ω_{0}). A detailed exposition of this deformation theory can be found in [Wew02].

This paper is divided into four sections. The first section contains the general theory of stable reduction of Galois covers, the monodromy action and deformation data. In §2 we apply this theory to the case of three point cover and prove our first main result, which essentially corresponds to Theorem 1 above. We also define the notion of a special G-map. In §3 we apply the deformation theory of [Wew02] in order to lift a special deformation datum to an auxiliary cover with certain good properties. This is preparatory work for §4, where we prove the lifting result corresponding to Theorem 3. At the end of §4 we draw some conclusions from this result, concerning the field of moduli, and we give some examples.

Acknowledgments. This work grew out of discussions with Michel Raynaud on my previous paper [Wew]. I would like to thank him heartily for his critical remarks and encouragement. The final version of this paper was written while I was a guest of the Max–Planck–Institut in Bonn.

1. The stable reduction of Galois covers

Let K be a field of characteristic zero equipped with a complete discrete valuation whose residue field k is algebraically closed and has characteristic p > 0. Let f : Y → X be a Galois cover of smooth projective curves over K. In this section we define and study the stable reduction of f. We make two crucial assumptions on the reduction of f. First, we assume that the marked curve (X, S) (where S is the branch locus of f) has good reduction. Second, we assume that f has mild reduction, a property which is automatically verified if p strictly divides the order of the Galois group of f.

We start by recalling some results of Raynaud [Ray99] on the structure of f̄ : Ȳ → X̄. Then we attach to each irreducible component of X̄ where the stable reduction f̄ : Ȳ → X̄ is inseparable a so-called deformation datum. This datum encodes infinitesimal information on the action of the Galois group of f : Y → X. The deformation data attached to the individual components of X̄ are related by certain compatibility conditions. The existence of a set of compatible deformation data imposes strong restrictions on the map f̄: Ȳ → X̄. In this section, we content ourself with working out those restrictions which are of a purely combinatorial nature. For instance, we obtain a new proof of Raynaud’s vanishing cycle formula [Ray99].

Most of the results of this section are already present in [Wew], in a special case.

1.1. The stable model and the monodromy action.

1.1.1. The following notation is fixed throughout §1 and §2. Let R_{0} be a complete discrete valuation ring, with quotient field K_{0} of characteristic 0 and residue field k of characteristic p > 0. We assume that k is algebraically closed. We fix an algebraic closure K̄ of K_{0}.

Let (X_{0}, S_{0}) be a smooth stably marked curve over R_{0}. Since R_{0} is strictly henselian this implies that S_{0} = { x_{R_{0},j} | j ∈ B_{0}}, where x_{R_{0},j} : Spec R_{0} → X_{0} are pairwise distinct sections, indexed by a finite set B_{0}. If K/K_{0} is a field extension then we write X_{K} and S_{K} instead of X_{0} ⊗_{R_{0}} K and S_{0} ⊗_{R_{0}} K. If K = K̄ then we simply write X and S instead of XK̄ or SK̄. The elements of S are denoted by x_{j}, j ∈ B_{0}. In §2 we will consider the case X_{0} = ℙ_{R_{0}}^{1} and S_{0} = {0, 1, ∞}.

Let G be a finite group and f : Y → X a G-cover of X with branch locus S. We consider the set of ramification points of the cover f as a marking on Y; this makes Y a stably marked curve. Let K be a finite extension of K_{0} and R the ring of integers of K. A model of f over K is a G-cover f_{K} : Y_{K} → X_{K} such that f = f_{K} ⊗_{K} K̄. A good model of f over K is a model which extends to a finite and tame cover f_{R} : Y_{R} → X_{0} ⊗_{R_{0}} R, ramified along S_{0} ⊗_{R_{0}} R and étale everywhere else. We say that f has good reduction if there exists a good model over some extension K/K_{0}. If f does not have good reduction, we say that it has bad reduction.

The following is well known, see [Bec89]. If f has good reduction then there exists a good model f_{K_{0}} : Y_{K_{0}} → X_{K_{0}} of f over K_{0}, unique up to isomorphism. Moreover, if p does not divide the order of the Galois group G, then f : Y → X has good reduction.

1.1.2. From now on, we assume that f : Y → X has bad reduction. Note that this implies that p divides the order of G. If K/K_{0} is a sufficiently large finite extension, then there exists a model f_{K} : Y_{K} → X_{K} of f over K such that Y_{K} extends to a stably marked curve Y_{R} over R. In particular, Y_{R} is a semistable model of Y_{K} over R such that the ramification points of the cover f_{K} specialize to pairwise disjoint, smooth points on the special fiber Ȳ := Y_{R} ⊗ k.

Since the stably marked model is unique, the action of the group G on Y_{K} extends to Y_{R}. Let X_{R} := Y_{R}/G be the quotient; then X_{R} is a semistable curve over R with generic fiber X_{K}, see [Ray90], Appendix. Moreover, the branch points x_{j} of f_{K} specialize to pairwise distinct smooth points x̄_{j} on the special fiber X̄ := X_{R} ⊗ k. We consider X̄ as a marked semistable curve (note, however, that X̄ is not stably marked). By [Knu83], there exists a canonical contraction morphism q_{R} : X_{R} → X_{0} ⊗_{R_{0}} R. Let X̄ denote the special fiber of X_{R}. There is a unique irreducible component of X̄ on which the induced map q̄ : X̄ → X̄_{0} := X_{0} ⊗_{R_{0}} k is an isomorphism. We may and will identify this component with X̄_{0}. All other components of X̄ are of genus 0 and are contracted by q̄ to a closed point of X̄_{0}.

Definition 1.1. The natural map f_{R} : Y_{R} → X_{R} is called the stable model of the Galois cover f_{K}. The induced map f̄: Ȳ → X̄ on the special fibers is called the stable reduction of f_{K}. The component X̄_{0} of X̄ is called the original component.

It is clear that the stable model is ‘stable’ under extension of the field K. In particular, the stable reduction does not depend on the choice of K. We remark that the stable model defined above differs slightly from the model used in [Ray99]. This difference is not essential, but certain results of [Ray99] have to be reformulated. This is done in Lemma 1.4 below.

1.1.3. For an element σ ∈ Gal(K̄/K_{0}) we denote by ^{σ}f : ^{σ}Y → ^{σ}X = X the conjugate G-cover. Let Γ^{in} be the subgroup of Gal(K̄/K_{0}) consisting of all elements σ such that f ≅ ^{σ}f, as G-covers. Similarly, let Γ^{ab} be the subgroup of Gal(K̄/K_{0}) consisting of elements σ such that ^{σ}f ≅ f as mere covers. Set

K^{in} := K̄^{Γ^{in}}, K^{ab} := K̄^{Γ^{ab}}.

The field K^{in} (resp. K^{ab}) is the field of moduli of the G-cover f (resp. of the mere cover f), relative to the extension K̄/K_{0}. See [DD97] for more details.

Let K/K_{0} be a Galois extension such that f has stable reduction over K, and let f_{R} : Y_{R} → X_{R} be the stable model. Also, let σ ∈ Γ^{ab}. By definition of Γ^{ab}, there exists a σ-linear automorphism κ_{K,σ} : Y_{K} →^{∼} Y_{K} which normalizes the action of G and induces the canonical σ-linear automorphism X_{K} →^{∼} X_{K}. The automorphism κ_{K,σ} is unique up to composition with an element of G. It extends to a σ-linear automorphism κ_{R,σ} : Y_{R} →^{∼} Y_{R}. Let κ̄_{σ} : Ȳ →^{∼} Ȳ be the restriction of κ_{R,σ} to the special fiber. This is a k-linear automorphism which normalizes the action of G. It is uniquely determined by σ ∈ Γ^{ab}, up to composition with an element of G. We obtain a continuous homomorphism

κ̄^{ab} : Γ^{ab} ⟶ Aut_{k}(Ȳ)/G,

which we shall call the (absolute) monodromy action. Note that κ̄_{σ} induces an automorphism of X̄ which is the identity on the original component. It follows from [DD97] that the set of K^{ab}-models of the mere cover f is in bijection with the set of homomorphisms Γ^{ab} → Aut_{k}(Ȳ) which lift κ̄^{ab}. If a model f_{K^{ab}} of f is specified, we will sometimes consider κ̄^{ab} as a homomorphism Γ^{ab} → Aut_{k}(Ȳ).

Let Aut_{G}(Ȳ) denote the group of k-linear automorphisms of Ȳ which commute with the G-action, and let C_{G} denote the center of G. The restriction of κ̄^{ab} to Γ^{in} yields a homomorphism

κ̄^{in} : Γ^{in} ⟶ Aut_{G}(Ȳ)/C_{G},

called the inner monodromy action. Similar to what we stated above, K^{in}-models of the G-cover f correspond one-to-one to lifts of κ̄^{in} to a homomorphism Γ^{in} → Aut_{G}(Ȳ).

Let Γ^{st} denote the subgroup of Γ^{ab} of elements σ such that κ̄σ^{ab} ∈ G, and set K^{st} := K̄^{Γ^{st}}. One shows that K^{st}/K_{0} is the smallest extension over which f has stable reduction. Compare with [Ray99], §2.2.

1.2. Mild reduction.

1.2.1. Let f : Y → X be a G-cover as in §1.1. We assume that f has bad reduction and denote by f̄ : Ȳ → X̄ its stable reduction. If η is a point on Ȳ (closed or generic), we denote by G_{η} (resp. I_{η}) the corresponding decomposition (resp. inertia) group. By definition, G_{η} is the set of elements of G which stabilize η, and I_{η} is the normal subgroup of G_{η} consisting of elements which act trivially on the residue field k(η). Definition 1.2. We say that f : Y → ℙ^{1} has mild reduction it has bad reduction and if the following two conditions hold for all points η ∈ Ȳ. Firstly, the p-part of I_{η} has at most order p. Secondly, if I_{η} ≠ 1 then the order of G_{η}/I_{η} is prime to p.

If f : Y → X has bad reduction and p strictly divides the order of the Galois group G, then the reduction of f is automatically mild. From now on, let us assume that f : Y → X has mild reduction.

1.2.2. Let us denote by (X̄_{v}) the family of irreducible components of the curve X̄. Definition 1.3. A component X̄_{v} of X̄ which is not the original component and which is connected to the rest of X̄ in a single point is called a tail. The point where a tail X̄_{v} meets the rest of X̄ is denoted by ∞_{v}. If a tail X̄_{v} contains one of the points x̄_{j}, j ∈ B_{0}, then X̄_{v} is called primitive. Otherwise, the tail X̄_{v} is called new. A component X̄_{v} which is not a tail is called an interior component.

For each component X̄_{v} of X̄, we choose an irreducible component Ȳ_{v} of Ȳ above X̄_{v}. We write G_{v} for the decomposition group of Ȳ_{v} and I_{v} for the inertia group of Ȳ_{v}. Lemma 1.4 (Raynaud). Suppose that f : Y → ℙ^{1} has mild reduction. Then the following holds.

(1) If X̄_{v} is a tail, then the inertia group I_{v} is trivial. In other words, the map Ȳ_{v} → X̄_{v} is a Galois cover, with Galois group G_{v}. If X̄_{v} is a primitive tail and contains the point x̄_{j}, j ∈ B_{0}, then the cover Ȳ_{v} → X̄_{v} is wildly ramified at ∞_{v}, tamely ramified at x̄_{j} and étale everywhere else. If X̄_{v} is a new tail, then Ȳ_{v} → X̄_{v} is wildly ramified at ∞_{v} and étale everywhere else.

(2) If X̄_{v} is not a tail, then the inertia group I_{v} is cyclic of order p. The cover Ȳ_{v} → X̄_{v} factors through the k-linear Frobenius map F : Ȳ_{v} → Z̄_{v} := Ȳ_{v}^{(p)}. The resulting map Z̄_{v} → X̄_{v} is a Galois cover, with Galois group H_{v} := G_{v}/I_{v}. It is étale outside the set of points which are either singular points of X̄ or belong to the set {x̄_{j} | j ∈ B_{0}}.

(3) For j ∈ B_{0}, let n_{j} be the ramification index of f : Y → X in x_{j}. Suppose that n_{j} is prime to p. Then the specialization x̄_{j} of x_{j} lies on a primitive tail X̄_{v}, and the ramification index of Ȳ_{v} → X̄_{v} at x̄_{j} is n_{j}.

(4) With the same notation as in (3), suppose that n_{j} = pn′_{j}. Then x̄_{j} lies on a component X̄_{v} which is not a tail. Moreover, the Galois cover Z̄_{v} → X̄_{v} has ramification index n′_{j} at x̄_{j}.

Proof. This is essentially proved in [Ray99], §2 and §3. There are two points where our assumptions differ from the assumptions made in [Ray99]. Firstly, it is assumed in [Ray99] that p exactly divides the order of G. However, one only needs that f has mild reduction. The second point is that (3) becomes false if we replace our definition of the stable reduction with the definition used in [Ray99]. It is proved in [BW00], Lemma 2.1.1 that (3) holds if we define f̄ as in Definition 1.1. □

1.3. Deformation data.

1.3.1. A priori, our definition of a deformation datum is independent of the study of stable reduction. So the following notation replaces, during §1.3.1, the notation introduced so far. Let p be a prime number, k an algebraically closed field of characteristic p and X a smooth projective and irreducible curve over k. Definition 1.5. A deformation datum over X is a pair (Z, ω), where Z → X is a tame (connected) Galois cover and ω is a meromorphic differential form on Z such that the following holds.

(1) The order of the Galois group H = Gal(Z/X) is prime to p.

(2) We have

β^{∗}ω = χ(β) · ω, for all β ∈ H,

where χ : H → 𝔽_{p}^{×} is a character with values in the prime field 𝔽_{p}.

(3) The differential ω is either logarithmic (i.e. of the form ω = du/u) or exact (i.e. ω = du). In the first case, the deformation datum (Z, ω) is called multiplicative, in the second case it is called additive.

The pair (H, χ) is called the type of the deformation datum (Z, ω).

Let ξ ∈ Z be a closed point and τ ∈ X its image. Denote the stabilizer of ξ by H_{ξ} ⊂ H. We attach to the triple (Z, ω, τ) the following three invariants:

(1)  m_{τ} := |H_{ξ}|,  h_{τ} := ord_{ξ}(ω) + 1,  σ_{τ} := h_{τ} /m_{τ} .

It is clear that m_{τ}, h_{τ} and σ_{τ} depend only on τ but not on ξ, and that (m_{τ} , h_{τ}) = (1, 1) for all but a finite number of points τ ∈ X. If (m_{τ} , h_{τ}) ≠ (1, 1) then τ is called a critical point for the deformation datum (Z, ω).

Given a deformation datum (Z, ω), we often use the following or a similar notation. Let (τ_{j})_{j∈J} be the family of critical points for (Z, ω), indexed by the finite set J. For j ∈ J, we write m_{j}, h_{j} and σ_{j} instead of m_{τ_{j}}, h_{τ_{j}} and σ_{τ_{j}}. The data (m_{j}; h_{j})_{j∈B} is called the signature of the deformation datum (Z, ω). A straightforward computation shows that

(2)  ∑_{j∈J} (σ_{j} − 1) = 2g_{X} − 2.

(where g_{X} denotes the genus of the curve X). Remark 1.6. Let Z̃ := Z/ Ker(χ). It is easy to see that ω descents to a differential ω̃ on Z̃ and that (Z̃, ω̃) is again a deformation datum, with critical points τ_{j} and invariants

m̃_{j} = \frac{m_{j}}{| Ker(χ)|},  h̃_{j} = \frac{h_{j}}{| Ker(χ)|},  σ̃_{j} = \frac{h̃_{j}}{m̃_{j}} = σ_{j}.

1.3.2. Let us now return to our original notation, concerning the Galois cover f : Y → X and its stable reduction f̄: Ȳ → X̄. Let X̄_{v} be an interior component of X̄ (see Definition 1.3). Following [Hen00] and [Saï01], we define a deformation datum (Z̄_{v}, ω_{v}) over X̄_{v}, as follows.

Choose an irreducible component Ȳ_{v} of Ȳ above X̄_{v}. By Lemma 1.4 (2), the inertia group I_{v} of Ȳ_{v} is cyclic of order p and the quotient group H_{v} := G_{v}/I_{v} is of prime-to-p order. Moreover, the cover Ȳ_{v} → X̄_{v} is the composition of a purely inseparable cover Ȳ_{v} → Z̄_{v} of degree p and an H_{v}-Galois cover Z̄_{v} → X̄_{v}. Let Ŷ_{v} := Spec Ô_{Y_{R},v} be the completion of Y_{R} at the generic point of the component Ȳ_{v}; thus Ô_{Y_{R},v} is a complete discrete valuation ring with residue field k(Ȳ_{v}). The action of I_{v} on Y_{R} induces an action on Ŷ_{v}. Let Ẑ_{v} := Ŷ_{v}/I_{v} be the quotient; we have Ẑ_{v} = Spec A for a complete discrete valuation ring A with residue field k(Z̄_{v}). We may assume that the field K contains the pth roots of unity. Let us choose an isomorphism I_{v} ≅ μ_{p}(K) between I_{v} and the group of pth roots of unity. By [Ray99], §1.2.1, the map Ŷ_{v} → Ẑ_{v} is a torsor under a finite flat R-group scheme G of rank p, such that G ⊗_{R} K = μ_{p} acts on Ŷ_{v} ⊗_{R} K via the isomorphism I_{v} ≅ μ_{p}(K). By Kummer theory, the μ_{p}-torsor Ŷ_{v} ⊗_{R} K → Ẑ_{v} ⊗_{R} K is given by a Kummer equation y^{p} = u, with u ∈ A.

To define the differential ω_{v}, we have to distinguish two cases. First, let us assume that G ⊗_{R} k ≅ μ_{p}. This happens if and only if we may choose the Kummer equation y^{p} = u such that the residue ū of u is not a pth power. In this case, we set ω_{v} := dū/ū and we say that the cover f has multiplicative reduction at Ȳ_{v}. Otherwise G ⊗_{R} k ≅ α_{p} and we say that f has additive reduction at Ȳ_{v}. In this case we can choose the equation y^{p} = u such that u = 1 + π^{p}w, with π ∈ R, w ∈ A and 0 < v_{R}(π) < v_{R}(p)/(p − 1), and such that the residue w̄ of w is not a pth power. We set ω_{v} := dw̄. It is clear that ω_{v} depends on the choice of the isomorphism I_{v} ≅ μ_{p}(K). Indeed, composing this isomorphism with the automorphism ζ_{p} ↦ ζ_{p}^{a} (where a ∈ 𝔽_{p}^{×}) amounts to multiplying ω_{v} by a. Let χ_{v} : H_{v} → 𝔽_{p}^{×} be the character describing the action of H_{v} on I_{v} by conjugation. It is clear that β^{∗}ω_{v} = χ_{v}(β) · ω_{v} for all β ∈ H_{v}.

We have defined the differential ω_{v} over the generic point of Z̄_{v}. However, we can extend this definition to a neighborhood of any point of Z̄ which lies over a smooth point of X̄. To be more precise, let τ ∈ X̄_{v} be a closed point which is smooth in X̄ and choose points η ∈ Ȳ_{v} and ξ ∈ Z̄_{v} above τ. Let Ŷ_{η} be the completion of Y_{R} at η and Ẑ_{ξ} = Ŷ_{η}/I_{v}. Then Ŷ_{η} ⊗_{R} k = Spec Ô_{Ȳ_{v},η} ≅ k[[y]] and Ẑ_{ξ} = Spec Ô_{Z̄_{v},ξ} ≅ k[[z]]. As above, the action of I_{v} on Ŷ_{η} gives rise to a differential ω_{ξ} of the field Frac(Ô_{Z̄_{v},ξ}) ≅ k((z)). One checks easily that ω_{ξ} is the pullback of ω_{v} to the completion of Z̄_{v} at ξ. Now the local definition of ω_{ξ} shows the following. The differential ω_{v} does not vanish at ξ. Moreover, if ω_{v} has a pole at ξ, then this pole is simple, ω_{ξ} is logarithmic and the point τ ∈ X̄_{v} is the specialization of a branch point of f with ramification index divisible by p. We summarize the discussion in the next proposition.

Proposition 1.7. Over each interior component X̄_{v} we obtain a deformation datum (Z̄_{v}, ω_{v}), with the following properties. If τ ∈ X̄_{v} is a critical point for (Z̄_{v}, ω_{v}) then one of the following two statements holds.

(1) The point τ ∈ X̄_{v} is a singular point of X̄.

(2) The point τ is the specialization of a point x_{j} ∈ S_{0} at which the cover f has ramification of order divisible by p.

In Case (1), we have σ_{τ} ∉ {0, 1}. In Case (2), we have σ_{τ} = 0 and (Z̄_{v}, ω_{v}) is multiplicative.

1.3.3. By Proposition 1.7, the stable reduction f̄ : Ȳ → X̄ of a G-cover f : Y → X with mild reduction is naturally equipped with a collection of deformation data (Z̄_{v}, ω_{v}). These deformation data are related among each other by certain compatibility conditions. To formulate these conditions, let τ be a singular point of X̄, and let X̄_{v_{1}}, X̄_{v_{2}} be the two components of X̄ which intersect in τ. For i = 1, 2, we define a pair (m_{i}, h_{i}) as follows.

• If X̄_{v_{i}} is an interior component, we define (m_{i}, h_{i}) as the pair of invariants attached to (Z̄_{v_{i}}, ω_{v_{i}}, τ).

• Suppose that X̄_{v_{i}} = X̄_{j} is a tail of X̄. Let η ∈ Ȳ_{j} be a point lying above τ and let I_{η} ⊂ G be the inertia group at η. We define m_{i} as the order of the maximal prime-to-p quotient of I_{η}. Furthermore, we let h_{i} be the conductor of I_{η}, i.e. the unique integer such that |I_{η,h_{i}}| = p and I_{η,h_{i}+1} = {1}. (Here (I_{η,ν})_{ν≥0} denotes the filtration of higher ramification groups.)

Proposition 1.8. We have m_{1} = m_{2} and h_{1} = −h_{2}.

Proof. The equality m_{1} = m_{2} is trivial. Let Ŷ := Spec Ô_{Y_{R},η} be the completion of Y_{R} at η. The R-scheme Ŷ is a formal annulus and is acted upon by I_{η}. The invariants h_{1} and h_{2} can be defined purely in terms of this local action, and the equality h_{1} = −h_{2} is proved in [Hen00], Proposition 1.10. (In [Hen00], the invariant h is called −m.) □

Remark 1.9. It seems plausible that the existence of a set of deformation data satisfying these compatibility conditions gives a necessary and sufficient condition for a map f̄: Ȳ → X̄ between semistable curves to occur as the stable reduction of a Galois cover between smooth curves in characteristic 0. That this is true for three point covers is proved in §4. In the case of p-cyclic covers, results in this direction are obtained in [Hen00] and [Saï01].

1.4. The vanishing cycle formula. The vanishing cycle formula, proved by Raynaud in [Ray99], is the key formula which controls the (local) invariants describing the bad reduction occuring over the tails by a global expression. In this subsection we give another proof. The main observation is that (2) gives a ‘local’ vanishing cycle formula for each interior component of X̄. The global vanishing cycle formula is then easily obtained from its local version using the compatibility relation of Proposition 1.8 and induction over the tree of components of X̄. We use this occasion to set up some more notation which may seem at bit technical at first but will pay off in §2.1.

1.4.1. Recall that an unordered graph is given by a quintupel (V, E, s, t, ¯ ). Here V is the set of vertices, E is the set of (ordered) edges, s, t : E → V are the source and the target map, and E →^{∼} E, e ↦ ē, is a fixed point free involution of E such that s(ē) = t(e) for all e ∈ E.

Let T′ = (V′, E′, s, t, ¯ ) be the dual graph of the semistable curve X̄. Thus, a vertex v ∈ V′ corresponds to an irreducible component X̄_{v} of X̄, and an edge e ∈ E′ corresponds to a triple (x̄, v_{1}, v_{2}) such that the (distinct) components X̄_{v_{1}} and X̄_{v_{2}} intersect in the (singular) point x̄. Let B_{wild} ⊂ B_{0} be the set of indices j such that ramification index of the Galois cover f : Y → X at x_{j} is divisible by p. We define T = (V, E, s, t, ¯ ) as the unordered graph obtained from T′ by adding a vertex for each point x_{j} with j ∈ B_{wild}. More precisely, V := V′ ∪ B_{wild} and a vertex j ∈ B_{wild} is connected to the vertex v ∈ V′ by an edge e ∈ E if and only if the point x_{j} ∈ S_{0} specializes to the component X̄_{v}. Note that T is a connected tree. Also, to each edge e of T corresponds a closed point x̄_{e} ∈ X̄, as follows. If the edge e joins two vertices v_{2}, v_{2} ∈ V′ then x̄_{e} is the point where X̄_{v_{1}} intersects X̄_{v_{2}}. If e joins two vertices j ∈ B_{wild} and v ∈ V′ then x̄_{e} := x̄_{j} ∈ X̄_{v} is the point to which x_{j} specializes.

We consider the vertex v_{0} corresponding to the original component X̄_{v_{0}} as the root of the tree T. Also, we denote by B the set of leaves of T, i.e. the set of vertices v ∈ V − {v_{0}} which are adjacent to a unique vertex v′. Thus, a vertex v ∈ V_{int} := V − B corresponds to an interior component X̄_{v}. By definition, B_{wild} ⊂ B. An element j in B_{prim} = B_{0} − B_{wild} corresponds to a ramification point x_{j} of f of order prime-to-p. By Lemma 1.4, x_{j} specializes to a point x̄_{j} on a primitive tail X̄_{v}. The resulting map B_{prim} → B is clearly injective. Therefore, we may and will consider B_{0} = B_{prim} ∪ B_{wild} as a subset of B. Then B_{new} := B − B_{0} is the set of leaves corresponding to the new tails of X̄.

1.4.2. We shall now attach integers m_{e}, h_{e} and a rational number σ_{e} to each edge e of the tree T. Suppose first that v := s(e) is an interior vertex. Let (Z̄_{v}, ω_{v}) be the deformation datum over X̄_{v}, as defined in §1.3.2. Note that x̄_{e} ∈ X̄_{v} is a critical point for (Z̄_{v}, ω_{v}), by Proposition 1.7. We define the numbers m_{e}, h_{e} and σ_{e} as the invariants attached to the triple (Z̄_{v}, ω_{v}, x̄_{e}). If v := s(e) is a leaf of T, then we set m_{e} := mē, h_{e} := −hē and σ_{e} := −σē. Equation (2) and Proposition 1.8 imply the following.

Proposition 1.10. For all interior vertices v of T we have

(3)  ∑_{s(e)=v} (σ_{e} − 1) = 2g_{v} − 2

(where g_{v} is the genus of X̄_{v}). Furthermore, for all edges e of T we have

(4)  σ_{e} + σē = 0.

For each leaf j ∈ B, let e_{j} be the unique edge with target j. We set m_{j} := m_{e_{j}}, h_{j} := m_{e_{j}} and σ_{j} := σ_{e_{j}}. If j ∉ B_{wild} then m_{j} is the prime-to-p part of the ramification index of the Galois cover Ȳ_{j} → X̄_{j} at x̄_{e} = ∞_{j} and h_{j} is the conductor of the wild ramification in that point, see Proposition 1.8. Therefore, the rational number σ_{j} = h_{j}/m_{j} is the ramification invariant introduced in [Ray99], §1.1. It is equal to the jump in the filtration of higher ramification groups for the branch point ∞_{j} ∈ X̄_{j}, with respect to the upper numbering, see [Ser68]. Note that we have defined σ_{j} = 0 for j ∈ B_{wild}. We shall call the tuple (σ_{j})_{j∈B} the signature of the stable reduction f̄: Ȳ → X̄. It is shown in [Ray99] that σ_{j} > 1 for j ∈ B_{new}. Moreover, we have the following.

Corollary 1.11 (The vanishing cycle formula). The signature (σ_{j})_{j∈B} of f̄ verifies

(5)  ∑_{j∈B_{prim}} σ_{j} + ∑_{j∈B_{new}} (σ_{j} − 1) = 2g_{X} − 2 + |B_{0}|.

Proof. This is proved in [Ray99], §3.4. Here is another proof, using Proposition 1.10. Let T_{0} ⊂ T_{1} ⊂ … ⊂ T_{n} be a chain of subtrees of T such that T_{0} = {v_{0}}, T_{i} = T_{i−1} ∪ {v_{i}} and T_{n} = T − B. Let E_{i} denote the set of edges of T such that s(e) ∈ T_{i} and t(e) ∉ T_{i}. We claim that ∑_{e∈E_{i}}(σ_{e} − 1) = 2g_{X} − 2 holds for all 0 ≤ i ≤ n. Indeed, Proposition 1.10 shows that the claim is true for i = 0 and then, by induction, for all i. For i = n, the claim is equivalent to (5). □

2. The case of three point covers

In this section we apply the results of the last section to the case of three point covers. In §2.1 we prove our first main result, which essentially states that the stable reduction of a three point cover with mild reduction is as simple as possible. The basic idea underlying the proof of this result is that f̄: Ȳ → X̄, together with the associated deformation data, should be a ‘rigid’ objects, i.e. should not allow any nontrivial deformation. This imposes strong restrictions on the deformation data (Z̄_{v}, ω_{v}) attached to f̄. We verify these restrictions coming from rigidity only indirectly, using the local and global vanishing cycle formulas. A key lemma from [Wew] shows that f̄ can support no additive deformation datum, and only a single multiplicative one. To make this proof work one has to exclude two exceptional cases, see §2.1.1. This is somewhat annoying but not serious, because all our final results apply to these exceptional cases as well.

In §2.2 we define the notion of a special G-deformation datum. To such an object we can associate a map f̄: Ȳ → X̄ which looks as if it were the stable reduction of a three point G-cover, and which we call a special G-map.

2.1. The structure of f̄.

2.1.1. We continue with the notation introduced in §1. In addition, we assume that X_{0} = ℙ_{R_{0}}^{1} and S_{0} = {0, 1, ∞}. Note that the right hand side of (5) is equal to 1. Definition 2.1. We say that the stable reduction of f is exceptional if one of the following two cases occurs.

(1) We have B_{prim} = {j_{0}}, B_{new} = ∅ and σ_{j_{0}} = 1.

(2) We have B_{prim} = ∅, B_{new} = {j_{0}} and σ_{j_{0}} = 2.

Proposition 2.2. Let f : Y → X = ℙ1 ; K̄ be a Galois cover ramified at {0, 1, ∞}, with exceptional reduction. Then the curve X̄ consists only of the original component X̄_{0} and the unique tail X̄_{j_{0}}.

Proof. We will prove the proposition only for Case (2) of Definition 2.1 and leave Case (1) to the reader. Let T′ be the dual graph of the curve X̄, considered as a subtree of T. By assumption, T′ has a unique leaf j_{0}. Therefore, T′ is simply a chain going from the root v_{0} to j_{0}. Let v be the unique vertex of T′ adjacent to the leaf j_{0} and let e be the edge with source v and target j_{0}. This means that the interior component X̄_{v} intersects the unique (new) tail X̄_{j_{0}} in the point x̄_{e}. We have to show that v = v_{0}, i.e. that X̄_{v} is the original component X̄_{0}. So, let us assume that v ≠ v_{0}. Then there exists a unique edge e′ ≠ e with source v, corresponding to a point x̄_{e′} where X̄_{v} intersects another component X̄_{v′}. Let us first suppose that there exists another edge e′′ (of the tree T) with source v besides e and e′. Then t(e′′) ∈ B_{wild}, i.e. the point x̄_{e′′} is the specialization of a branch point of f with ramification index divisible by p. Since X̄_{v} is not the original component, there can be at most one branch point specializing to X̄_{v}. Therefore, e, e′, e′′ are the only edges with source v. Since σ_{e} = 2 and σ_{e′′} = 0, (3) shows that σ_{e′} = −1. This means that the differential ω_{v} has a pole of order ≥ 2 at each point of Z̄_{v} lying above x̄_{e′}. This shows that ω_{v} cannot be logarithmic. On the other hand, ω_{v} has a simple pole at each point above x̄_{e′′}, so ω_{v} cannot be exact. This is a contradiction. We may therefore assume that e and e′ are the only edges with source v. It follows that the cover Z̄_{v} → X̄_{v} is ramified at most at two points; in particular, the genus of Z̄_{v} and therefore the genus of Ȳ_{v} as well, is zero. Moreover, Ȳ_{v} intersects the rest of Ȳ in two points and no ramification point specializes to Ȳ_{v}. Thus, the ‘three point condition’ does not hold for Ȳ, in contradiction with the definition of the stable model in §1.1. The proposition follows. □

Remark 2.3. It may happen that the Galois cover f : Y → X has bad reduction but that the curve Y has good reduction. If this is the case then the stable reduction of f is exceptional.

2.1.2. For the rest of §2.1 we assume that the reduction of f is not exceptional. The vanishing cycle formula (5), together with the inequalities σ_{j} > 0 for j ∈ B_{prim} and σ_{j} > 1 for j ∈ B_{new}, immediately imply the inequalities

(6)  0 < σ_{j} < 1 for j ∈ B_{prim},  1 < σ_{j} < 2 for j ∈ B_{new}.

Moreover, |B_{prim} ∪ B_{new}| ≥ 2. For an edge e of T we define ν_{e} := ⌊ σ_{e} ⌋ to be the largest integer less than or equal to the rational number σ_{e}. Note that we have σ_{e} = ν_{e} + ⟨σ_{e}⟩, where ⟨σ_{e}⟩ ≥ 0 denotes the fractional part of σ_{e}. For a leaf j ∈ B, we write ν_{j} := ⌊ σ_{j} ⌋. By (6) we have ν_{j} = 0 for j ∈ B_{0} and ν_{j} = 1 for j ∈ B_{new}. Since |B_{0}| = 3, the vanishing cycle formula (5) implies

(7)  ∑_{j∈B} ⟨σ_{j}⟩ = 1.

Recall that T′ denotes the dual graph of the curve X̄, considered as subtree of T. Let E′ denote the set of edges of T′. Moreover, let E′_{int} ⊂ E′ be the set of edges connecting two interior vertices. Proposition 2.4.

(1) For all edges e ∈ E′ we have ν_{e} + νē = −1.

(2) For all interior vertices v ∈ V_{int} we have

∑_{s(e)=v} ⟨σ_{e}⟩ = 1  and  ∑_{s(e)=v} (ν_{e} − 1) = −3.

Proof. For e ∈ E′ define

d_{e} := ⟨σ_{e}⟩ + ⟨σē⟩ − 1 = −ν_{e} − νē − 1.

The last equality follows from (4). Clearly, d_{e} = dē, d_{e} ∈ {−1, 0} and d_{e} = −1 if and only if σ_{e} ∈ ℤ. For v ∈ V_{int} define

c_{v} := −1 + ∑_{s(e)=v}⟨σ_{e}⟩ = −3 + ∑_{s(e)=v}(ν_{e} − 1).

The last equality is a consequence of (3). It is easy to see that c_{v} ∈ {−1, 0, 1, …} and that c_{v} = −1 if and only if d_{e} = −1 for all e ∈ E′ such that s(e) = v. We have to show that d_{e} = c_{v} = 0.

It is convenient to introduce the following terminology. A vertex v of T is said to precede an edge e (notation: v ≺ e) if v lies in the connected component of T − {e} which contains s(e). An edge e′ is said to precede another edge e (notation: e′ ≺ e) if t(e′) ≺ e. Note that ‘≺’ is a partial order on the set E of edges.

Suppose that there exists an edge e ∈ E′ with d_{e} ≠ 0. We may suppose that e is minimal with this property, i.e. d_{e′} = 0 for all edges e′ ∈ E′ which precede e. We claim that c_{v} ≥ 0 for all v ∈ V_{int} which preceed e. Indeed, since X̄ has at least two tails, there exist at least two (distinct) edges in E′ with target v. At least one of them precedes e. Hence there exists an edge e′ ∈ E′ with d_{e′} = 0 and t(e) = v. This proves the claim. As a consequence, we obtain the inequality

\begin{aligned}0 ≤ ∑ ; \begin{matrix}v ∈ V_{int} \\ v ≺ e \end{matrix} c_{v} & = ⟨σ_{e}⟩ + ( ∑ ; \begin{matrix}e′ ∈ E′_{int} \\ e′ ≺ e \end{matrix} d_{e′} ) + ( ∑ ; \begin{matrix}j ∈ B \\ j ≺ e \end{matrix} ⟨σ_{j}⟩) − 1 \\ & = ⟨σ_{e}⟩ + ( ∑ ; \begin{matrix}j ∈ B \\ j ≺ e \end{matrix} ⟨σ_{j}⟩) − 1. \end{aligned} ; (8)

Since the curve X̄ has at least two tails, there exists j ∈ B_{prim} ∪ B_{new} such that j does not precede e. Moreover σ_{j} > 0. Therefore, (8) implies ⟨σ_{e}⟩ > 0. But this is a contradiction to the assumption d_{e} ≠ 0. We conclude that d_{e} = 0 for all e ∈ E′. It remains to prove c_{v} = 0 for all v ∈ V_{int}. The same argument we used above together with d_{e} = 0 for all e ∈ E′ shows that c_{v} ≥ 0 for all v ∈ V_{int}. Now the proposition follows from

∑_{v∈V_{int}} c_{v} = ( ∑_{e∈E′_{int}} d_{e}) + ( ∑_{j∈B} ⟨σ_{j}⟩) − 1 = 0.

□

Lemma 2.5. Suppose that f̄ is not exceptional, and let e be an edge of T. Then

−2 ≤ ν_{e} ≤ 1.

Moreover, ν_{e} ≥ 0 if and only if the root v_{0} precedes e. In particular, if v is an interior vertex unequal to v_{0} then there exists a unique edge e with source v such that ν_{e} < 0.

Proof. (Compare to [Wew], Lemma 2.6.) Using Proposition 2.4 and induction in the tree T one shows that

ν_{e} = 1 − | { j ∈ B_{0} | j > e } |.

Note that |B_{0}| = 3 and that the root v_{0} is the median of the set of leaves B_{0}. The lemma follows immediately. □

2.1.3. We can now prove our first main result. It applies in particular if p strictly divides the order of G. Theorem 2.6. Let f : Y → X = ℙ1 ; K̄ be a Galois cover ramified at {0, 1, ∞}, with bad reduction. If f has mild reduction (Definition 1.2) then the curve X̄ consists only of the original component X̄_{0} and the tails X̄_{j}, j ∈ B_{prim} ∪ B_{new}.

Proof. If B_{wild} = ∅ then this follows from [Wew], Theorem 2.1, using Raynaud’s construction of the auxiliary cover. The proof we give below follows the same lines, but avoids the construction of the auxiliary cover.

We prove the theorem by contradiction. Assume that there exists an interior vertex v of T other than the root v_{0}. By Proposition 2.2, the reduction of f is not exceptional. We may assume that v is adjacent to the root v_{0}, i.e. there exists an edge e_{0} with source v and target v_{0}. This means that the component X̄_{v} intersects the original component X̄_{0} in the point x̄_{e_{0}}. Let (Z̄_{v}, ω_{v}) be the deformation datum over X̄_{v} as defined in §1.3.2. For the rest of this proof, e runs over the set of edges with source v. By Proposition 2.4 we have ∑_{e}⟨σ_{e}⟩ = 1 and ∑_{e}(ν_{e} −1) = −3. Also, Lemma 2.5 implies that −2 ≤ ν_{e} ≤ 1 for all e and ν_{e} < 0 if and only if e = e_{0}. In this situation, [Wew], Lemma 2.8 says that ω_{v} cannot be exact (actually, in [Wew] one assumes that the type of (Z̄_{v}, ω_{v}) has an injective character χ_{v}; the general case follows from Remark 1.6). Therefore, ω_{v} is logarithmic. On the other hand, since ν_{e_{0}} < 0, the differential has a pole of order ≥ 2 at each point of Z̄_{v} lying above x̄_{e_{0}}. Therefore, ω_{v} cannot be logarithmic. This gives the desired contradiction and proves the theorem. □

2.2. Special G-deformation data. In this section we define special G-deformation data. To a special G-deformation datum we then associate a special G-map f̄: Ȳ → X̄. Briefly, a special G-map is a map between semistable curves over k which looks like the stable reduction of a three point G-cover with mild reduction.

2.2.1. Let k be an algebraically closed field of characteristic p > 0 and X̄ a smooth projective curve over k of genus 0. Let (Z̄, ω) be a deformation datum over X̄, of type (H, χ). As usual, we denote by (τ_{j})_{j∈B} the set of critical points for (Z̄, ω). Also, for j ∈ B we denote by m_{j}, h_{j} and σ_{j} the invariants attached to (Z̄, ω, τ_{j}). See §1.3. Definition 2.7. The deformation datum (Z̄, ω) is called special if for all j ∈ B we have σ_{j} < 2 and σ_{j} ≠ 1 and if there exists a subset B_{0} ⊂ B with exacly three elements such that σ_{j} < 1 if and only if j ∈ B_{0}. Remark 2.8. In case the character χ is injective and σ_{j} > 0 we recover the definition of a special deformation datum given in [Wew].

In the following we assume that (Z̄, ω) is special. Let B_{wild} (resp. B_{prim}, resp. B_{new}) denote the set of all indices j ∈ B such that σ_{j} = 0 (resp. 0 < σ_{j} < 1, resp. 1 < σ_{j} < 2). Then B_{0} = B_{wild} ∪ B_{prim}, and it follows from (3) that

(9)  ∑_{j∈B} ⟨σ_{j}⟩ = 1.

We call the special deformation datum (Z̄, ω) normalized if X̄ = ℙ_{k}^{1} and {τ_{j} | j ∈ B_{0}} = {0, 1, ∞}.

Their connection to three point covers suggests that special deformation data are ‘rigid’ objects, i.e. there should not exist any nontrivial families of special deformation data. This is indeed true; a precise statement is formulated and proved in [Wew02], §5.4. See also §3 below.

2.2.2. In this subsection, X̄ will denote a smooth projective curve over k of genus 0, together with a distinguished closed point ¯∞ ∈ X̄. We also fix a finite group G. Definition 2.9. A G-tail cover is a (not necessarily connected) G-cover f̄: Ȳ → X̄ such that the following holds.

(1) The cover f̄ is wildly ramified above ¯∞, of order pm with (p, m) = 1.

(2) The restriction of f̄ to X̄ −{ ¯∞} is tame and branched at most at one point. A pointed G-tail cover is a G-tail cover together with a point η ∈ Ȳ above ¯∞ and a subgroup H of the inertia group I_{η} such that I_{η} = I_{η,1} ⋊ H. An (outer) automorphism of a pointed G-tail cover is an automorphism γ : Ȳ →^{∼} Ȳ which normalizes the action of G (and hence induces an automorphism of X̄), centralizes the action of H ⊂ G and fixes the point η. Such an automorphism is called inner if it centralizes the action of G.

Let (f̄: Ȳ → X̄, η, H) be a pointed G-tail cover. By Condition (1), I_{η,1} is cyclic of order p and H cyclic of order prime-to-p. Let m denote the order of H and h the conductor of I_{η,1}. The pair (m_{j}, h_{j}) is called the ramification type and the rational number σ := h/m the ramification invariant of f̄. Note that h > 0 and (h, p) = 1. Moreover, m|(p − 1)h, by the Hasse–Arf Theorem (see [Ser68], IV §3). Definition 2.10. A tail cover f̄ : Ȳ → X̄ is called special if the following two conditions hold.

(1) We have σ < 2 and σ ≠ 1.

(2) If σ > 1 then the restriction of f̄ to X̄ − { ¯∞} is étale. If σ < 1 then we say that the tail cover f̄ is primitive, otherwise we say that it is new.

By [Ray99], Proposition 1.1.6, the converse of Condition (2) holds automatically: if σ < 1 then there exists a unique point ¯0 ∈ X̄ − { ¯∞} above which f̄ is tamely ramified. The reason is this: if f̄ were étale over X̄ − { ¯∞} and σ < 1, then the Riemann–Hurwitz formula would produce a negative number for the genus of Ȳ.

It is clear from the results of §2.1 that the tail covers obtained from the stable reduction of a three point Galois cover are special (provided that the reduction is not exceptional). We can therefore expect that a special tail cover f̄: Ȳ → X̄ is ‘rigid’. In fact, using the methods of [BM00] one can show that the curve Ȳ does not admit any nontrivial G-equivariant deformations. What is more important for us is that all special G-tail covers with a given ramification type (m, h) are ‘locally isomorphic’, in the following sense. Definition 2.11. Let (f̄_{1} : Ȳ_{1} → X̄, η_{1}, H) and (f̄_{2} : Ȳ_{2} → X̄, η_{2}, H) be two pointed G-tail covers with the same subgroup H ⊂ G and the same ramification type (m, h). Let X̄ ¯∞ denote the generic point of the completion of X̄ at ¯∞ and set Ȳ_{i, ¯∞} := Ȳ_{i} ×X̄ X̄ ¯∞. A local isomorphism between f̄_{1} and f̄_{2} is a G-equivariant isomorphism

φ ¯∞ : Ȳ_{1, ¯∞} ⟶^{∼} Ȳ_{2, ¯∞}

with the following properties.

(1) The automorphism ψ ¯∞ : X̄ ¯∞ →^{∼} X̄ ¯∞ induced by φ ¯∞ extends to a global automorphism ψ : X̄ →^{∼} X̄ (which necessarily fixes ¯∞).

(2) We have φ ¯∞(η_{1}) = η_{2}. Moreover, if f̄_{1} is ramified at ¯0_{1} ∈ X̄ − { ¯∞} then f̄_{2} is also ramified at some point ¯0_{2} ∈ X̄ − { ¯∞}, and the automorphism ψ in (1) can be chosen in such a way that ψ(¯0_{1}) = ¯0_{2}.

Lemma 2.12. All special pointed G-tail covers with given ramification type (m, h) and subgroup H ⊂ G are locally isomorphic.

Proof. This is a special case of the results of [Pri02], §4. For the convenience of the reader, we sketch a proof.

Let (f̄ : Ȳ → X̄, η, H) be a special pointed G-tail cover, of ramification type (m, h). We will show that f̄: Ȳ → X̄ is uniquely determined, up to local isomorphism, by the datum (G, H, m, h). First of all, by the definition of local isomorphism it suffices to look at tail covers f̄ which are totally ramified above ¯∞. In other words, we may assume that G = I ⋊ H, where I ⊂ G is cyclic of order p and H is cyclic of order m. Second, we shall assume that the tail cover f̄ is new. Hence we can write h = m + a, with 0 < a < m. Note that m|a(p − 1) and that a is prime to p. We will indicate later how to modify the proof in the case that f̄ is primitive.

Let x be a global coordinate for X̄ ≅ ℙ_{k}^{1} such that X̄ ¯∞ = Spec k((x^{−1})). By elementary Galois theory, the localized cover Ȳ ¯∞ → X̄ ¯∞ is given by two equations

(10)  z^{m} = x,  y^{p} − y = z^{a} (b_{0}x + b_{1} + b_{2}x^{−1} + …),

with b_{i} ∈ k and b_{0} ≠ 0. The action of G on Ȳ ¯∞ is given as follows. We may choose a generator α of I ≅ ℤ/p such that α^{∗}z = z and α^{∗}y = y + 1. Furthermore, for any β ∈ H we have β^{∗}z = ψ(β) · z and β^{∗}y = χ(β) · y, where ψ : H → 𝔽̄_{p}^{×} is a character of order m such that χ = ψ^{a}. See e.g. [Pri02], Lemma 1.4.1.

A local isomorphism amounts to a change of coordinates

(11)  x ↦ c(x + d),  z ↦ c^{1/m}z(1 + dx^{−1})^{1/m},  y ↦ y + g,

where c, d ∈ k, c ≠ 0 and g = z^{a}(e_{1}x^{−1} + e_{2}x^{−2} + …) ∈ k[[z^{−1}]]. An easy computation shows that for a suitable choice of the constants c and d, and after transforming the equations (10) by the change of coordinate (11), we may assume that b_{0} = 1 and b_{1} = 0. It is also easy to see that we may choose e_{1}, e_{2}, … in such a way that

(12)  g^{p} − g = −z^{a} (b_{2}x^{−1} + b_{3}x^{−2} + …)

(here we use that m|a(p − 1)). With this choice, equations (10) becomes

z^{m} = x, y^{p} − y = z^{h},

which visibly depends only on the ramification type (m, h). This proves the lemma for a new tail cover f̄. The proof for a primitive tail cover is similar. The main difference is that we are only allowed to do a homothety x ↦ cx instead of a general affine transformation x ↦ c(x + d). However, since 0 < h < m the proof goes through. □

2.2.3. As before, k denotes an algebraically closed field and G a finite group. Suppose we are given the following data.

• A subgroup G_{0} ⊂ G, of the form G_{0} = I_{0} ⋊ H_{0}, where I_{0} ⊂ G_{0} is cyclic of order p and H_{0} ⊂ G_{0} is of prime-to-p order. Let χ_{0} : H_{0} → 𝔽_{p}^{×} denote the

character defined by the rule βαβ^{−1} = α^{χ_{0}(β)}, where α ∈ I_{0}, β ∈ H_{0}.

• A normalized special deformation datum (Z̄_{0}, ω_{0}) over X̄_{0} := ℙ_{k}^{1} of type (H_{0}, χ_{0}). Let (τ_{j})_{j∈B} denote the set of critical points, m_{j}, h_{j} and σ_{j} the

invariants attached to (Z̄_{0}, ω_{0}, τ_{j}).
• For all j ∈ B − B_{wild} a point ξ_{j} ∈ Z̄_{0} above τ_{j} ∈ X̄_{0}.

• For all j ∈ B − B_{wild} a subgroup G_{j} ⊂ G and a connected pointed G_{j}-tail

cover (f̄_{j} : Ȳ_{j} → X̄_{j}, η_{j}, H_{j}).

By abuse of notation, we shall write (Z̄_{0}, ω_{0}; f̄_{j}) to denote all of the data introduced above. Definition 2.13. The datum (Z̄_{0}, ω_{0}; f̄_{j}) is called a special G-deformation datum if the following conditions hold.

(1) The group G is generated by its subgroups G_{0} and G_{j}.

(2) The tail cover f̄_{j} : Ȳ_{j} → X̄_{j} has ramification of type (m_{j}, h_{j}). Moreover,

I_{η_{j}} = I_{0} ⋊ H_{j}, and H_{j} = H_{0} ∩ I_{η_{j}}.

It is clear from the results of §2.1 that the stable reduction of a three point cover with mild reduction gives rise to a special G-deformation datum. To make this statement more precise, we introduce the following terminology.

Definition 2.14. A stable G-map over a scheme S is a homomorphism f : Y → X of marked semistable curves over S together with an S-linear action of G on Y such that Y is stably marked and f commutes with the G-action. Two stable G-maps f_{i} : Y_{i} → X_{i} are said to be isomorphic if there exist isomorphisms φ : Y_{1} →^{∼} Y_{2} and ψ : X_{1} → X_{2} of marked curves such that f_{2} ∘ φ = ψ ∘ f_{1} and such that φ is G-equivariant.

If f : Y → X is a G-cover over K̄, as in §1.1, then we may consider the stable reduction f̄: Ȳ → X̄ of f as a stable G-map over k.

Fix a special G-deformation datum (Z̄_{0}, ω_{0}; Ȳ_{j}). We will show that (Z̄_{0}, ω_{0}; Ȳ_{j}) gives rise to a stable G-map f̄: Ȳ → X̄ over k. Define

X̄ := ( X̄_{0} ∐ ∐_{j} X̄_{j} ) /_{∼}.

Here the equivalence relation ‘∼’ identifies the critical point τ_{j} ∈ X̄_{0} with the distinguished point ¯∞_{j} ∈ X̄_{j}. The topological space X̄ carries a unique structure of a semistable k-curve such that X̄_{0} and X̄_{j} are the irreducible components of X̄. We define marked points x̄_{j} on X̄ (indexed by the set B_{0} = B_{prim} ∪ B_{wild}), as follows. For j ∈ B_{prim} we let x̄_{j} ∈ X̄_{j} be the unique tame branch point of the tail cover f̄_{j} (see the remark after Definition 2.10). For j ∈ B_{wild} we set x̄_{j} := τ_{j}.

To define the curve Ȳ, let Ȳ_{0} be the smooth projective curve with function field k(Z̄_{0})^{1/p}. The action of H_{0} on Z̄_{0} extends uniquely to an action on Ȳ_{0}. We let G_{0} act on Ȳ_{0} via the quotient map G_{0} → H_{0}. Furthermore, we define

Ȳ := ( Ind_{G_{0}}^{G}(Ȳ_{0}) ∐ ∐_{j} Ind_{G_{j}}^{G}(Ȳ_{j}) ) /_{∼}.

Here the equivalence relation ‘∼’ is defined as follows: for η′ ∈ Ind_{G_{0}}^{G}(Ȳ_{0}) and η′′ ∈ Ind_{G_{j}}^{G}(Ȳ_{j}) we have η′ ∼ η′′ if and only if there exists an element α ∈ G such that α(η′) ∈ Ȳ_{0} lies above ξ_{j} ∈ Z̄_{0} and α(η′′) = η_{j} ∈ Ȳ_{j}. (This is an equivalence relation because of Condition (2) of Definition 2.13.) Again, Ȳ carries a canonical structure of a semistable curve over k. It follows from Condition (1) of Definition 2.13 that Ȳ is connected. Let f̄: Ȳ → X̄ be the unique map which commutes with the natural G-action on Ȳ and whose restriction to Ȳ_{0} (resp. to Ȳ_{j}) is equal to f̄_{0} (resp. to f̄_{j}). The marked points of Ȳ are, by definition, those which lie over a marked point of X̄. Thus, a point on Ȳ is marked either if it is the G-translate of a ramification point of a tail cover f̄_{j} or if it is a G-translate of a point on Ȳ_{0} which lies above a wild critical point. One checks that f̄: Ȳ → X̄ is a stable G-map. We call X̄_{0} = ℙ_{k}^{1} the original component of f̄. Definition 2.15. A stable G-map f̄: Ȳ → X̄ over k is called special if there exists a special G-deformation datum (Z̄_{0}, ω_{0}; Ȳ_{j}) such that f̄ is isomorphic to the stable G-map constructed above.

We can summarize the results of §1 and §2.1 as follows. Proposition 2.16. Let f : Y → X = ℙ1 ; K̄ be a G-Galois cover branched at 0, 1 and ¯∞, with mild reduction. Then the stable reduction of f is either exceptional or special.

In §4.1 we will prove the ‘converse’ of this proposition: every special G-map arises as the stable reduction of a three point cover f : Y → X = ℙ1 ; K̄ with mild reduction.

2.2.4. Let us fix a special G-deformation datum (Z̄_{0}, ω_{0}; Ȳ_{j}) and let f̄ : Ȳ → X̄ be the associated special G-map. Let us denote by Aut_{G}^{0}(f̄) the group of automorphisms of the special G-map f̄ which induce the identity on the original component X̄_{0}. If f̄ is the stable reduction of a three point G-cover f : Y → X then the image of the inner monodromy action κ̄^{in} associated to f is contained in Aut_{G}^{0}(f̄)/C_{G}. We are interested in a description of Aut_{G}^{0}(f̄) in terms of the special deformation datum (Z̄_{0}, ω_{0}; Ȳ_{j}).

Lemma 2.17. There is a natural bijection between the group Aut_{G}^{0}(f̄) and the set of tuples (γ_{0}; γ_{j}), where γ_{0} ∈ G and γ_{j} : Ȳ_{j} →^{∼} Ȳ_{j} is an (outer) automorphism of the pointed G_{j}-tail cover (f̄_{j}, η_{j}, H_{j}) (see Definition 2.9) such that the following holds.

(1) The element γ_{0} ∈ G normalizes I_{0} and centralizes H_{0}.

(2) The equation

γ_{0}^{−1} ∘ α ∘ γ_{0} = γ_{j} ∘ α ∘ γ_{j}^{−1}

holds for all α ∈ G_{j} and j ∉ B_{wild}.

Proof. Let γ : Ȳ →^{∼} Ȳ be an element of Aut_{G}^{0}(f̄). It is clear that we can find an element γ_{0} ∈ G such that γ_{0}^{−1} ∘ γ induces the identity on Ȳ_{0}. It is also clear that γ_{0} normalizes G_{0} and that conjugation by γ_{0} induces the identity on G_{0}/I_{0} ≅ H_{0}. Since the order of H_{0} is prime to p, the Schur–Zassenhaus Theorem implies that the subgroup γ_{0}H_{0}γ_{0}^{−1} ⊂ G_{0} is conjugate to H_{0} by an element of G_{0}. Using the fact that the action of H_{0} on I_{0} is irreducible, one shows that there exists a unique element α ∈ I_{0} such that

γ_{0} ∘ β ∘ γ_{0}^{−1} = α^{−1} ∘ β ∘ α

for all β ∈ H_{0}. After replacing γ_{0} by α ∘γ_{0}, we may thus assume that γ_{0} centralizes H_{0}. For all j ∉ B_{wild}, set γ_{j} := γ_{0}^{−1} ∘ γ|Ȳ_{j}. One easily checks that γ_{j} is an (outer) automorphism of the pointed G_{j}-tail cover (f̄_{j}, η_{j}, H_{j}) and that the tuple (γ_{0}; γ_{j}) satisfies Condition (1) and (2).

Conversely, given a tuple (γ_{0}; γ_{j}) such that (1) and (2) hold, we define an automorphism γ : Ȳ →^{∼} Ȳ of f̄, as follows. If y is a point on Ȳ_{0} and α is an element of G, we set

γ(α(y)) := α ∘ γ_{0}(y).

This is well defined because of Condition (1). If y is a point on Ȳ_{j} and α ∈ G, we set

γ(α(y)) := α ∘ γ_{0} ∘ γ_{j}(y).

This is well defined because γ ∘ γ_{j} commutes with G_{j}, by Condition (2). Moreover, the two definitions respect the equivalence relation ‘∼’ used in the definition of Ȳ in §2.2.3. Hence we obtain an automorphism γ : Ȳ →^{∼} Ȳ, which is easily seen to be an element of Aut_{G}^{0}(f̄). This proves the lemma. □

The association γ ↦ γ_{0} is a group homomorphism Aut_{G}^{0}(f̄) → G. Let us denote by n the order of the image of this homomorphism in G/C_{G}. Note that n|(p − 1). Let us denote by Aut_{G}(f̄_{j}) the group of inner automorphisms of the pointed G_{j}-tail covers (f̄_{j}, η_{j}, H_{j}). The lemma shows that

| Aut_{G}^{0}(f̄)| = n · |C_{G}| · ∏_{j∉B_{wild}} | Aut_{G}(f̄_{j})|.

Remark 2.18. Suppose that the character χ : H_{0} → 𝔽_{p}^{×} is injective. (In particular, H_{0} is cyclic). Then the integer n defined above is a multiple of

n′ := gcd_{j}(m_{j}).

In fact, let γ be an element of H_{0} which is contained in the stabilizer H_{j} of ξ_{j}, for all j ∈ B_{prim} ∪ B_{new}. Then the tuple (γ_{0}; γ_{j}) with γ_{j} := γ_{0}^{−1}|Ȳ_{j} represents an element of Aut_{G}^{0}(f̄) which does not lie in C_{G} and which acts trivially on the vertical components of Ȳ.

3. The auxiliary cover

Let (Z̄, ω) be a special deformation datum of type (H, χ). By a lift of (Z̄, ω) we mean a Galois cover f : Y → X = ℙ^{1} with Galois group G := ℤ/p ⋊_{χ} H such that the deformation datum occuring over the original component of the stable reduction of f is equal to (Z̄, ω). As we will see below, such a lift is uniquely determined by the deformation datum (Z̄, ω) and the set of branch points (x_{j}) of f (which lift the critical points (τ_{j}) of (Z̄, ω)).

Following [Wew], we say that the lift f : Y → X is special if the stable reduction of f is as simple as possible, i.e. if the curve X̄ is a comb. For instance, if f : Y → X is the auxiliary cover associated to a three point cover with mild reduction, then f is special by Theorem 2.6 (note that f is in general not a three point cover itself).

In [Wew] we showed that a lift f of (Z̄, ω) is special if each branch point of f is ‘sufficiently closed’ to a K_{0}-rational point. The main result of this section – Theorem 3.9 – says that the converse is true as well. This theorem will play a crucial role in §4.

The proof of Theorem 3.9 relies on the results of [Wew02], which concern the deformation theory of a certain curve with group scheme action associated to (Z̄, ω). The connection is as follows. If f : Y → X is a lift of the deformation datum (Z̄, ω) then there exist a certain natural R-model f_{R} : Y_{R} → X_{R} = ℙ_{R}^{1} (which is not the stable model) such that the G-action on Y extends to an action of the group scheme G := μ_{p} ⋊ H on Y_{R}. The special fiber Ȳ of Y_{R} is a singular curve with an action of G; it depends on (Z̄, ω) but not on the lift f : Y → X. Therefore, a lift of (Z̄, ω) is the same thing as an equivariant deformation of Ȳ.

3.1. The G-cover associated to a multiplicative deformation datum.

3.1.1. Throughout this section, we fix an odd prime p and an algebraically closed field k of characteristic p. We denote by W (k) the ring of Witt vectors over k.

Let X̄ be a smooth projective curve over k and let (Z̄, ω) be a multiplicative deformation datum over X̄ of type (H, χ), see Definition 1.5. In this section we show that (Z̄, ω) corresponds naturally to a (not necessarily smooth) curve Ȳ over k together with an action of a certain group scheme G such that X̄ = Ȳ /G.

We define G as a group scheme over W (k), as follows. If R is a W (k)-algebra, an element of G(R) is given by a tuple (ζ, β), where ζ ∈ R satisfies ζ^{p} = 1 and β ∈ H. Multiplication is defined as follows:

(ζ_{1}, β_{1}) · (ζ_{2}, β_{2}) := (ζ_{1}ζ_{2}^{χ(β_{1})}, β_{1}β_{2}).

In other words: G is the semidirect product μ_{p} ⋊ H determined by the action of H on μ_{p} via χ. If R is a W (k)-algebra and Y a scheme over R, we say that G acts on Y if we are given an R-linear action of G ⊗ R on Y.

Let (τ_{j})_{j∈B} be the set of critical points for the deformation datum (Z̄, ω). For j ∈ B, let m_{j}, h_{j} be the invariants attached to (Z̄, ω, τ_{j}), see §1.3. A critical point τ_{j} is called a tame (resp. wild) branch point if m_{j} > 1 (resp. if h_{j} = 0). We denote by B_{tame} and B_{wild} the corresponding subsets of B and set B_{ram} := B_{tame} ∪ B_{wild}. In general we have B_{tame} ∩ B_{wild} ≠ ∅ and B_{ram} ≠ B.

Let Ū ⊂ X̄ be the complement of the wild branch points and let V̄ ⊂ Z̄ be its inverse image. Then ω is a regular and logarithmic differential form on V̄. Let W̄ → V̄ be the corresponding μ_{p}-torsor, see e.g. [Mil80], p. 129. The definition of W̄ is roughly the following. Locally on V̄ we can write ω = du/u for a unit u. Then y^{p} = u is a local equation for the μ_{p}-torsor W̄ → V̄.

Proposition 3.1. Let Ȳ be the projective compactification of W̄ such that the complement Ȳ − W̄ is contained in the smooth locus of Ȳ. The action of μ_{p} on W̄ extends uniquely to an action of G on Ȳ such that Z̄ = Ȳ /μ_{p} and X̄ = Ȳ /G.

Proof. Let λ : μ_{p} × W̄ → W̄ denote the action of μ_{p} on W̄. We have to show that λ extends to all of Ȳ. Let η ∈ Ȳ − W̄ be a point in the complement of W̄ and let ξ ∈ Z̄ be its image. By construction, the point ξ lies above a wild critical point τ_{j}, so ω has a simple pole at ξ. Write ω = a du/u, where u is a regular function in a neighborhood of ξ with a simple zero at ξ and a ∈ 𝔽_{p}^{×} is the residue of ω at ξ. By definition of Ȳ there exists a rational function y on Ȳ such that y^{p} = u and λ^{∗}y = ζ^{b} ⊗ y, for some b ∈ 𝔽_{p}^{×}. It follows also from the definition of Ȳ that η is a smooth point of Ȳ. Therefore, the integral equation y^{p} = u shows that y ∈ O_{Ȳ ,η}. Moreover, since u has a simple zero at ξ, we have O_{Ȳ ,η} = O_{Z̄,ξ}[y | y^{p} = u]. This shows that the action λ extends to Ȳ. It is obviously that this extension is unique.

It remains to show that λ extends to an action of G on Ȳ which induces the canonical action of H on Z̄. But this is clear from the rule β^{∗}ω = χ(β) · ω and the definition of Ȳ. □

The map Ȳ → X̄ together with the action of G on Ȳ will be called the G-cover associated to (Z̄, ω). Note that the curve Ȳ is integral and has isolated, ‘cusp-like’ singularities precisely above the critical points which are not wild branch points, see [Ray90].

3.1.2. Let ℭ_{k} be the category of Noetherian Artinian local W (k)-algebras with residue field k. An equivariant deformation of Ȳ over R ∈ ℭ_{k} is a deformation Y of the curve Ȳ over R together with an action of G on Y such that the natural isomorphism Ȳ ≅ Y ⊗_{R} k is G-equivariant. Let Def (Ȳ , G) denote the functor which associates to a ring R ∈ ℭ_{k} the set of isomorphism classes of equivariant deformations of Ȳ over R.

Given an equivariant deformation Y_{R} of Ȳ, we denote by Z_{R} := Y_{R}/μ_{p} and X_{R} := Y_{R}/G the quotient schemes by μ_{p} and G, respectively. The R-curve X_{R} (resp. Z_{R}) is a smooth deformation of the curve X̄ (resp. Z̄). The association Y_{R} ↦ X_{R} gives rise to a morphism of deformation functors Def (Ȳ , G) → Def (X̄). Here Def (X̄) denotes the functor which classifies deformations of X̄. Let us write Def (X̄; τ_{j} | j ∈ B_{ram}) for the functor which classifies deformations of the pointed curve (X̄; τ_{j} | j ∈ B_{ram}). We claim that the morphism Def (Ȳ , G) → Def (X̄) factors in a natural way over the forgetful morphism Def (X̄; τ_{j} | j ∈ B_{ram}) → Def (X̄), thus inducing a morphism

(13)  Def (Ȳ , G) ⟶ Def (X̄; τ_{j} | j ∈ B_{ram}).

Let Y_{R} be an equivariant deformation of Ȳ and set X_{R} := Y_{R}/G, Z_{R} := Y_{R}/μ_{p}. To prove the claim we have to endow, in a functorial way, the curve X_{R} with sections τ_{j,R} : Spec R → X_{R} lifting the branch points τ_{j}, for all j ∈ B_{ram}. It is clear that the natural map Z_{R} → X_{R} is a tame H-Galois cover lifting Z̄ → X̄. By the theory of tame ramification, the branch divisor of Z_{R} → X_{R} consists of sections τ_{j,R} : Spec R → X_{R} lifting the tame branch points τ_{j} ∈ X̄, for all j ∈ B_{tame}. Now let j ∈ B_{wild} and choose a point ξ ∈ Z̄ above τ_{j}. We can write ω = du/u with ord_{ξ}(u) = 1. By construction, the restriction of Ȳ → Z̄ to a neighborhood of ξ is a μ_{p}-cover given by the Kummer equation y^{p} = u. Hence the deformation Y_{R} → Z_{R} is, locally around ξ, still a μ_{p}-cover given by a Kummer equation y^{p} = u_{R}, where u_{R} lifts u. The equation u_{R} = 0 defines a section ξ_{R} : Spec R → Z_{R} lifting ξ. We define τ_{j,R} : Spec R → X_{R} as the image of ξ_{R}. Using the H-action, it is easy to see that our two definitions of τ_{j,R} agree if j ∈ B_{tame} ∩ B_{wild}. This proves the claim.

We say that Y_{R} → X_{R} is a deformation of the G-cover Ȳ → X̄, and we call the tuple (τ_{j,R})_{j∈B_{ram}} the branch locus of Y_{R} → X_{R}. It is clear that a deformation of the G-cover Ȳ → X̄ is the same thing as an equivariant deformation of Ȳ.

3.2. The case of a special deformation datum.

3.2.1. We continue with the notation of §3.1. In addition, we assume from now on that the deformation datum (Z̄, ω) is special, see Definition 2.7. We use the notation B_{prim}, B_{new} and B_{0} as in §2.2.1. It follows immediately from the definitions that we have B_{tame} = B_{prim} ∪ B_{new} and B_{wild} = B_{0} − B_{prim}. In particular, we have B_{tame} ∩ B_{wild} = ∅ and B = B_{ram}. Furthermore, we assume that the special deformation datum (Z̄, ω) is normalized, i.e. we have X̄ = ℙ_{k}^{1} and {τ_{j} | j ∈ B_{0}} = {0, 1, ∞}.

Let Y_{R} → X_{R} be a deformation of the G-cover Ȳ → X̄ with branch locus (τ_{j,R})_{j∈B}. There exists a unique isomorphism X_{R} ≅ ℙ_{R}^{1} which reduces to the identity on X̄ = ℙ_{k}^{1} and such that { τ_{j,R} | j ∈ B_{0} } = {0, 1, ∞}. Hence we may and will identify X_{R} with ℙ_{R}^{1} and the sections τ_{j,R}, for j ∈ B_{new}, with the corresponding elements of R. A B-tuple (τ_{j,R}) of R-points of X_{R} = ℙ_{R}^{1} such that τ_{j,R} lifts τ_{j} for all j ∈ B and { τ_{j,R} | j ∈ B_{0} } = {0, 1, ∞} is called a normalized B-tuple over R. Proposition 3.2. The morphism of deformation functors

Def (Ȳ , G) ⟶ Def (X̄; τ_{j})

is an isomorphism. In other words: given R ∈ ℭ_{k} and a normalized B-tuple (τ_{j,R}) over R, there exists a deformation Y_{R} → X_{R} = ℙ_{R}^{1} of the G-cover Ȳ → X̄ = ℙ_{k}^{1} with branch locus (τ_{j,R}). Moreover, Y_{R} → X_{R} is unique up to unique isomorphism.

Proof. This is a special case of [Wew02], Theorem 5.7. □

For j ∈ B_{new}, let T_{j} be an indeterminate; set R̃ := W (k)[[ T_{j} | j ∈ B_{new} ]]. For j ∈ B_{new}, let [τ_{j}] ∈ W (k) denote the Teichmüller lift of τ_{j} (in fact, any lift of τ_{j} to an element of W (k) would do as well). Let (τ̃_{j}) be the normalized B-tuple over R̃ which has τ̃_{j} = [τ_{j}]+T_{j} for j ∈ B_{new}. Let X denote the formal completion along the special fiber of the projective line over R̃. It is clear that for any R ∈ ℭ_{k} there is a one-to-one correspondence between local W (k)-algebra homomorphism R̃ → R and normalized B-tuples over R. Therefore, (X; τ̃_{j}) is the universal formal deformation of the pointed curve (X̄; τ_{j}). Hence Proposition 3.2 implies: Corollary 3.3. Let Y → X be the (unique) formal deformation of the G-cover Ȳ → X̄ with branch locus (τ̃_{j}). Then Y is a universal formal equivariant deformation of Ȳ.

3.2.2. For j ∈ B, we denote by X̂_{j} the completion of X̄ at τ_{j} and set Ŷ_{j} := Ȳ ×X̄ X̂_{j}. The action of G on Ȳ induces an action on Ŷ_{j} such that X̂_{j} = Ŷ_{j}/G. Given a deformation Y_{R} → X_{R} of the G-cover Ȳ → X̄, we denote by X̂_{j,R} the completion of X_{R} at τ_{j} and set Ŷ_{j,R} := X̂_{j} ×_{X_{R}} X̂_{j,R}. Then Ŷ_{j,R} is a G-equivariant deformation of Ŷ_{j}. The association Y_{R} ↦ (Ŷ_{j,R})_{j∈B} defines a morphism of deformation functors

(14)  Φ : Def (Ȳ , G) ⟶ ∏_{j∈B} Def (Ŷ_{j}, G),

Here Def (Ŷ_{j}, G) denotes the deformation functor which classifies G-equivariant formal deformations of Ŷ_{j}. Following [BM00], we call Φ the local-global morphism. Proposition 3.4. (1) The functor Def (Ŷ_{j}, G) admits a miniversal deforma-

tion over the ring

R̃_{j} := \{ \begin{matrix}W (k), & for j ∈ B_{0}, \\ W (k)[[t_{j}]], & for j ∈ B_{new}. \end{matrix}

Here t_{j} is an indeterminate.

(2) The local-global morphism Φ is an isomorphism. Therefore,

R̃ ≅ ˆ⊗_{W (k)} R̃_{j} = W (k)[[ t_{j} | j ∈ B_{new} ]].

(3) For every j ∈ B_{new} there exists a unit w_{j} ∈ R̃ such that

t_{j} ≡ w_{j}T_{j} (mod p).

Proof. Assertion (1) and (2) are a special case of [Wew02], Theorem 5.11. Assertion (3) corresponds to [Wew02], Proposition 5.14. □

3.2.3. Let Y → X be the universal deformation over R̃ of the G-cover Ȳ → X̄, as defined in §3.2.1. Fix an index j ∈ B_{new}. We denote by X̂_{j} the completion of X at τ_{j} and set Ŷ_{j} := Y×_{X} X̂_{j}. Being an equivariant deformation of Ŷ_{j} over R̃, Ŷ_{j} is induced from the miniversal deformation of Proposition 3.4 via a homomorphism of W (k)- algebras φ_{j} : R̃_{j} → R̃. The homomorphism φ_{j} is unique up to an automorphism of W (k)-algebra R̃_{j}. Since φ_{j} is injective by Proposition 3.4 (2), we may identify the ring R̃_{j} with the image of φ_{j}. By Proposition 3.4 (1) the ring R̃_{j} is a formal power series over W (k) in one variable t_{j}. It is clear that Assertion (3) of Proposition 3.4 does not depend on the choice of the parameter t_{j} as long as R̃_{j} ≅ W (k)[[t_{j}]].

The quotient Z := Y/μ_{p} is a formal H-equivariant deformation of Z̄. Let us choose a point ξ ∈ Z̄ above τ_{j} and denote by Ẑ_{j} the completion of Z at ξ_{j}. Since Z is formally smooth over R̃ we have Ẑ_{j} = Spec R̃[[z]] for some parameter z. We may assume

(15)  β^{∗}z ≡ χ_{j}(β) · z (mod p),

for all β ∈ H_{j} and some character χ_{j} : H_{j} → k^{×}. Actually, we have χ_{j}^{h_{j}} = χ|_{H_{j}}. (Of course, we could have choosen z in such a way that the above congruence would become an equality in R̃[[z]], but this would give a stronger restriction on the choice of z than we are are willing to make.)

The natural map Y → Z is a μ_{p}-torsor in a neighborhood of ξ. Therefore, the pullback to the completion Ẑ_{j} is defined by a Kummer equation

(16)  y^{p} = u = ∑_{μ≥0} c_{μ} z^{μ},

with c_{μ} ∈ R̃ and c_{0} ∈ R̃^{×}. The unit u ∈ R̃[[z]]^{×} is unique up to a pth power. As we have already remarked above, the μ_{p}-torsor Ŷ_{j} → Ẑ_{j} is the pullback of a μ_{p}-torsor defined over R̃_{j}. In other words, we can choose the parameter z and the Kummer equation (16) in such a way that all coefficients c_{μ} lie in R̃_{j} ⊂ R̃. In the following, we will assume that c_{μ} ∈ R̃_{j} and we pretend that Ŷ_{j} → Ẑ_{j} is actually defined over R̃_{j}. Lemma 3.5. Let a_{j} := h_{j} − m_{j}. We may choose the parameter z, the Kummer equation (16) and a parameter t_{j} for R̃_{j} such that

u ≡ 1 + t_{j} z^{a_{j}} + z^{h_{j}} mod ( p, t_{j} z^{a_{j}+1}, z^{h_{j}+1} ).

Proof. Given an element c ∈ R̃_{j}, we write c′ (resp. c̄) for the image of c in R′ := R̃_{j}/(p) (resp. in k = R̃_{j}/(p, t_{j})). We claim that c′_{0} ∈ (R′^{×})^{p}. To prove this claim, let β be a generator of H_{j} ≅ ℤ/m_{j} and 0 < a < p an integer such that a ≡ χ(β) mod p. The subgroup scheme G_{j} := μ_{p} · H_{j} ⊂ G acts on Ŷ_{j}, extending the action of H_{j} on Ẑ_{j} given by (15). Therefore, there exists a unit w ∈ R̃_{j}[[z]]^{×} such that β^{∗}y = wy^{a}. Taking pth powers on both sides we obtain β^{∗}u = w^{p}u^{a}. Using (15) shows that c_{0}^{1−a} is congruent to a pth power modulo p. Since σ_{j} = h_{j}/m_{j} is not an integer, we have a ≢ 1 mod p. The claim follows. In particular, after multiplying u with a pth power in R̃_{j}[[z]] we may assume that c′_{0} = 1. We may also assume that c′_{μ} = 0 whenever p|μ.

Let ω′ := du′/u′. This is a logarithmic differential which lifts ω from Ẑ_{j} to Ẑ_{j} ⊗ R′; it classifies the μ_{p}-torsor Ŷ_{j} ⊗ R′ → Ẑ_{j} ⊗ R′ up to isomorphism. Since the action of μ_{p} on Ŷ_{j} extends to an action of G_{j}, we have β^{∗}ω′ = χ(β) · ω′ for all β ∈ H_{j}. This and (15) shows that

ω′ = ∑_{μ≥0} b′_{μ} z^{μm_{j}+a_{j}−1} dz

with b′_{μ} ∈ R′ (note that 0 < a_{j} < m_{j} and χ_{j}^{a_{j}} = χ|_{H_{j}}). Since ω has a zero of exact order h_{j} − 1 at ξ_{j}, we have b̄_{0} = 0 and b̄_{1} ≠ 0. After a short computation, we conclude that c′_{μ} = 0 for 0 < μ < a_{j} and c̄_{μ} = 0 for a_{j} < μ < h_{j}.

So far, we have shown that

u ≡ 1 + ct_{j} z^{a_{j}} + z^{h_{j}} mod ( p, t_{j} z^{a_{j}+1}, z^{h_{j}+1} )

for some element c ∈ R̃_{j}. It remains to prove that c̄ ≠ 0. For then we can replace t_{j} by c^{−1}t_{j} and obtain the desired congruence. Let k[ϵ] be the ring of dual numbers over k and κ : R_{j} → k[ϵ] the unique W (k)-algebra homomorphism with κ(t_{j}) = ϵ. We can write

κ(u) = ū + ϵ · v̄ = (1 + z^{h_{j}} + ∑_{μ≥h_{j}} c̄_{μ} z^{μ}) + ϵ · ( ∑_{μ≥a_{j}} d̄_{μ} z^{μ}),

with c̄_{μ}, d̄_{μ} ∈ k. We have to show that d̄_{a_{j}} ≠ 0.

By the proof of [Wew02], Lemma 5.12, there exists an H_{j}-invariant derivation of k[[z]][z^{−1}]

θ_{j} = (ē_{0} + ē_{1} x + …) \frac{d}{dx} = \frac{z^{1−m_{j}}}{m_{j}} ( ē_{0} + ē_{1} z^{m_{j}} + …) \frac{d}{dz}

(which is the same thing as a derivation of k[[x]], with x := z^{m_{j}}) such that the following holds:

(a) v̄ = θ_{j}(ū),

(b) we have ē_{0} = 0 if and only if the pullback of Ŷ_{j} via the homomorphism

κ : R̃_{j} → k[ϵ] is the trivial deformation of Ŷ_{j}. In fact, the class of θ_{j} modulo k[[x]] x \frac{d}{dx} represents the isomorphism class of the equivariant deformation Ŷ_{j} ⊗ k[ϵ], via a certain isomorphism

Ext_{G}^{1}(ΩȲ , OȲ)_{τ_{j}} ≅ T_{X̄,τ_{j}}/x · T_{X̄,τ_{j}},

see [Wew02], §5.3. But since Ŷ_{j} is a miniversal deformation of Ŷ_{j} and κ is not constant, Ŷ_{j} ⊗ k[ϵ] must be nontrivial. Therefore,

d̄_{a_{j}} = \frac{h_{j} ē_{0}}{m_{j}} ≠ 0.

This proves the lemma. □

Corollary 3.6. We may choose the parameter z and the Kummer equation (16) in such a way that

u ≡ 1 + w_{j}T_{j} z^{a_{j}} + z^{h_{j}} mod ( p^{2}, T_{j} z^{a_{j}+1}, z^{h_{j}+1} ).

for some unit w_{j} ∈ R̃^{×}.

Proof. It follows from Proposition 3.4 (3) and Lemma 3.5 that we may assume

u ≡ 1 + w_{j}T_{j} z^{a_{j}} + z^{h_{j}} + p · v mod ( p^{2}, T_{j} z^{a_{j}+1}, z^{h_{j}+1} )

for a unit w_{j} ∈ R̃ and some v ∈ R̃[[z]]. Replacing the parameter z by z′ := z − pv(h_{j} + a_{j}w_{j}T_{j})^{−1}, we obtain the stronger congruence, modulo p^{2}. □

3.3. The stable reduction of the auxiliary cover.

3.3.1. Let k be an algebraically closed field of characteristic p > 0 and let (Z̄, ω) be a normalized special deformation datum over X̄ := ℙ_{k}^{1} of type (H, χ). Let Ȳ → X̄ be the G-cover associated to (Z̄, ω), as defined in §3.1.1. Let K_{0} denote the fraction field of W (k) and let K/K_{0} be a finite extension. We assume that K contains the pth roots of unity. The integral closure R of W (k) inside K is a complete mixed characteristic discrete valuation ring with residue field k.

Let (τ_{j,R}) be a normalized B-tuple over R, see §3.2.1. By Corollary 3.3 there exists a unique formal deformation Y_{R} → X_{R} = ℙ_{R}^{1} of the G-cover Ȳ → X̄ with branch locus (τ_{j,R}). By Grothendieck’s Existence Theorem, this formal deformation is effective, i.e. we may regard Y_{R} → X_{R} as a finite morphism of schemes over R. Write Y_{K} → X_{K} = ℙ_{K}^{1} for the generic fiber of Y_{R} → X_{R} and x_{j} := τ_{j,K} for the K-rational point on X_{K} corresponding to the section τ_{j,R}, for j ∈ B. Let

G := G(K) ≅ ℤ/p ⋊_{χ} H

be the group of K-rational points of G. The group G acts on Y_{K}, and it is easy to see that Y_{K} → X_{K} is a G-Galois cover with branch locus (x_{j}). The ramification index of this cover at the branch point x_{j} is equal to m_{j} for j ∈ B_{tame} = B_{prim} ∪B_{new} and equal to p for j ∈ B_{wild}. Write Y → X := ℙ1 ; K̄ for the pullback of Y_{K} → X_{K} to K̄.

Definition 3.7. The G-cover f : Y → X is called the lift of the special deformation datum (Z̄, ω) with branch locus (x_{j}).

This definition is justified by the next proposition.

Proposition 3.8. Let G′ be a finite group, isomorphic to the semi-direct product ℤ/p ⋊ H determined by the character χ. Let W → X be a G′-Galois cover. Then W → X and Y → X are isomorphic as mere covers if and only if the following conditions hold.

(1) The branch locus of W → X is equal to {x_{j} | j ∈ B}.

(2) The cover W → X has bad reduction and the deformation datum occuring over the original component of the stable reduction is isomorphic to (Z̄, ω).

Proof. Let W → X be a G′-Galois cover satisfying Conditions (1) and (2) of the proposition. After replacing K by a larger field, if necessary, we may suppose that W → X has a model W_{K} → X_{K} over K which extends to a stable model WR^{st} → XR^{st}. Let W_{R} denote the normalization of X_{R} = ℙ_{R}^{1} in the function field of W_{K}. We obtain a canonical map WR^{st} → W_{R} which is an isomorphism except in the singular points of W̄. By Condition (2), the action of G′ on W̄ has a kernel I′ ⊲ G′, cyclic of order p. By the assumption on G′ and W → X, we have G′/I′ ≅ H and there exists a finite map W̄ → Z̄, purely inseparable of degree p and H-equivariant. Clearly, the isomorphism G′/I′ ≅ H extends to an isomorphism G′ ≅ G.

Let G′ be the schematic closure of G′ = Aut(W_{K}/X_{K}) inside the R-group scheme Aut(W_{R}/X_{R}). By [Ray74], G′ is a finite flat group scheme over R with generic fiber G′. We claim that the isomorphism G′ ≅ G chosen above extends to an isomorphism G′ ≅ G ⊗_{W (k)} R of R-group schemes. Indeed, it suffices to show that the closure of I′ inside G′ is isomorphic to μ_{p} (because then G′ is the minimal prolongation of G′ over R, see [Ray74]). Therefore, the claim follows from Condition (2).

It is no loss of generality to identify G′ with G ⊗_{W (k)} R. So G acts on W̄ such that Z̄ = W̄ /μ_{p} and X̄ = W̄ /G. Moreover, the restriction of the map W̄ → Z̄ to a sufficiently small open subset is a μ_{p}-cover corresponding to the differential ω. We conclude that there exists a G-equivariant isomorphism W̄ →^{∼} Ȳ, see the proof of Proposition 3.1. Via this isomorphism, we may regard W_{R} as an equivariant deformation of Ȳ. By Condition (1), the two G-covers Y_{R} → X_{R} ad W_{R} → X_{R} have the same branch locus (τ_{j,R}). Hence the proposition follows from Corollary 3.3. □

3.3.2. Let (τ_{j,R}) be a normalized B-tuple over R, x_{j} := τ_{j,K} ∈ X := ℙ1 ; K̄, and let Y → X be the lift of (Z̄, ω) with branch locus (x_{j}). After replacing K by a finite extension, we may assume that the natural K-model Y_{K} → X_{K} extends to the stable model YR^{st} → XR^{st}. We denote by Ȳ^{st} → X̄^{st} the stable reduction and by x̄_{j} ∈ X̄^{st} the specialization of the branch point x_{j}. The stable model is obtained as a blow-up of the natural R-model Y_{R} → X_{R}, i.e. we have a commutative diagram

(17)  \begin{matrix}YR^{st} & ⟶ & XR^{st} \\ ↓ & & ↓ \\ Y_{R} & ⟶ & X_{R} \end{matrix}

in which the vertical arrows are modifications which are isomorphisms away from the critical points of X̄ resp. the singular points of Ȳ. In particular, the map X̄^{st} → X̄ = ℙ^{1} induces an isomorphism between X̄ and the original component X̄_{0} of X̄^{st}, and contracts all other components to one of the critical points τ_{j}, j ∈ B_{prim} ∪ B_{new}. By construction of Y_{R}, the deformation datum over X̄_{0} ≅ X̄ associated to Ȳ^{st} → X̄^{st} is isomorphic to (Z̄, ω).

Here is the main result of this section. Theorem 3.9. The following two conditions are equivalent:

(1) The curve X̄^{st} consists only of the original component X̄_{0} and, for each j ∈ B_{tame}, a tail X̄_{j} which contains x̄_{j} and intersects X̄_{0} ≅ X̄ in τ_{j}.

(2) For all j ∈ B_{new} we have

val(τ_{j,R} − [τ_{j}]) ≥ \frac{p m_{j}}{(p − 1) h_{j}} · val(p).

(Recall that [τ_{j}] ∈ W (k) ⊂ R denotes the Teichmüller lift of τ_{j}.) Remark 3.10. Under the additional assumption B_{wild} = ∅, the implication ‘(2) ⇒ (1)’ is already proved in [Wew].

Proof. The element τ_{j,R} − [τ_{j}] ∈ R is the image of the parameter T_{j} (defined in §3.2.2) under the classifying morphism R̃ → R of the equivariant deformation Y_{R}. By abuse of notation, we write T_{j} := τ_{j,R} − [τ_{j}] as an element of R.

Let us first prove the implication ‘(2) ⇒ (1)’, i.e. we assume that (1) holds. To find the modification YR^{st} → Y_{R} which leads to the stably marked model of Y_{R}, it suffices to consider the completion of Y_{R} at a singular point of Ȳ. Let η_{j} ∈ Ȳ be one of the singular points, lying above the critical point τ_{j} ∈ X̄ for j ∈ B_{tame}. Let ξ_{j} ∈ Z̄ be the image of η_{j}. Let Ŷ_{R,j} denote the completion of Y_{R} at η_{j} and Ẑ_{R,j} the completion of Z_{R} at ξ_{j}. Condition (1) of the theorem means that the germ Ŷ_{R,j} has potentially good reduction. We can write Ẑ_{R,j} = Spec R[[z]]. The μ_{p}-torsor Ŷ_{R,j} → Ẑ_{R,j} is given by a Kummer equation y^{p} = u, with u ∈ R[[z]]^{×}.

Let us first treat the case j ∈ B_{new}. By Lemma 3.5 we may assume that

(18)  y^{p} = u ≡ 1 + w_{j} T_{j} z^{a_{j}} + z^{h_{j}} mod ( p^{2}, T_{j} z^{a_{j}+1}, z^{h_{j}+1} ),

with w_{j} ∈ R^{×}. We also may assume that R contains an element λ such that λ^{h_{j}(p−1)} = −p. Let y = 1 + λ^{h_{j}}y′ and z = λ^{p}z′. The new coordinates y′ and z′ give rise to a diagram

Y′_{R,j} ⟶ Z′_{R,j}
↓  ↓
Ŷ_{R,j} ⟶ Ẑ_{R,j}

in which the vertical arrows are blow-ups centered in the closed point of the germs Ŷ_{R,j} and Ẑ_{R,j}, respectively. Let Ȳ′_{j} (resp. Z̄′_{j}) be the exceptional divisor of these blowups. The curve Z̄′_{j} is a projective line with parameter z′. To obtain the equation for the cover Ȳ′_{j} → Z̄′_{j}, rewrite (18) in terms of z′ and y′, subtract 1, divide by λ^{ph_{j}} and reduce modulo the maximal ideal of R. Using the assumption (p − 1)h_{j} val(T_{j}) ≥ pm_{j} val(p), one obtains an Artin-Schreier equation

(19)  y′^{p} − y′ = w̄_{j} z′^{a_{j}} + z′^{h_{j}}

with conductor h_{j}. It follows that Ȳ′_{j} is smooth and that Y′_{R,j} is a semistable model of the germ Ŷ_{R,j}. In other words, the germ Ŷ_{R,j} has potentially good reduction. See e.g. [Ray99], Lemme 4.3.3 or [Leh01], §3. We conclude that Condition (1) holds for the given index j ∈ B_{new}.

Condition (1) holds for j ∈ B_{prim} as well, without any assumption. Indeed, if j ∈ B_{prim} then 0 < h_{j} < m_{j}. An easy version of the proof of Lemma 3.5 shows that (for the right choice of parameter z) the μ_{p}-torsor Y_{R,j} → Z_{R,j} is given by a Kummer equation of the form

y^{p} = u ≡ 1 + z^{h_{j}} mod ( p^{2}, z^{h_{j}+1} ).

After a change of coordinate y = 1 + λ^{h_{j}}y′ and z′ = λ^{p}z′ we again find an Artin– Schreier equation with conductor h_{j}, which shows that the germ Ŷ_{R,j} has potentially good reduction. This finishes the proof of the implication ‘(2) ⇒ (1)’.

It remains to prove the implication ‘(1) ⇒ (2)’. Suppose that Condition (2) is violated for some j ∈ B_{new}. We may also assume that R contains an element λ with λ^{pm_{j}} = T_{j}. After substituting y = 1 + λ^{h_{j}}y′ and z = λ^{p}z′ into (18) and reducing modulo the maximal ideal of R, we find that the cover Ȳ′_{j} → Z̄′_{j} has equation

y′^{p} = w̄_{j} z′^{a_{j}} + z′^{h_{j}}.

It follows that the cover Ȳ′_{j} → Z̄′_{j} is an α_{p}-torsor (away from the point z′ = ∞) corresponding to the exact differential ω′ = d(w̄_{j}z′^{a_{j}} + z′^{h_{j}}). The differential ω′ has a zero of order a_{j} − 1 at z′ = 0 and m_{j} further simple zeros. Above each point where ω′ has a zero, Ȳ′_{j} has a singularity. We conclude that the germ Ŷ_{R,j} does not have potentially good reduction. In other words, Condition (1) of Theorem 3.9 does not hold. This finishes the proof of Theorem 3.9. □

The proof shows that, if both the conditions of the theorem hold, then the closed rigid disk

D_{j} := \{ x | val(x − [τ_{j}]) ≥ \frac{p m_{j}}{(p − 1) h_{j}} · val(p) \} ⊂ X_{K}^{rig}

is equal to the set of points on X_{K} which specialize to the tail X̄_{j} ⊂ XR^{st}. We remark at this point that, since \frac{pm_{j}}{(p−1)h_{j}} < 1 for j ∈ B_{new}, the family of disks (D_{j})_{j∈B} is independent of the choice of (τ_{j,R}), as long as Condition (2) of Theorem 3.9 holds.

3.3.3. Let us now assume that the branch points x_{j} of the lift f : Y → X of (Z̄, ω) are all K_{0}-rational. This is equivalent to the condition that τ_{j,R} ∈ W (k). Then Condition (2) of Theorem 3.9 is automatically verified. Moreover, the cover f : Y → X has a natural model f_{K_{0}} : Y_{K_{0}} → X_{K_{0}} = ℙ_{K_{0}}^{1} over K_{0} (which is not Galois, since K_{0} does not contain the pth roots of unity). Let

κ̄^{ab} : Gal(K̄/K_{0}) ⟶ Aut_{k}(Ȳ^{st})

be the absolute monodromy action associated to the K_{0}-model f_{K_{0}} of f, see §1.1.3. The goal of this subsection is to describe this action in detail.

First some notation. Let Ȳ_{0} be the unique irreducible component of Ȳ^{st} which lies above the original component X̄_{0} ⊂ X̄^{st}. We may identify Ȳ_{0} with the normalization of Ȳ or, what amounts to the same thing, the smooth projective model of the function field k(Z̄)^{1/p}. Let us choose, for all j ∈ B_{tame}, a point η_{j} ∈ Ȳ_{0} above τ_{j} ∈ X̄. We denote by ξ_{j} ∈ Z̄ the image of η_{j}, by H_{j} ⊂ H the stabilizer of ξ_{j} and by I ⊂ G the cyclic subgroup of order p. Set G_{j} := I · H_{j} ⊂ G. Let Ȳ_{j} ⊂ Ȳ^{st} (resp. X̄_{j} ⊂ X̄^{st}) be the component which intersects Ȳ_{0} in η_{j} (resp. intersects X̄ in τ_{j}). By Theorem 3.9, X̄^{st} = X̄ ∪ ∪_{j}X̄_{j}. Furthermore, the proof of the theorem shows that (Ȳ_{j} → X̄_{j}, η_{j}, H_{j}) is a pointed G_{j}-tail cover with ramification type (m_{j}, h_{j}) which is totally ramified above τ_{j} ∈ X̄_{j}, see §2.2.2.

Let us denote by Aut_{k}^{0}(Ȳ^{st}) the group of k-linear automorphisms of Ȳ^{st} which normalize the action of G, commute with the action of H ⊂ G and induce the identity on Ȳ_{0}. Similarly, let Aut_{k}^{0}(Ȳ_{j}) denote the group of (outer) automorphisms of the pointed G_{j}-tail cover f̄_{j}. Recall that these are k-linear automorphisms of Ȳ_{j} which normalize the action of G_{j}, commute with the action of H_{j} ⊂ G_{j} and fix the point η_{j}. For each j we have a natural homomorphism

(20)  Aut_{k}^{0}(Ȳ^{st}) ⟶ Aut_{k}^{0}(Ȳ_{j}).

Let α be a generator of I ⊲ G. For φ ∈ Aut_{k}^{0}(Ȳ) we have φ ∘ α ∘ φ^{−1} = α^{ρ} for some ρ ∈ 𝔽_{p}^{×}; this defines a character

(21)  ρ : Aut_{k}^{0}(Ȳ) ⟶ 𝔽_{p}^{×}.

Similarly, one defines characters ρ_{j} : Aut_{k}^{0}(Ȳ_{j}) → 𝔽_{p}^{×} for all j ∈ B_{tame}. The composition of ρ with κ̄^{ab} is the cyclotomic character modulo p. The kernel of the character ρ is the subgroup Aut_{G}^{0}(Ȳ^{st}) ⊂ Aut_{k}^{0}(Ȳ^{st}) of automorphisms which commute with the G-action. Compare with §2.2.4. Proposition 3.11. (1) We have a canonical isomorphism

Aut_{k}^{0}(Ȳ^{st}) ≅ { (ρ; φ_{j}) ∈ 𝔽_{p}^{×} × ∏_{j} Aut_{k}^{0}(Ȳ_{j}) | ρ = ρ_{j}(φ_{j}) ∀j }.

(2) The group Aut_{k}^{0}(Ȳ_{j}) is cyclic of order (p − 1)h_{j}. Therefore, the group Aut_{k}^{0}(Ȳ^{st}) has order (p − 1) ∏_{j} h_{j}.

(3) The restriction of κ̄^{ab} to the component Ȳ_{j} is a surjective homomorphism

κ̄j^{ab} : Gal(K̄/K_{0}) ⟶ Aut_{k}^{0}(Ȳ_{j}).

(4) The image of κ̄^{ab} is a cyclic subgroup of Aut_{k}^{0}(Ȳ^{st}), of order

N := (p − 1) · lcm_{j}(h_{j}).

Proof. Part (1) is left to the reader. We also leave it to the reader to check that the image of the monodromy action κ̄^{ab} is contained in Aut_{k}^{0}(Ȳ^{st}).

The explicit description of the change of coordinates y = 1 + λ^{h_{j}}y′, z = λ^{p}z′ in the proof of Theorem 3.9 shows that the restriction of κ̄^{ab} to Ȳ_{j} is cyclic of order (p − 1)h_{j} (it corresponds to the cyclic Galois extension K_{0}(λ)/K_{0} where λ^{(p−1)h_{j}} = −p). In particular, the group Aut_{k}^{0}(Ȳ_{j}) has a cyclic subgroup of order (p − 1)h_{j}. Therefore, in order to prove the proposition it suffices to show that the group Aut_{k}^{0}(Ȳ_{j}) has exact order (p − 1)h_{j}.

First, we claim that Aut_{k}^{0}(Ȳ_{j}) is cyclic. Indeed, the natural action of Aut_{k}^{0}(Ȳ_{j}) on Ȳ_{j} descents to a faithful action on the quotient X̄_{j} = Ȳ_{j}/G_{j} which fixes the two points τ_{j} and x̄_{j}. But X̄_{j} has genus 0, so the claim follows. Now assume that Aut_{k}^{0}(Ȳ_{j}) has order (p−1)h_{j}n. Let φ ∈ Aut_{k}^{0}(Ȳ_{j}) be an element of order h_{j}n. Since φ lies in the kernel of the character ρ_{j} it commutes with the action of G_{j} on Ȳ_{j}. Let Ȳ′_{j} := Ȳ_{j}/⟨φ⟩ → X̄′_{j} := X̄_{j}/⟨φ⟩ be the induced G_{j}-cover. Recall that the action of I ≅ ℤ/p on Ȳ_{j} has conductor h_{j} at the point η_{j}. Let η′_{j} denote the image of η_{j} on Ȳ′_{j}. A classical calculation (see e.g. [Ray99], Lemme 1.1.2) shows that the action of I on Ȳ′_{j} has conductor 1/n at η′_{j}. We conclude that n = 1. This finishes the proof of the proposition. □

The proposition implies that the G-cover f : Y → X has stable reduction over the unique tame extension K^{st}/K_{0} of degree N.

4. Construction of three point covers by lifting

Let f̄ : Ȳ → X̄ be a special G-map, see Definition 2.15. A lift of f̄ is a three point G-cover with stable reduction f̄. Let L(f̄) denote the set of isomorphism classes of all lifts of f̄. The Galois group Gal(K̄/K_{0}) acts on L(f̄) in a natural way. In this section we determine the structure of L(f̄) as a Gal(K̄/K_{0})-set. In particular, we prove that L(f̄) is nonempty and that the action of Gal(K̄/K_{0}) is tamely ramified.

At the end, we derive various consequences of the above result, concerning the field of moduli of three point covers, and we discuss a aprticular class of examples, called genus zero dessins of prime degree.

4.1. The set of lifts of f̄.

4.1.1. Let k be an algebraic closure of the finite field 𝔽_{p}, W (k) the ring of Witt vectors over k and K_{0} the fraction field of W (k). We fix an algebraic closure K̄ of K_{0}. In this section, K/K_{0} always denotes a finite field extension contained in K̄ and R denotes the ring of integers of K. Let (Z̄_{0}, ω_{0}; Ȳ_{j}) be a special G-deformation datum and let f̄: Ȳ → X̄ be the associated special G-map, see §2.2. Definition 4.1. A lift of f̄ over R is a stable G-map f_{R} : Y_{R} → X_{R} over R (see Definition 2.14), together with an isomorphism f̄ ≅ f_{R} ⊗_{R} k of stable G-maps and an isomorphism X_{R} ⊗_{R} K →^{∼} (ℙ_{K}^{1}, {0, 1, ∞}) of marked curves. Given a lift f_{R} : Y_{R} → X_{R}, we will usually identify f̄ with the special fiber of f_{R} and ℙ_{K}^{1} with the general fiber of X_{R}. Two lifts f_{R} : Y_{R} → X_{R} and f′_{R} : Y′_{R} → X′_{R} over R are said to be isomorphic if there exists an isomorphism f_{R} →^{∼} f′_{R} of stable G-maps which induces the identity on the special fiber and on X_{K} = X′_{K} = ℙ_{K}^{1}.

If f_{R} : Y_{R} → X_{R} is a lift of f̄, then the general fiber f_{K} : Y_{K} → X_{K} = ℙ_{K}^{1} of f_{R} is a three point G-cover and f_{R} is its stable model. In particular, f_{K} has mild reduction and f̄ is the stable reduction of f_{K}. If f_{R} and f′_{R} are isomorphic lifts of f̄ then the isomorphism f_{R} →^{∼} f′_{R} is unique. (This is a general property of deformations of stably marked curves, see [Knu83].)

4.1.2. Let L(f̄, K) denote the set of isomorphism classes of lifts of f̄ over R, the ring of integers of K. If K′/K is a finite extension then we have a natural inclusion L(f̄, K) ↪ L(f̄, K′). Let

L(f̄) := ∪_{K}L(f̄, K)

be the union, taken over all finite extensions K/K_{0} contained in K̄. It is clear that L(f̄) is a finite set. A priori, L(f̄) may be empty; we will see later that it is not.

There is a natural continuous action of Gal(K̄/K_{0}) on L(f̄) from the left, defined as follows. Let f_{R} : Y_{R} → X_{R} be a lift of f̄ over R, the ring of integers of K, and let σ ∈ Gal(K̄/K_{0}). We may assume that K/K_{0} is Galois. Let ^{σ}f_{R} : ^{σ}Y_{R} → ^{σ}X_{R} be the pullback of f_{R} by the automorphism σ̃ : Spec R →^{∼} Spec R induced by σ. Since σ induces the identity on the residue field k, we have a canonical isomorphism f_{R} ⊗_{R} k →^{∼} ^{σ}f_{R} ⊗_{R}k. We may therefore consider ^{σ}f_{R} as a deformation of f̄. This defines the action of Gal(K̄/K_{0}) on L(f̄).

Recall that Aut_{G}^{0}(f̄) was defined in §2.2.4 as the group of automorphisms of f̄ which induce the identity on the original component. There is a natural right action of Aut_{G}^{0}(f̄) on L(f̄), defined as follows. If γ ∈ Aut_{G}^{0}(f̄) and f_{R} : Y_{R} → X_{R} is a lift of f̄, then we define f_{R}^{γ} as the stable G-map f_{R} : Y_{R} → X_{R} together with the isomorphism Ȳ →^{∼} Y_{R} ⊗_{R} k which is the composition with γ : Ȳ →^{∼} Ȳ with the canonical isomorphism Ȳ →^{∼} Y_{R} ⊗_{R} k. This defines an action of Aut_{G}^{0}(f̄) on L(f̄) from the right which commutes with the action of Gal(K̄/K_{0}). We define

L̃(f̄) := L(f̄)/ Aut_{G}^{0}(f̄).

It is clear that elements of L̃(f̄) correspond naturally to isomorphism classes of three point G-covers f : Y → X whose stable reduction is isomorphic to f̄. By definition we have:

Proposition 4.2. Let f_{R} : Y_{R} → X_{R} be a lift of f̄ and denote by f : Y → X the geometric generic fiber of f_{R}.

(1) The stabilizer in Gal(K̄/K_{0}) of f_{R}, considered as an element of L(f̄), is equal to Γf^{st}, as defined in §1.1.3.

(2) The stabilizer in Gal(K̄/K_{0}) of f_{R}, considered as an element of L̃(f̄), is equal to Γf^{in}, as defined in §1.1.3.

4.1.3. The auxiliary cover. Let f_{R} : Y_{R} → X_{R} be a lift of f̄. For j ∈ B_{prim}∪B_{new}, let D_{j} ⊂ X_{K}^{rig} denote the closed rigid disk corresponding to the tail X̄_{j}. By definition, a K-rational point x ∈ D_{j} specializes to a point x̄ ∈ X̄_{j}. In particular, for j ∈ B_{prim} the branch point x_{j} ∈ {0, 1, ∞} of f lies on D_{j}. Note that the model X_{R} of X_{K} = ℙ_{K}^{1} is uniquely determined by the family of disks (D_{j})_{j}. A priori, the disks D_{j} (and therefore the model X_{R}) may depend on the lift f_{R}. We are now going to show that they are in fact independent of f_{R}. This is an important steps towards our main result.

Let us choose, for all j ∈ B_{new}, a K-rational point x_{j} ∈ D_{j}. We call these points the auxiliary branch points. By [Ray99], §3.2, the choice of the auxiliary branch points yields a certain G_{0}-cover f^{aux} : Y^{aux} → X, called the auxiliary cover of f. By definition, f^{aux} has branch locus {x_{j} | j ∈ B} and the same type of bad reduction as f. To be more precise, let X̂_{R} denote the formal completion of X_{R} along X̄ and let D_{j} ⊂ X̂_{R}, for j ∈ B_{prim} ∪ B_{new}, denote the open subset corresponding to X̄_{j} − {τ_{j}} ⊂ X̄, i.e. D_{j} is the closed formal disk whose generic fiber is D_{j}. Let U^{∘} ⊂ X̂_{R} denote the open subset corresponding to X̄ − ∪_{j}X̄_{j} ⊂ X̄. Finally, let X_{j} denote the completion of X_{R} at τ_{j} ∈ X̄ and set U := U^{∘} ∪ ∪_{j}X_{j}. Recall that part of our special G-deformation datum (Z̄_{0}, ω_{0}; Ȳ_{j}) is a subgroup G_{0} = I_{0} · H_{0}. The auxiliary cover of the lift f_{R} : Y_{R} → X_{R} is a G_{0}-cover f^{aux} : Y^{aux} → X, characterized by the following properties, see [Ray99], §3.2.

(a) The cover f^{aux} has a stable model fR^{aux} : YR^{aux} → XR^{aux} over R. Moreover, we have a (unique) isomorphism X_{R} ≅ XR^{aux} extending the identity on the generic fiber.

(b) For j ∉ B_{wild}, the restriction of fR^{aux} to the formal disk D_{j} is a tame cover of formal schemes, ramified along the closure of x_{j} in D_{j}.

(c) There exists a G-equivariant isomorphism

(22)  Ŷ_{R} ×X̂_{R} U ≅ Ind_{G_{0}}^{G}(ŶR^{aux} ×X̂_{R} U)

of formal U-schemes. It follows that the branch locus of f^{aux} : Y^{aux} → X is precisely the set of points {x_{j} | j ∈ B}. Proposition 4.3. (1) The cover f^{aux} is the (unique) lift of (Z̄_{0}, ω_{0}) with branch

points (x_{j}), see §3.3.1. In particular, as a mere cover, f^{aux} does not depend on the lift f_{R} of f̄.

(2) The disks D_{j} are independent of the lift f_{R} : Y_{R} → X_{R}. Moreover, each disk D_{j} contains a K_{0}-rational point.

Proof. It follows immediately from Property (c) that the deformation datum over the original component associated to the stable reduction of f^{aux} can be identified with (Z̄_{0}, ω_{0}). Hence (1) follows from Proposition 3.8. By Property (a), the cover f^{aux} satisfies Condition (1) of Theorem 3.9. Therefore, (2) follows from Theorem 3.9. □

We remark that the proof of Proposition 4.3 is the only place where we use the subtle implication ‘(1)⇒(2)’ of Theorem 3.9.

4.2. Patching data. Lifting the special G-map f̄ to a three point G-cover f : Y → X can be done in three steps. The first step consists in lifting the tail covers f̄_{j} : Ȳ_{j} → X̄_{j} to Galois covers E_{j} → D_{j} of the formal disks D_{j} corresponding to D_{j} ⊂ X. Since the restriction of f̄_{j} to X̄_{j} − {τ_{j}} is tame, such a lift exists and is unique. The second step consists in lifting the (inseparable) cover Ȳ_{0} → X̄_{0} to a Galois cover V → U, where U is the ‘formal neighborhood’ of X̂_{R} − ∪_{j}D_{j} defined in §4.1.3. The existence and uniqueness of such a lift follows from the results of §3.1. In the third step we patch the covers E_{j} → D_{j} and V → U together along the boundaries B_{j} of the disks D_{j} and obtain the desired G-cover f : Y → X. Unlike the first two steps, the third step is not unique; it depends on the choice of what we call a patching datum for f̄. The construction we have just sketched amounts to the existence of a bijection

L(f̄) ⟶^{∼} P (f̄)

between the set of lifts of f̄ and the set P (f̄) of all patching data. Essentially by construction, this bijection is Gal(K̄/K_{0})-invariant. Moreover, the results of §3.3.3 can be used to determine the complete structure of P (f̄) as Gal(K̄/K_{0})-set. Intuitively, the Galois group Gal(K̄/K_{0}) acts on P (f̄) by ‘Dehn-twists’ along the boundaries B_{j}.

4.2.1. For each index j ∈ B, let us now choose, once and for all, a K_{0}-rational point x_{j} ∈ X such that the closure of x_{j} in X_{0} = ℙ_{W (k)}^{1} intersects the special fiber X̄_{0} = ℙ_{k}^{1} in the critical point τ_{j}. As usual, we assume that x_{j} ∈ {0, 1, ∞} for j ∈ B_{0}. Let f^{†} : Y^{†} → X = ℙ1 ; K̄ be the lift of the special deformation datum (Z̄_{0}, ω_{0}) with branch points (x_{j})_{j∈B}, see §3.3. By definition, f^{†} is a G^{†}-cover, where G^{†} := μ_{p}(K̄) ⋊ H_{0}. If f_{R} is a lift of f̄, then Proposition 4.3 shows that f^{†} is isomorphic, as a mere cover, to the auxiliary cover f^{aux} associated to f_{R}. In the following, we will not identify f^{†} with f^{aux}, because we want to emphasize the fact that such an identification is not canonical.

Let K/K_{0} be a finite extension over which f^{†} has stable reduction and let fR^{†} : YR^{†} → XR^{†} (resp. f̄^{†} : Ȳ^{†} → X̄^{†}) denote the stable model (resp. the stable reduction) of f^{†}. By Theorem 3.9, the curve X̄^{†} consists of the original component X̄_{0} = ℙ_{k}^{1} and the tails X̄j^{†}, where j runs over the set B_{prim} ∪B_{new}. Moreover, the deformation datum over the original component associated to f̄^{†} is canonically isomorphic to (Z̄_{0}, ω_{0}). We may and will identify the (unique) component Ȳ0^{†} of Ȳ^{†} over the original component with the distinguished component Ȳ_{0} of Ȳ (by definition, Ȳ_{0} = Z̄_{0}^{1/p}, see §2.2.3). Recall that we have distinguished points η_{j} ∈ Ȳ_{0} above τ_{j} ∈ X̄_{0}; they are part of the G-deformation datum (Z̄_{0}, ω_{0}; Ȳ_{j}). For j ∉ B_{wild} we let Ȳj^{†} denote the irreducible component of Ȳ^{†} which intersects Ȳ_{0} in η_{j}. Write ηj^{†} ∈ Ȳj^{†} for the point where Ȳj^{†} intersects Ȳ_{0}. Let Gj^{†} ⊂ G^{†} be the subgroup generated by μ_{p}(K̄) and H_{j}. One checks that (f̄j^{†} : Ȳj^{†} → X̄j^{†}, ηj^{†}, H_{j}) is a pointed Gj^{†}-tail cover with ramification type (m_{j}, h_{j}), totally ramified above τ_{j}. See §2.2.2.

Let Aut_{k}^{0}(Ȳ^{†}) denote the group of k-linear automorphisms of Ȳ^{†} which normalize the action of G^{†}, commute with the action of H_{0} ⊂ G^{†} and restrict to the identity on Ȳ_{0}. We have seen in §3.3.3 that the absolute monodromy action on f̄^{†} is a homomorphism

κ̄^{†} : Gal(K̄/K_{0}) ⟶ Aut_{k}^{0}(Ȳ^{†}).

Similarly, let Aut_{k}^{0}(Ȳj^{†}) denote the group of k-linear automorphisms of Ȳj^{†} which normalize the action of Gj^{†}, commute with the action of H_{j} ⊂ Gj^{†} and fix the point η_{j}. By Proposition 3.11, the group Aut_{k}^{0}(Ȳj^{†}) is cyclic of order (p − 1)h_{j}; furthermore, restricting κ̄^{†} to Ȳj^{†} yields a surjective homomorphism

κ̄j^{†} : Gal(K̄/K_{0}) ⟶ Aut_{k}^{0}(Ȳj^{†}) ≅ ℤ/(p − 1)h_{j}.

4.2.2. Fix j ∉ B_{wild}. We denote by X̄_{j, ¯∞} (resp. X̄j, ¯∞^{†}) the generic point of the completion of the tail X̄_{j} (resp. the tail X̄j^{†}) at the point τ_{j}. We set Ȳ_{j, ¯∞} :=

Ȳ_{j} ×X̄_{j} X̄_{j, ¯∞} and Ȳj, ¯∞^{†} := Ȳj^{†} ×X̄_{j}^{†} X̄j, ¯∞^{†}. Let P_{j}(f̄) be the set of pairs (λ, φ ¯∞)

consisting of the following objects:

• λ : G^{†} →^{∼} G_{0} is an isomorphism which induces the identity on H_{0}, and • φ ¯∞ : Ind_{G_{j}}^{G}(Ȳ_{j, ¯∞}) →^{∼} Ind_{G_{j}^{†},λ}^{G}(Ȳj, ¯∞^{†}) is a G-equivariant isomorphism such

that the following holds.

(1) We have φ ¯∞(η_{j}) = ηj^{†}.

(2) The induced isomorphism ψ ¯∞ : X̄_{j, ¯∞} →^{∼} X̄j, ¯∞^{†} extends to an isomor-

phism ψ : X̄_{j} →^{∼} X̄j^{†}.

(3) If j ∈ B_{prim} then ψ(x̄_{j}) = x̄j^{†}.

(The notation ‘Ind_{G_{j}^{†},λ}^{G}’ means that we embed Gj^{†} into G via λ : G^{†} →^{∼} G_{0} ⊂

G.)

The group Aut_{k}^{0}(Ȳj^{†}) acts on the set P_{j}(f̄) from the left, as follows. Let (λ, φ ¯∞) be an element of P_{j}(f̄) and γ an element of Aut_{k}^{0}(Ȳj^{†}). Let inn(γ^{−1}) : G^{†} →^{∼} G^{†} denote the unique automorphism of G^{†} which sends an element α ∈ I^{†} to γ^{−1} ∘ α ∘ γ and induces the identity on H_{0} ⊂ G^{†}. Set λ′ := λ∘inn(γ^{−1}). One checks that γ extends uniquely to a G-equivariant isomorphism

γ̃ : Ind_{G_{j}^{†},λ}^{G}(Ȳj^{†}) ⟶^{∼} Ind_{G_{j}^{†},λ′}^{G}(Ȳj^{†}).

Let φ′ ; ¯∞ be the composition of φ ¯∞ with the local isomorphism induced by γ̃. Then (λ′, φ′ ; ¯∞) is an element of P_{j}(f̄). This defines the action of Aut_{j}^{0}(Ȳj^{†}) on P_{j}(f̄). It is easy to see that this action makes P_{j}(f̄) a principal homogeneous Aut_{k}^{0}(Ȳj^{†})-space. Via the monodromy action κ̄j^{†}, we obtain a transitive Galois action of Gal(K̄/K_{0}) on P_{j}(f̄).

Define

P (f̄) := { (λ; φ_{j, ¯∞}) | (λ, φ_{j, ¯∞}) ∈ P_{j}(f̄) ∀j }.

An element of P (f̄) is called a patching datum. The actions of the groups Aut_{k}^{0}(Ȳj^{†}) on P_{j}(f̄) extend uniquely to an action of Aut_{k}^{0}(Ȳ^{†}) on P (f̄) (use Proposition 3.11 (1)). Again, this action is transitive and fixed point free. We consider P (f̄) as a Gal(K̄/K_{0})-set, via the monodromy action κ̄^{†}. Proposition 4.4. The set P (f̄) of patching data has exactly (p−1)·∏_{j} h_{j} elements. The Gal(K̄/K_{0})-orbits have length

N := (p − 1) · lcm_{j}(h_{j}).

In particular, the action of Gal(K̄/K_{0}) on P (f̄) is tamely ramified.

Proof. By Proposition 3.11 and the freeness of the action of Aut_{k}^{0}(Ȳ^{†}) on P (f̄), it suffices to show that P (f̄) is nonempty. But this follows from Lemma 2.12. □

4.2.3. We shall now define a Gal(K̄/K_{0})-equivariant map

(23)  L(f̄) ⟶ P (f̄),

as follows. Let f_{R} : Y_{R} → X_{R} be a lift of f̄. Let f^{aux} : Y^{aux} → X be the auxiliary cover associated to f and the set (x_{j})_{j∈B}. Let fR^{aux} : YR^{aux} → X_{R} denote the stable model of the auxiliary cover f^{aux}. By Proposition 4.3 (1), there exists a K-linear isomorphism φ_{K} : YK^{aux} →^{∼} YK^{†} which induces the identity on X_{K}. By the uniqueness of the stable model, φ_{K} extends to an isomorphism φ_{R} : YR^{aux} →^{∼} YR^{†}. Let φ̄ : Ȳ^{aux} →^{∼} Ȳ^{†} denote the restriction of φ_{R} to the special fiber. After composing φ_{K} with an element of G^{†}, we may assume that φ_{K} commutes with the action of H_{0} (which is a subgroup of G_{0} and G^{†}) and that φ̄ induces the identity on Ȳ_{0} (which is a component of Ȳ^{aux} and Ȳ^{†}). Note that this determines φ_{K} uniquely. Let λ : G_{0} →^{∼} G^{†} be the isomorphism α ↦ φ_{K} ∘ α ∘ φ_{K}^{−1}. Fix j ∉ B_{wild}. Let Ȳj^{aux} denote the component of Ȳ^{aux} which intersects Ȳ_{0} in η_{j}, and set Ȳj, ¯∞^{aux} := Ȳj^{aux} ×X̄_{j} X̄_{j, ¯∞}. The isomorphism (22) induces a G-equivariant isomorphism

(24)  Ind_{G_{j}}^{G}(Ȳ_{j, ¯∞}) ⟶^{∼} Ind_{G_{0}}^{G}(Ȳj, ¯∞^{aux}).

Let φ_{j, ¯∞} : Ind_{G_{j}}^{G}(Ȳ_{j, ¯∞}) →^{∼} Ind_{G^{†}}^{G}(Ȳj, ¯∞^{†}) be the composition of this isomorphism with the isomorphism induced by φ̄. By construction, the tuple (λ; φ_{j, ¯∞}) is a patching datum. This defines the map (23).

We claim that (23) is Gal(K̄/K_{0})-equivariant. Indeed, for any σ ∈ Gal(K̄/K_{0}), we may identify the auxiliary cover of the twisted cover ^{σ}f_{R} := f_{R} ⊗_{R}^{σ} R with the σ-twist ^{σ}(fR^{aux}) := fR^{aux}⊗_{R}^{σ}R of the auxiliary cover of f_{R}. Let κR,σ^{†} : ^{σ}(YR^{†}) →^{∼} YR^{†} be the unique extension of the canonical K-linear isomorphism ^{σ}YK^{†} →^{∼} YK^{†} induced from the K_{0}-model YK_{0}^{†}. The restriction of κ_{R,σ} to the special fiber is equal to κ̄^{†}(σ), i.e. the monodromy action of σ on Ȳ^{†}. The composition φ′_{R} := κR,σ^{†} ∘ φ_{R} is the unique isomorphism which induces an H_{0}-equivariant isomorphism of mere covers fK^{aux} ≅ fK^{†} and whose restriction to the special fiber is the identity on Ȳ_{0}. It follows that the map (23) maps the class of the twisted lift ^{σ}f_{R} to the image of (λ; φ_{j, ¯∞}) under the action of κ̄^{†}(σ) ∈ Aut_{k}^{0}(Ȳ^{†}). This proves the claim that (23) is Gal(K̄/K_{0})-equivariant. Theorem 4.5. The map (23) is an isomorphism of Gal(K̄/K_{0})-sets.

Proof. We will show that (23) is an isomorphism by constructing an inverse P (f̄) → L(f̄). Let (λ; φ̄_{j, ¯∞}) ∈ P (f̄) be a patching datum. We have to construct a lift f_{R} : Y_{R} → X_{R} of f̄ which is mapped to (λ; φ̄_{j, ¯∞}) by (23).

By definition, the local isomorphism ψ̄_{j, ¯∞} : X̄_{j, ¯∞} →^{∼} X̄j, ¯∞^{†} extends to a global isomorphism ψ̄_{j} : X̄_{j} →^{∼} X̄j^{†}. Moreover, if j ∈ B_{prim} then ψ_{j}(x̄_{j}) = x̄j^{†}. Let ψ̄ : X̄ →^{∼} X̄^{†} denote the isomorphism which is equal to the identity on the original component and which is equal to ψ̄_{j} on the tails. We define X_{R} := XR^{†} as a marked curve over R. However, we identify the special fiber of X_{R} with X̄ via the isomorphism ψ̄.

Let X̂_{R} denote the formal completion of X_{R} along X̄. For j ∉ B_{wild}, let D_{j} ⊂ X̂_{R} be the open formal subscheme corresponding to X̄_{j} − {τ_{j}} ⊂ X̄ and let X_{j} be the completion of X_{R} at τ_{j}. Let U^{∘} ⊂ X̂_{R} be the open formal subscheme corresponding to X̄_{0} − {τ_{j} | j ∈ B_{prim} ∪ B_{new}} ⊂ X̄. Let U denote the disjoint union of U^{∘} with all the X_{j}. The fiber product

B_{j} := U ×X̂_{R} D_{j} ≅ Spec R[[T]]{T^{−1}}

is the ‘boundary’ of the formal disk D_{j}. Its special fiber B_{j} ⊗_{R} k is canonically isomorphic to X̄j, ¯∞^{†} and isomorphic to X̄_{j, ¯∞} via ψ̄_{j, ¯∞}. Define

V := Ind_{G^{†},λ}^{G}(ŶR^{†} ×X̂_{R} U).

Furthermore, let E_{j} → D_{j} be the (unique) tame lift of the (not necessarily connected) G-cover

Ȳ ×X̄ (X̄_{j} − {τ_{j}}) ⟶ X̄_{j} − {τ_{j}} ⟶^{ψ̄_{j}} X̄j^{†} − {τ_{j}}

which is unramified away from the section x_{j,R} : Spec R → D_{j}. The local isomorphism φ_{j, ¯∞} yields a G-equivariant isomorphism

(E_{j} ×_{D_{j}} B_{j}) ⊗_{R} k ≅ Ind_{G_{j}}^{G}(Ȳ_{j, ¯∞}) ⟶^{φ̄_{j, ¯∞}} Ind_{G^{†},λ}^{G}(Ȳj, ¯∞^{†}) ≅ (V ×_{U} B_{j}) ⊗_{R} k

of étale X̄j, ¯∞^{†}-schemes. It lifts canonically to a G-equivariant isomorphism

φ_{j} : E_{j} ×_{D_{j}} B_{j} ⟶^{∼} V ×_{U} B_{j}

of étale formal B_{j}-schemes. By a theorem of Ferrand–Raynaud [FR70] (see also [Pri00]), there exists a formal X̂_{R}-scheme Y with G-action such that E_{j} = Y ×X̂_{R} D_{j} and V = Y ×X̂_{R} U. By Grothendieck’s Existence Theorem, Y is the formal completion of a projective R-curve Y_{R}. By construction, the natural map f_{R} : Y_{R} → X_{R} is a lift of f̄ which is mapped by (23) to the patching datum (λ; φ̄_{j, ¯∞}). This completes the proof of the theorem. □

4.3. Some applications of Theorem 4.5. Let k be an algebraically closed field of characteristic p > 0. We denote by K_{0} the fraction field of W (k) and by K̄ an algebraic closure of K_{0}. Theorem 4.5 together with Proposition 4.4 implies: Corollary 4.6. Let f̄ : Ȳ → X̄ be a special G-map defined over k. Then there exists a three point G-cover f : Y → X := ℙ1 ; K̄ whose stable reduction is isomorphic to f̄.

This corollary is used in [BW02] to determine the stable reduction of all three point covers with Galois group PSL_{2}(p). In particular, this gives a new proof for the result of Deligne–Rapoport on the reduction of the modular curves X_{0}(p) and X_{1}(p).

In addition to the existence of three point covers with given stable reduction, Theorem 4.5 and Proposition 4.4 yield interesting arithmetic information about a given three point cover. Corollary 4.7. Let f : Y → X = ℙ1 ; K̄ be a three point Galois cover with mild reduction. Let (m_{j}, h_{j})_{j∈B} be the signature of the stable reduction of f. The cover f has stable reduction over the unique tame extension K^{st}/K_{0} of degree

N := (p − 1) · lcm_{j∉B_{wild}}(h_{j}).

In particular, any three point Galois cover f : Y → X whose degree is not divisible by p^{2} can be defined over a tame extension of K_{0}.

Proof. Let f_{R} : Y_{R} → X_{R} be a stable model of f and f̄ : Ȳ → X̄ the stable reduction. If f̄ is not exceptional, then the corollary follows from Proposition 3.11 and Theorem 4.5. We may therefore assume that f̄ is exceptional. We will also assume that we are in Case (2) of Definition 2.1, leaving Case (1) for the reader. By Proposition 2.2, the curve X̄ consists of the original component and a single new tail X̄_{j_{0}}.

Let fR^{aux} : YR^{aux} → X_{R} denote the auxiliary cover associated to f_{R}, see §4.1.3. The Galois group of fR^{aux} is a subgroup G_{0} ⊂ G, which is a direct product G_{0} = I_{0} × H_{0}. Here I_{0} ≅ ℤ/p is the inertia group of a component Ȳ_{0} of Ȳ over X̄_{0} and H_{0} is the Galois group of the cover Z̄_{0} = Ȳ_{0}^{(p)} → X̄_{0}. Let W_{R} := YR^{aux}/H_{0}. Then W_{R} → X_{R} is a stable model of the p-cyclic Galois cover W_{K} → X_{K} = ℙ_{K}^{1}, which is ramified at 0, 1 and ∞. Therefore, the curve W_{K} has an equation of the form

y^{p} = f(x) := c x^{a}(x − 1)^{b},

with integers a, b such that a, b, a+b ≢ 0 mod p and a unit c ∈ R^{×}. After enlarging the field K if necessary and a change of coordinates we may assume that c ∈ W (k)^{×} and f(a/(a + b)) ∈ (W (k)^{×})^{p}. It is known from [CM88] (see also [Leh01]) that the model X_{R} which leads to the stable reduction of W_{K} → X_{K} is the blowup of ℙ_{R}^{1} in the disk

D := \{ x | val(x − \frac{a}{a + b}) ≥ \frac{p val p}{2(p − 1)} \}.

Moreover, the stable model W_{R} → X_{R} descents to a stable model over the ring

R′ := W (k)[ p^{1/2(p−1)}, f(\frac{a}{a + b})^{1/p}] = W (k)[ p^{1/2(p−1)} ].

Note that the (admissible) H_{0}-cover Ȳ^{aux} → W̄ := W_{R} ⊗_{R} k is ramified at the unique singular point of W̄ of order m = h/2. Therefore, the stable model fR^{aux} : YR^{aux} → X_{R} descents to a stable model over the ring R′′ := W (k)[p^{1/h(p−1)}]. Using the method of the proof of Theorem 4.5, it is not hard to show that f_{R} : Y_{R} → X_{R} descents to a stable model over R′′. This finishes the proof of the corollary. □

The last corollary can be refined in order to obtain a formula for the ramification index of the field of moduli of a three point cover in terms of its stable reduction. Corollary 4.8. Let f̄ : Ȳ → X̄ be a special G-map, with signature (m_{j}, h_{j})_{j∈B}. Let P ⊂ G denote a p-Sylow of G and let n := [N_{G}(P) : C_{G}(P)] be the index of the centralizer of P inside the normalizer of P. Recall that L̃(f̄) denotes the set of isomorphism classes of three point G-covers defined over K̄ whose stable reduction is isomorphic to f̄. There exists a positive integer n′ dividing n which depends only on f̄ such that

|L̃(f̄)| = \frac{p − 1}{n′} ∏_{j∉B_{wild}} \frac{h_{j}}{| Aut_{G}(f̄_{j})|}.

Let f : Y → X be a lift of f̄. Then the field of moduli K^{in} of f (relative to the extension K̄/K_{0}) is the unique tame extension of K_{0} of degree

N′ := \frac{p − 1}{n′} lcm_{j∉B_{wild}} \frac{h_{j}}{| Aut_{G}(f̄_{j})|}.

Proof. We have seen in §4.1.2 that L̃(f̄) = L(f̄)/ Aut_{G}^{0}(f̄). Moreover, the stabilizer in Aut_{G}^{0}(f̄) of any element of L(f̄) is equal to C_{G}. Let n′ be the order of the image of the homomorphism Aut_{G}^{0}(f̄) → G/C_{G} which is defined by Lemma 2.17. It is easy to see that n′ divides n. We see that Corollary 4.8 follows from Proposition 3.11, Theorem 4.5 and Lemma 2.17. □

4.4. Genus zero dessins of degree p. Let p be a prime and G ⊂ S_{p} be a primitive transitive permutation group of degree p. We write N ⊂ G for the stabilizer of 1. Let f : Y → X := ℙ1 ; ℚ̄ be a three point Galois cover, with Galois group G. Then g : Z := Y/N → X is a three point cover of degree p, and f : Y → X is the Galois closure of g. We make the following assumptions. Assumption 4.9. (1) The subgroup N ⊂ G is self-normalizing and G is

center-free.

(2) The curve Z has genus 0.

The first assumption is made only for simplicity. It ensures that the G-cover f : Y → X and the (non-Galois) cover g : Z → X have the same field of moduli, and that this field of moduli is also the minimal field of definition for both covers, see e.g [DD97]. The second assumption is more restrictive and implies that the stable reduction of f : Y → X to characteristic p is rather easy to describe.

Let K be the field of moduli of f : Y → X, relative to the extension ℚ̄/ℚ. Let 𝔭 be a prime ideal of K dividing p. There exists a finite extension K′/K and a prime 𝔭′ of K′ over 𝔭 such that f has a K′-model f_{K′} : Y_{K′} → X_{K′} which extends to a stable model over the local ring O_{K′,𝔭′}. Let f̄: Ȳ → X̄ be the stable reduction of f′_{K} at 𝔭′. One checks that f̄: Ȳ → X̄ depends only on the field K and the prime 𝔭. Hence we may call f̄: Ȳ → X̄ the stable reduction of f : Y → X at 𝔭.

Let us assume that f : Y → X has bad reduction at 𝔭. Note that p strictly divides the order of G. Therefore, by Theorem 2.6 the stable reduction f̄: Ȳ → X̄ is a special G-map, associated to a special G-deformation datum (Z̄_{0}, ω_{0}; Ȳ_{j}), see §2.2.3. In the following, we use freely the notation introduced in §2.2.3.

Lemma 4.10. (1) The character χ : H_{0} → 𝔽_{p}^{×} is injective. In particular,

H_{0} ≅ ℤ/m with m|(p − 1).

(2) There are no new tails.

(3) For j ∈ B_{prim}, let x_{j} ∈ {0, 1, ∞} denote the branch point of f : Y → X which specializes to the primitive tail X̄_{j}. The ramification invariant of the primitive tail cover f̄_{j} : Ȳ_{j} → X̄_{j} is given by the formula

σ_{j} = \frac{|g^{−1}(x_{j})| − 1}{p − 1}.

Proof. Part (1) is an immediate consequence of the fact that the p-cyclic subgroup I_{0} of S_{p} is self-centralizing.

To prove (2), suppose that f_{j} : Ȳ_{j} → X̄_{j} is a new tail, with Galois group G_{j}. Let N_{j} := N ∩ G_{j} and Z̄_{j} := Ȳ_{j}/N_{j}. It follows from Assumption 4.9 (2) that the curve Z̄_{j} has genus 0. Since N has index p in G and G_{j} contains a p-Sylow of G, the index of N_{j} in G_{j} is also p. We see that ḡ_{j} : Z̄_{j} → X̄_{j} is a cover of degree p between curves of genus 0, totally branched at ¯∞_{j} ∈ X̄_{j} and étale elsewhere. This means that in suitable coordinates ḡ_{j} is given by a monic polynomial q ∈ k[z] of degree p such that the derivative q′ has degree 0. Hence q = z^{p} + az + b, and after a further normalization we get q = z^{p} − z. It follows that the tail cover f̄_{j} : Ȳ_{j} → X̄_{j} is an Artin–Schreier cover with conductor 1, i.e. σ_{j} = 1. This contradicts the results of §2. We conclude that f̄: Ȳ → X̄ has no new tails, proving (2).

For j ∈ B_{prim}, let ḡ_{j} : Z̄_{j} → X̄_{j} be as before. The same argument as before shows that ḡ_{j} is a cover between smooth curves of genus 0 of degree p, totally ramified at ¯∞_{j} ∈ X̄_{j}. But now ḡ_{j} is tamely ramified at x̄_{j} and étale over X̄_{j} − { ¯∞_{j}, x̄_{j}} (recall that x̄_{j} ∈ X̄_{j} denotes the specialization of the branch point x_{j} ∈ {0, 1, ∞}). Since the stable model of f is a tame cover in a neighborhood of X̄_{j} − { ¯∞_{j}} we have |ḡ_{j}^{−1}(x̄_{j})| = |g^{−1}(x_{j})|. Now (3) follows from a calculation with the Riemann– Hurwitz formula and the fact that g(Z̄_{j}) = 0. See e.g. [BP], Lemma 2.3.1. □

The lemma shows that the signature of f̄ : Ȳ → X̄ is a tripel (σ_{1}, σ_{2}, σ_{3}) of rational numbers, with σ_{j} = h_{j}/m_{j}, 0 ≤ h_{j} < m_{j} and (h_{j}, m_{j}) = 1. It satisfies the vanishing cycle formula

(25)  σ_{1} + σ_{2} + σ_{3} = 1.

The special deformation datum (Z̄_{0}, ω_{0}) is essentially determined by the signature. Indeed, the curve Z̄_{0} is isomorphic to the smooth projective model of the plane curve with equation

(26)  z^{m} = x^{h_{1}}(x − 1)^{h_{2}},

where m is the least common multiple of the m_{j}, and the differential ω_{0} has the form

(27)  ω_{0} = ϵ \frac{z dx}{x(x − 1)},

for some ϵ ∈ 𝔽_{p}^{×}. Conversely, given a tripel (σ_{1}, σ_{2}, σ_{3}) satisfying all of the above conditions, one checks that (26) and (27) define a special deformation datum (Z̄_{0}, ω_{0}). Therefore, the existence of a special deformation datum with a given signature imposes no restriction on the stable reduction f̄: Ȳ → X̄.

Example 4.11. Let p = 7 and G := S_{7}. Suppose furthermore that the branch cycle description of the cover g : Z → X is (6, 6, 2-2) (each entry designates a conjugacy class of S_{7}, e.g. ‘2-2’ denotes the product of two 2-cycles). From the table computed by Malle [Mal94] we see that there are exactly 4 non-isomophic three point covers with monodromy group S_{7} and this branch cycle description. Moreover, the absolute Galois group Gal(ℚ̄/ℚ) permutes these four covers transitively, with permutation group S_{4}. Therefore, the field of moduli of the G-cover f : Y → X is a degree 4 extension K/ℚ.

Let 𝔭 be a prime ideal of K over 7. Suppose that f : Y → X has bad reduction at 𝔭 and let f̄: Ȳ → X̄ denote the stable reduction. It follows from Lemma 4.10 (3) that the signature of f̄ is equal to (1/6, 1/6, 2/3). For j = 1, 2, the Riemann– Hurwitz formula implies that the Galois group G_{j} of the primitive tail cover f̄_{j} : Ȳ_{j} → X̄_{j} is isomorphic to the semi-direct product 7 : 6. It follows that Ȳ_{j} → X̄_{j} is given by the two equations

(28)  z^{6} = x,  y^{7} − y = z.

The third tail cover f̄_{3} : Ȳ_{3} → X̄_{3} has inertia invariant σ_{3} = 2/3 and branch cycle description 2-2 at the tame branch point. We say that f̄_{3} is a primitive tail cover of type (2/3, 2-2). The Galois group G_{3} of f̄_{3} is a subgroup of A_{7} containing a semidirect product 7 : 3 and a permutation of type 2-2. By elementary group theory, G_{3} is isomorphic to A_{7} or to L_{2}(7) := PSL_{2}(7).

It is a priori not clear how many non-isomorphic primitive tail covers of type (2/3, 2-2) exist, and what their Galois groups are. We claim that there exists in fact a unique primitive tail cover of type (2/3, 2-2), with Galois group L_{2}(7). The existence of one such cover is proved in [BW02]. (It is obtained by reducing a certain three point cover with Galois group L_{2}(7).) To prove uniqueness, we use our lifting result, Theorem 4.5.

Given f̄_{3} : Ȳ_{3} → X̄_{3} as above, one shows that there exists, up to isomorphism, a unique special G-map f̄: Ȳ → X̄ with signature (1/6, 1/6, 2/3) whose third tail cover is isomorphic to f̄_{3} : Ȳ_{3} → X̄_{3}. By Corollary 4.8 there exist exactly

N′ := \frac{12}{n′ · | Aut_{G_{3}}(f̄_{3})|}

nonisomorphic three point covers f′ : Y′ → X′ whose stable reduction (at a given prime 𝔭̃ of ℚ̄) is isomorphic to f̄. Moreover, all of these covers are conjugate under the action of the inertia group of 𝔭̃. Using Lemma 2.17, Remark 2.18 and Equation (28) one shows that n′ = 3. Thus, N′ is equal to either 2 or 4.

As we have mentioned before, all three point covers with Galois group S_{7} and branch cycle description (6, 6, 2-2) are conjugate. Hence we may assume that f = f′. We conclude that for each primitive tail cover f̄_{3} of type (2/3, 2-2) there exists a prime 𝔭 in K above 7 with e(𝔭/7) ∈ {2, 4}. This prime 𝔭 is uniquely determined by the property that the stable reduction of f : Y → X at 𝔭 has a primitive tail cover isomorphic to f̄_{3}. By Malle’s table, we have 7 = 𝔭_{1}^{2}𝔭_{2}𝔭_{3} in K. Therefore, our analysis of the stable reduction implies:

• The cover f : Y → X has bad reduction at 𝔭_{1} and good reduction at 𝔭_{2} and 𝔭_{3}. The stable reduction of f : Y → X at 𝔭_{1} occurs after an extension which is ramified at 𝔭_{1} of order 2.

• There exists a unique primitive tail cover of type (2/3, 2-2). Its Galois group is L_{2}(7).

We remark that good reduction at 𝔭_{2} and 𝔭_{3} does not follow from the results of [Ray99], because S_{7} has only one conjugacy class of elements of order 7.

Example 4.12. As before, let p = 7 and G = S_{7}. But now suppose that the branch cycle description of the cover f : Y → X is (2-3, 2-3, 7). By Malle’s table there are 9 such covers, up to isomorphism. Note that they all have bad reduction to characteristic 7, because of the ramification index 7.

It follows from Lemma 4.10 (3) that the signature of f̄ : Ȳ → X̄ is equal to (1/2, 1/2, 0). For j = 1, 2, the Galois group G_{j} of the tail cover f̄_{j} : Ȳ_{j} → X̄_{j} contains a 7-cycle and a 2-cycle. Hence, by a theorem of Jordan we have G_{j} = S_{7}.

We claim that there exists, up to isomorphism, a unique primitive tail cover f̄_{1} : Ȳ_{1} → X̄_{1} of type (1/2, 2-3) with Galois group S_{7}. We have already shown its existence. To prove uniqueness, we use again Theorem 4.5.

Let us choose a point η_{1} ∈ Ȳ_{1} above the wild branch point ¯∞_{1} ∈ X̄_{1}. Since σ_{1} = 1/2 the inertia group at η_{1} is a dihedral group G_{0} = I_{0} ⋊ H_{0}, with |I_{0}| = 7 and H_{0} = |2|. Recall that the triple (f̄_{1}, η_{1}, H_{0}) is called a pointed S_{7}-tail cover. Let f̄_{2} : Ȳ_{2} → X̄_{2} be a copy of the S_{7}-tail cover f̄_{1}. It is easy to see that there are exactly three points on Ȳ_{2} above ¯∞_{2} with inertia group G_{0}. Let η_{2} be one of them. Then (f̄_{2}, η_{2}, H_{0}) is another pointed S_{7}-tail cover. It is isomorphic to (f̄_{1}, η_{1}, H_{0}) (by an isomorphism commuting with the action of S_{7}) if and only if η_{1} = η_{2}. Finally, let (Z̄_{0}, ω_{0}) be the (unique) special deformation datum with signature (1/2, 1/2, 0). One checks that (Z̄_{0}, ω_{0}, Ȳ_{1}, Ȳ_{2}) gives rise to a special G-deformation datum and hence to a special G-map f̄: Ȳ → X̄, see §2.2.3. Moreover, by varying the point η_{2} we obtain exactly three non-isomorphic special G-maps with signature (1/2, 1/2, 0) and whose primitive tail covers are both isomorphic to f̄_{1} : Ȳ_{1} → X̄_{1}.

Choose one of the three special G-maps f̄ : Ȳ → X̄ constructed from f̄_{1} : Ȳ_{1} → X̄_{1}. By Corollary 4.8 there exist exactly N′ = 6/n′ non-isomorphic three point covers f′ : Y′ → X with Galois group S_{7} whose stable reduction (at a given place 𝔭̃ above 7) is isomorphic to f̄. Moreover, they form a single orbit under the action of the inertia group at 𝔭̃. Using Remark 2.18 we see that 2|n′. Using the fact that S_{7} has no outer automorphism one further shows that n′ = 2 and hence N′ = 3. Therefore, for each primitive tail cover f̄_{1} : Ȳ_{1} → X̄_{1} we obtain, by our lifting result, exactly 9 non-isomorphic three point covers with Galois group S_{7} and branch cycle description (2-3, 2-3, 7). However, there are only 9 such covers altogether. This proves that f̄_{1} : Ȳ_{1} → X̄_{1} is indeed unique, up to isomorphism. As a by-product, we have also shown that the absolute ramification index of any prime ideal 𝔭 dividing 7 in the field of moduli of f : Y → X is equal to 3.

By [Mal94] the set of 9 isomorphism classes form two Galois orbits, of length 3 and 6 respectively, corresponding to extensions K_{1}/ℚ and K_{2}/ℚ. In K_{1} we have 7 = 𝔭_{1}^{3} and in K_{2} we have 7 = 𝔭_{2}^{3}𝔭_{3}^{3}. This confirms our prediction.

Remark 4.13. One can use Malle’s table [Mal94] to verify our results in many more examples. In particular, one gets a numerical verification of Theorem 1 of the Introduction for all genus zero dessins of degree ≤ 10. One also sees that the condition that p strictly divides the order of G is necessary for Theorem 1 to hold. For instance, one finds several three point covers of degree n ≥ 10, with monodromy group S_{n} or A_{n}, such that 5 is wildly ramified in the field of moduli. Remark 4.14. The relation between the existence of three point covers with bad reduction on the one hand and tail covers with low ramification invariant on the other hand is exploited more systematically in [BP]. In [Zap02] Zapponi studies the stable reduction of genus zero dessins of degree p which are totally ramified at one point, by analyzing the defining equation. His results confirm our results on dessins of genus 0.

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Mathematisches Institut, Beringstr. 1, 53115 Bonn, Germany

   E-mail address: wewers@math.uni-bonn.de