Reduction of the Hurwitz space of metacyclic covers

Irene I. Bouw

Abstract

We compute the stable reduction of some Galois covers of the projective line branched at three points. These covers are constructed using Hurwitz spaces parameterizing metacyclic covers. The reduction is determined by a certain hypergeometric differential equation. This generalizes the result of Deligne–Rapoport on the reduction of the modular curve X(p).

Introduction

Recently, much progress has been made in understanding the reduction of Galois covers of the projective line from characteristic zero to characteristic p > 0. The starting point is a series of papers by Raynaud, [19], [20], [21] in which the theory of the stable reduction of a Galois cover is developed. In [21], one finds a criterion for good reduction for Galois covers of the projective line branched at three points, in case the order of the Galois group is strictly divisible by p. Raynaud’s Criterion implies, for example, that many covers which have been constructed with the method of rigidity have good reduction to characteristic p.

Raynaud’s results have been extended in [29] and [30]. Let f : Y → ℙ^{1} be a G-Galois cover branched at three points x_{1}, x_{2}, x_{3}, defined over a field K of characteristic zero. Suppose that p > 2 is a prime which strictly divides the order of G. It is proved that the stable reduction of f at a prime of K above p is of a very easy kind. This theorem implies that to describe the stable reduction f̄ of f, it suffices to compute the associated deformation datum (Definition 1.1) for which there are only finitely many possibilities. However, in general, it is not known how to compute f̄.

In the literature, the stable reduction of a Galois cover with bad reduction has been computed in very few cases. Most attention has been focused on cyclic covers of the projective line. Coleman and McCallum [8] computed the reduction of cyclic covers of the projective line branched at three points. In [10], [22], [15], [16] the reduction of p-cyclic covers of ℙ^{1} branched at r ≥ 3 points is studied. But even in this case the results are far from complete.

Other known results use the reduction of modular curves. For example, let X(N) be the modular curve parameterizing elliptic curves with full level-N structure. Then f : X(2p) → X(2) is a Galois cover with Galois group PSL_{2}(p) branched at three points of order p. Its stable reduction can be deduced from [9] and [14].

In this paper, we compute the stable reduction of Galois covers of the projective line branched at three points with Galois group PSL_{2}(p), SL_{2}(p) or PGL_{2}(p). The construction of these covers goes back to Völklein, [27]. Essentially, the covers we consider arise as Hurwitz spaces H(a) parameterizing covers of the projective line branched at four points. More precisely, choose m|(p−1) and let N be an extension of ℤ/m by ℤ/p. Then H(a) parameterizes N-Galois covers of ℙ^{1} branched at 0, 1, ∞, λ with ramification of order prime-to-p described by the type a = (a_{1}, a_{2}, a_{3}, a_{4}), Definition 2.1. Write π(a) : H(a) → ℙ_{λ}^{1} for the cover which sends g to the branch point λ. The cover we consider is the Galois closure ϖ(a) of π(a). It is expected that all covers of the projective line which are branched at three points and whose Galois group is PSL_{2}(p) or SL_{2}(p) arise via a construction like this, Section 8.

Theorem 4.2. The cover ϖ(a) has good reduction if and only if a_{1} + a_{2} + a_{3} + a_{4} ≠ 2m.

In Section 8 we illustrate that this theorem does not follow from Raynaud’s criterion for good reduction. For m = 2 the cover ϖ(a) is just the cover X(2p) → X(2).

In case the cover ϖ(a) has bad reduction, we compute its stable reduction in Section 6. We do not give a modular interpretation for the stable model, therefore our approach differs from the approach of [9] and [14]. However, also in our approach, the key ingredient for computing the reduction of ϖ(a) is the study of the reduction of the metacyclic covers which the Hurwitz space parameterizes. It turns out that the stable reduction is determined by a so-called a-Hasse invariant Φ_{a}. For m = 2 this is just the classical Hasse invariant. An important property of Φ_{a} is that it is the solution to a hypergeometric differential equation, Section 5. This hypergeometric differential equation determines all combinatorial invariants which are associated to the stable reduction, see Section 6 for a precise statement.

The organization of this paper is as follows. In Section 1 we define the stable reduction and collect some results on its structure. In Section 2 we introduce the metacyclic covers and study their reduction. In Section 3 we define the Hurwitz space H(a) in characteristic zero and describe the map π(a) : H(a) → ℙ_{λ}^{1} and its Galois closure. In Section 4 we prove the good reduction theorem. Sections 5 and 6 study the stable reduction of the covers with bad reduction. First we define the a-Hasse invariant, Section 5. Then we show how this gives the stable reduction, Section 6. The rest of the paper contains complements and examples.

1 The stable model

In this section we recall the definition of the stable model. We refer to [30] for proofs and more details. Suppose that G is a finite group whose order is strictly divisible by p. Let R be a complete discrete valuation ring with fraction field K of characteristic zero and residue field k = k̄ of characteristic p > 2. Let f : Y → X = ℙ_{K}^{1} be a G-Galois cover branched at x = (x_{1}, …, x_{r}), where the x_{i} are distinct K-rational points. Write 𝔹_{0} = {1, …, r} for the set indexing the branch points. We assume that r ≥ 3. We assume moreover that (X; x) has good reduction, i.e. there exists a model X_{0,R} = ℙ_{R}^{1} of X over R such that the x_{i} extend to pairwise disjoint sections Spec(R) → X_{0,R}.

Denote the ramification points of f by y = (y_{1}, … y_{s}). Let (Y_{R}; y) be the unique extension of (Y; y) to a stably pointed curve, which exists after replacing R by a finite extension, if necessary. Recall that (Y_{R}; y) is called stably pointed if Y_{R} is a semistable curve such that the usual “three point condition” is fulfilled for the union of the marked points and the singular points of Ȳ := Y_{R}⊗_{R} k. The action of G extends to Y_{R}; denote the quotient of Y_{R} by G by X_{R}. Then f_{R} : Y_{R} → X_{R} is called the stable model of f. The special fiber f̄: Ȳ → X̄ of f_{R} we call the stable reduction of f. We say that f has good reduction if and only if the stable reduction f̄: Ȳ → X̄ is a separable morphism. This is equivalent to X_{R} being smooth. If f does not have good reduction, we say it has bad reduction.

Suppose that f has bad reduction. Let f̄: Ȳ → X̄ be the stable reduction of f. We denote the dual graph of X̄ by T. The vertices 𝕍 of T correspond to the irreducible components of X̄ and the (oriented) edges 𝔼 to the singular points. The source and target of an edge e are denoted by s(e) and t(e).

Since (X; x) has good reduction, there exists a natural map X_{R} → X_{0,R}. This map restricts to the identity on a unique irreducible component of X̄ and contracts all other components of the special fiber to a point. Therefore we may (and will) consider X̄_{0} as an irreducible component of X̄, which we call the original component. The dual graph T of X̄ is a tree. We denote the vertex of T corresponding to the original component X̄_{0} by v_{0} and consider T to be oriented from v_{0}.

An edge e is called positive if its orientation coincides with the orientation of T. For every v ∈ 𝕍, we denote the corresponding component of X̄ by X_{v}. We choose a component Y_{v} of Ȳ above X_{v}. For every v ∈ 𝕍 we denote by ∞_{v} the unique point of X_{v} which is singular in X̄ and such that the corresponding edge e with s(e) = v is negative. If e is an edge with s(e) = v, we write x_{e} for the singular point of X_{v} corresponding to e.

An irreducible component X_{v} of X̄ is called a tail if it is different from the original component and meets the rest of X̄ in just one point. We denote by 𝔹 the subset of 𝕍 corresponding to the tails of X̄. A tail is called primitive if one of the branch points specializes to this component; otherwise the tail is called new. Let 𝔹_{prim} (resp. 𝔹_{new}) be the subset of 𝔹 corresponding to the primitive (resp. new) tails. We identify 𝔹_{prim} with a subset of 𝔹_{0} = {1, …, r}: we write X_{i} for the primitive tail to which the branch point x_{i} specializes. The complement 𝔹_{wild} := 𝔹_{0} − 𝔹_{prim} corresponds to the branch points whose ramification index is divisible by p. We use the notation x_{i} to denote both the point on X and its specialization to X̄.

For every b ∈ 𝔹, the map Y_{b} → X_{b} is separable. The inertia group of a point above ∞_{b} has order pm_{b}, for some m_{b} prime-to-p. Denote the corresponding conductor by h_{b}. We define the ramification invariant to be σ_{b} = h_{b}/m_{b}. (This is the jump in the filtration of the higher ramification groups in the upper numbering.) If b ∈ 𝔹_{new}, the cover Y_{b} → X_{b} is unbranched outside ∞_{b}. If i ∈ 𝔹_{prim}, then Y_{i} → X_{i} is tamely branched at x_{i} and unbranched outside x_{i} and ∞_{i}. If v ∈ 𝕍 − 𝔹, then the restriction of f̄ to X_{v} is inseparable.

Choose an irreducible component Ȳ_{0} of Ȳ above the original component X̄_{0}. Since f has bad reduction, the inertia group I_{0} if Ȳ_{0} has order p. The decomposition group D_{0} of Ȳ_{0} is the extension of a group H_{0} of order prime-to-p by I. The map Ȳ_{0} → X̄_{0} is the composition of a purely inseparable map Ȳ_{0} → Z̄_{0} and an H_{0}-Galois cover Z̄_{0} → X̄_{0}. Write Ȳ′_{0} (resp. Z̄′_{0}) for the quotient of Ȳ_{0} (resp. Z̄_{0}) by the prime-to-p part of the center of D_{0}. Then Z̄′_{0} → X̄_{0} is an m-cyclic cover, for some m dividing p − 1. Let χ : ℤ/m → 𝔽_{p}^{×} be an injective character. We choose generators ϕ and ψ of the decomposition group D′_{0} of Ȳ′_{0} such that ϕ^{p} = ψ^{m} = 1 and ψ^{i}ϕψ^{−i} = ϕ^{χ(i)}.

The restriction of Ȳ_{0} → Z̄_{0} to some open U of Z̄_{0} is an α_{p} or μ_{p}-torsor. We say that f has multiplicative (bad) reduction if Ȳ_{0} → Z̄_{0} restricts to a μ_{p}-torsor. If Ȳ_{0} → Z̄_{0} restricts to an α_{p}-torsor, we say that f has additive (bad) reduction.

To an α_{p} or μ_{p}-torsor, we can associate a differential ω, [17, Proposition 4.14]. Let W be the inverse image of U in Ȳ_{0}. There exists an open covering {U_{j}}_{j} of U and elements f_{j} ∈ Γ(U_{j}, O) such that the torsor restricted to U_{j} is given by an equation v^{p} = f_{j}.

Suppose first that W → U is a μ_{p}-torsor, then f_{j} ∈ Γ(U_{j}, O)^{∗}. To the torsor we associate the differential

ω = \frac{df_{j}}{f_{j}} ∈ H^{0}(U, Ω^{1}).

Here we assume that we fixed an isomorphism ℤ/p ≃ μ_{p}(K). Note that ω does not depend on j or the choice of f_{j}. In case W → U is an α_{p}-torsor, we associate to it the differential

ω = df_{j} ∈ H^{0}(U, Ω^{1}).

Moreover, ψ^{∗}ω = χ(1)ω. See [30] for more details.

Definition 1.1 We call the tuple (Z̄′_{0}, ω) the deformation datum corresponding to f.

Lemma 1.2 (a) The differential ω is regular outside x_{b} for b ∈ 𝔹_{wild}. For b ∈ 𝔹_{wild}, we have ord_{z_{b}}(ω) = −1. Here z_{b} ∈ Z̄_{0} is a point above x_{b}.

(b) The differential ω has a zero at z ∈ Z̄_{0} if and only if the image of z in X̄_{0} is a singular point of X̄.

Proof: First suppose that w ∈ Z̄_{0} is a point whose image in X̄_{0} is x_{b} for b ∈ 𝔹_{wild}. Let t be a local parameter at w. Then Ȳ_{0} → Z̄_{0} is given by y^{p} = t^{n} mod t^{n+1}, locally around w, for some n > 0 which is prime-to-p. It follows that ω is a logarithmic differential which has a simple pole at w.

Let w ∈ Z̄_{0} be a point which does not map to x_{b} for b ∈ 𝔹_{wild}. Let t be a local parameter at w. Then the natural map Ȳ_{0} → Z̄_{0} is given by y^{p} = 1 + t^{n} mod t^{n+1}, locally around w, [19, p. 192]. It follows from the definition of ω that ω does not have a pole at w and that ω has a zero at w if and only if w is singular in Z̄. □

Definition 1.3 Let 𝔼_{0} = {e ∈ 𝔼 | s(e) = v_{0}}. Let z_{e} ∈ Z̄_{0} be a point above x_{e}. For e ∈ 𝔼_{0}, the conductor of e is defined as

h_{e} = ord_{z_{e}}(ω) + 1.

Suppose that e ∈ 𝔼_{0}. Let m_{e} be the ramification index of z_{e} in Z̄_{0} → X̄_{0}. Define

ν_{e} := [\frac{h_{e}}{m_{e}} ] , a_{e} := m (\frac{h_{e}}{m_{e}} − ν_{e}) .

Lemma 1.4 Let e ∈ 𝔼_{0}.

(a) If the subtree of T with root t(e) contains a new tail, then ν_{e} ≥ 1.

(b)

∑_{e∈𝔼_{0}} ν_{e} = |𝔼_{0}| − 2 − \frac{1}{m} ∑_{b∈𝔹} a_{b}.

Proof: The lemma is proved in [30]. Part (a) is a reformulation of [21, Proposition 3.3.5]. Part (b) follows from the vanishing cycle formula [21, Section 3.4.2]. □

Theorem 1.5 Let G be a group whose order is strictly divisible by p. Let f : Y → X = ℙ_{K}^{1} be a G-Galois cover over K branched only at 0, 1, ∞. Suppose that f has bad reduction.

(a) The cover f has multiplicative reduction.

(b) Every irreducible component of X̄ other than X̄_{0} is a tail.

(c) Let e ∈ 𝔼_{0} with t(e) = b ∈ 𝔹. Choose a point z_{e} of Z̄′_{0} above x_{e}. Then z_{e} is a ramification point of Z̄′_{0} → X̄_{0}. Write m_{e} for its ramification index. The ramification invariant σ_{b} of the tail X_{b} is equal to h_{e}/m_{e}.

Proof: [30]. □

Remark 1.6 We use the notation of Theorem 1.5. Lemma 1.4 and Theorem 1.5 imply that ν_{e} = 1 if t(e) ∈ 𝔹_{new} and ν_{e} = 0 if t(e) ∈ 𝔹_{prim}. Moreover, we have that ∑_{b∈𝔹} a_{b} = m. One easily checks from the proof of Lemma 1.2 that ψ^{a_{b}} is the canonical generator of the inertia group of z_{e} in Z̄′_{0} → X̄_{0} (with respect to the character χ). In the terminology of Definition 2.1, we say that the type of Z̄′_{0} → X̄_{0} is (x_{e}, a_{b}), where e ∈ 𝔼_{0} and b = t(e).

2 The reduction of metacyclic covers

In this section, we study metacyclic covers of the projective line branched at four points. We start by defining the class of metacyclic covers we are interested in in this paper. We determine the different types of reduction that can occur. In Definition 2.8, we distinguish the three different cases that may occur. These play an important role in the rest of the paper.

We fix the following data.

• Let p ≠ 2 be a prime number and m > 1 an integer such that p ≡ 1 mod m.

• Let R be a complete discrete valuation ring whose fraction field K has characteristic zero and whose residue field k is algebraically closed of characteristic p.

• Fix a character χ : ℤ/m → 𝔽_{p}^{×} of order m. Put ζ_{m} = χ(1). We choose a lift χ_{K} : ℤ/m → K^{×} of χ. We denote χ_{K} also by χ.

• Let N be an extension of ℤ/m by ℤ/p. We fix once and for all generators ϕ and ψ of N. We suppose that ϕ^{p} = ψ^{m} = 1 and ψ^{i}ϕψ^{−i} = ϕ^{χ(i)}.

Definition 2.1 Let x = (x_{1}, … x_{r}) be r distinct points of X = ℙ1 ; K̄. Let a = (a_{1}, …, a_{r}) be an r-tuple of integers with 0 < a_{i} < m and ∑ a_{i} ≡ 0 mod m. Let f : Y → X be an N-Galois cover. We say that f is of type (x; a) (with respect to χ) if the following holds.

(a) The cover f is branched only at x_{1}, …, x_{r}.

(b) The element ψ^{a_{i}} is the canonical generator with respect to χ of some point y_{i} of Y above x_{i}. In other words, if u_{i} is a local parameter of y_{i} then

(ψ^{a_{i}})^{∗}u_{i} ≡ χ(1)^{(a_{i},m)}u_{i} (mod u_{i}^{2}).

In concrete terms this means the following. Let f : Y → X be a metacyclic cover of type (x, a) and suppose that none of the x_{i} is equal to ∞, for simplicity. Let Z be the quotient of Y by the normal subgroup of N of order p. Then Y → Z is étale and Z is the complete nonsingular curve given by the equation

z^{m} = ∏_{i}(x − x_{i})^{a_{i}},

where ψ^{∗}z = χ(1)z. We also use the terminology ‘of type (x; a)’ for the (unique) m-cyclic cover Z → X.

Definition 2.2 Let g : Z → X be the m-cyclic cover of type (x; a). Write V := Hom(π_{1}(Z), ℤ/p). Then ℤ/m acts on V and we may consider V_{χ} := {ξ ∈ V | ψξ = χ(1)ξ}. Suppose f : Y → X is a metacyclic cover of type (x; a). Then f factors through g. We define ξ_{f} ∈ V as the element corresponding to the exact sequence

1 → π_{1}(Y) → π_{1}(Z) → Gal(Y, Z) = ℤ/p → 1.

One checks that ψξ_{f} = χ(1)ξ_{f} and therefore ξ_{f} ∈ V_{χ}.

Lemma 2.3 With notations as above:

dim_{𝔽_{p}} V_{χ} = r − 2.

Proof: This follows from the main result of [7]. See also [27, Section 2.4]. □

Two metacyclic covers f_{i} : Y_{i} → X of type (x; a) are isomorphic if there exists an N-equivariant isomorphism h : Y_{1} → Y_{2} such that f_{1} = f_{2} ∘ h. Note that such h, if it exists, is unique, since the center of N is trivial. Two metacyclic covers f_{1}, f_{2} of type (x; a) are isomorphic if and only if the corresponding elements ξ_{i} ∈ V_{χ} are in the same orbit under the action of ℤ/m on V_{χ}. Therefore Lemma 2.3 implies that, up to isomorphism, there are (p^{r−2} −1)/m metacyclic covers of X of type (x; a).

Since we suppose that (X; x) has good reduction, the m-cyclic cover Z → X extends to an m-cyclic cover Z_{0,R} → X_{0,R} of smooth curves branched at the x_{i}. Let Z̄_{0} and X̄_{0} be the special fibers of Z_{0,R} and X_{0,R}. Let V̄ = Hom(π_{1}(Z̄_{0}), ℤ/p) and let V̄_{χ} be the χ-isotypical part of V̄. Since every étale p-cyclic cover of Z̄_{0} can be uniquely lifted to an étale p-cyclic cover of Z, we obtain a canonical injection V̄_{χ} ↪ V_{χ}. This proves the following criterion for good reduction.

Lemma 2.4 Let f : Y → X be a metacyclic cover of type (x; a) and let ξ ∈ V_{χ} be the corresponding element. Then f has potentially good reduction if and only if ξ ∈ V̄_{χ}.

Artin–Schreier Theory implies that there exists a canonical isomorphism V̄ ≃ H^{1}(Z̄, OZ̄)^{F}, where F denotes the absolute Frobenius. Write

H^{1}(Z̄_{0}, O) = ⊕_{i=1}^{m−1}L_{χ^{i}},

where L_{χ^{i}} = {ξ|ψξ = χ^{i}(1)ξ}. Then V̄_{χ} = L_{χ}^{F}.

Lemma 2.5 The dimension of L_{χ} as a k-vector space is (a_{1} + ⋯ + a_{r} − m)/m.

Proof: [4, Lemma 4.3]. □

Note that 0 ≤ dim_{k} L_{χ} ≤ r − 2. In fact, dim_{k} L_{χ} + dim_{k} L_{χ^{−1}} = r − 2. This gives an upper bound on the number of metacyclic covers of type (x; a) in characteristic p. If dim L_{χ} < r −2 there are strictly less N-covers of type (x; a) in characteristic p than in characteristic zero. In principle one can compute dim_{𝔽_{p}} V̄_{χ} by using the explicit basis for L_{χ} and the matrix of F with respect to this basis as computed in [4, Section 5].

Suppose that f : Y → X is a metacyclic cover of type (x; a) with bad reduction. Lemma 1.2 implies that ω is a regular differential on Z̄_{0}, i.e. an element of H^{0}(Z̄_{0}, Ω^{1}). The vector space H^{0}(Z̄_{0}, Ω^{1}) is dual to H^{1}(Z̄_{0}, OZ̄_{0}). The transpose of the absolute Frobenius F : H^{1}(Z̄_{0}, OZ̄_{0}) → H^{1}(Z̄_{0}, OZ̄_{0}) is called the Cartier operator C : H^{0}(Z̄_{0}, Ω^{1}) → H^{0}(Z̄_{0}, Ω^{1}), [23]. An element ω ∈ H^{0}(Z̄_{0}, Ω^{1}) defines a μ_{p}-torsor if Cω = ω and an α_{p}-torsor if Cω = 0, [17, Proposition 4.14]. Let V̄ = H^{1}(Z̄_{0}, OZ̄_{0}) and V̄_{χ} = {ξ ∈ V̄ | ψξ = χ(1)ξ}. Note that this definition coincides with the definition we gave in Section 2.

Define M̄ = H^{0}(Z̄_{0}, Ω^{1})^{C} and let M̄_{χ} = {ω ∈ M̄ | ψω = χ(1)ω}. Note that M̄_{χ} is an 𝔽_{p}[ℤ/m]- module. Since C : H^{0}(Z̄_{0}, Ω^{1}) → H^{0}(Z̄_{0}, Ω^{1}) is the transpose of F : H^{1}(Z̄_{0}, OZ̄_{0}) → H^{1}(Z̄_{0}, OZ̄_{0}), we conclude that

M̄_{χ} = V̄χ^{−1}^{dual},

where ‘dual’ means duality of 𝔽_{p}[ℤ/m]-modules.

Proposition 2.6 There exists a surjective morphism of 𝔽_{p}[ℤ/m]-modules

V_{χ} ↠ M̄_{χ}.  (1)

Proof: Since M̄_{χ} = V̄χ^{−1}^{dual}, there is an inclusion M̄χ^{dual} ↪ V_{χ^{−1}}. The Weil pairing defines a map

V_{χ} × V_{χ^{−1}} → μ_{p}.

The choice of an isomorphism ℤ/p ≃ μ_{p}(K) identifies therefore V_{χ} and Vχ^{−1}^{dual}. This defines (1). □

In more concrete terms we can describe the surjection of Proposition 2.6 as follows. Let f : Y → X be a metacyclic cover of type (x; a) and let ξ be the corresponding element of V_{χ}. If f has multiplicative reduction, then ξ maps to ω_{0} in M̄_{χ}. Otherwise, ξ maps to zero. In case H^{0}(Z̄, Ω_{1})^{C=0} = 0 there are no metacyclic covers with additive reduction and the following sequence is exact:

0 → V̄_{χ} → V_{χ} → M̄_{χ} → 0.

This happens, for example, when Z̄_{0} is ordinary. Namely, in that case F : L_{χ^{i}} → L_{χ^{i}} is a bijection for every i, in particular for i = 1, −1. Therefore dim_{𝔽_{p}} V̄_{χ} = dim_{k} L_{χ} and dim_{𝔽_{p}} M̄_{χ} = dim_{k} L_{χ^{−1}}. Therefore H^{0}(Z̄, Ω_{χ})^{C=0} = 0.

Suppose that r = 4. Define a := dim_{𝔽_{p}} V̄_{χ} = dim_{𝔽_{p}} L_{χ}^{F} and b := dim_{𝔽_{p}} M̄_{χ} = dim_{𝔽_{p}} L_{χ^{−1}}^{F}. Note that a, b ≤ 2. Lemma 2.5 implies that dim_{k} L_{χ^{i}} = r − 2 = 2, therefore a, b ≤ 2. The following proposition lists the possibilities for (a, b).

Proposition 2.7 Suppose r = 4. Then (a, b) ∈ {(2, 0), (0, 2), (1, 1), (0, 0)}.

Proof: The 𝔽_{p}[ℤ/m]-module V̄ is equal to Hom(μ_{p}, J(Z̄_{0})[p]), where ‘Hom’ should be considered in the category of finite flat group schemes. The p-divisible group J(Z̄_{0})[p^{∞}] decomposes into eigenspaces of the automorphism ψ, since ℤ_{p} contains the mth roots of unity. Write

J(Z̄_{0})[p^{∞}] = ⊕_{i}J(Z̄_{0})[p^{∞}]_{χ^{i}}

for this decomposition. We conclude that the finite flat group scheme J(Z̄_{0})[p]_{χ} = J(Z̄_{0})[p^{∞}]_{χ}/p is a direct summand of the group scheme J(Z̄_{0})[p]. Recall that there is a direct sum decomposition

J(Z̄_{0})[p] = (ℤ/p)^{σ} × (μ_{p})^{σ} × Λ,

where σ is the p-rank of Z̄_{0} and Λ is some local-local finite flat group scheme. Since J(Z̄_{0}) is an abelian variety, J(Z̄)[p] does not have any direct summand α_{p}.

Since the dimension of J(Z̄_{0})[p]_{χ} is two, we conclude that it is isomorphic to either (ℤ/p)^{2}, (μ_{p})^{2}, ℤ/p ⊕ μ_{p} or it is a local-local group scheme. The definition of a and b implies that

J(Z̄_{0})[p]_{χ} ≃ (ℤ/p)^{b} × (μ_{p})^{a} × Λ_{χ},

where Λ_{χ} is local-local. This proves the proposition. □

It is easy to check that all possibilities listed in Proposition 2.7 occur; we give explicit conditions for each possibility to occur further on.

Definition 2.8 Distinguish three cases.

• The multiplicative case: a_{1} + a_{2} + a_{3} + a_{4} = m.

• the mixed case: a_{1} + a_{2} + a_{3} + a_{4} = 2m.

• the étale case: a_{1} + a_{2} + a_{3} + a_{4} = 3m.

Proposition 2.9 Suppose a_{1}+a_{2}+a_{3}+a_{4} = m and λ ≢ 0, 1, ∞ mod p. Then all metacyclic covers of type (x; a) have multiplicative bad reduction. Therefore (a, b) = (0, 2), for all λ ∈ ℙ_{k}^{1} −{0, 1, ∞}.

Proof: The proposition is proved in [29, Proposition 1.3] for every r ≥ 3. □

Proposition 2.10 Suppose a_{1} + a_{2} + a_{3} + a_{4} = 2m. Then F : L_{χ} → L_{χ} is an isomorphism, for sufficiently general λ̄ ∈ k.

Proof: In fact, a similar statement holds without assumption on r and the type. In [4, lemma 4.6] it is shown that the locus U where F : L_{χ} → L_{χ} is an isomorphism is open. The fact that U is non-empty follows from [4, Proposition 7.4]. □

Let U ⊂ ℙ_{k}^{1} − {0, 1, ∞} be the locus where F : L_{χ} → L_{χ} is an isomorphism. In case m = 2, this U is the subset of λ ∈ ℙ_{k}^{1} − {0, 1, ∞} such that the elliptic curve with Weierstrass equation y^{2} = x(x − 1)(x − λ) is ordinary. Therefore, in the mixed case, U is in general strictly smaller than ℙ_{k}^{1} − {0, 1, ∞}, in contrast to what happens in the étale case (see below).

For λ ∈ U we have (a, b) = (1, 1). Lemma 2.4 implies that (p − 1)/m metacyclic cover of type (x; a) have good reduction and the other p(p − 1)/m have bad reduction. There are finitely many λ ∈ ℙ_{k}^{1} for which F : L_{χ} → L_{χ} is the zero morphism. For these λ we have (a, b) = (0, 0) and all metacyclic covers of type (x; a) have additive bad reduction.

Proposition 2.11 Let f : Y → X be a metacyclic cover of type (x; a). Suppose that a_{1}+⋯+a_{4} = 3m and λ ≢ 0, 1, ∞ mod p. Then f has good reduction and (a, b) = (2, 0).

Proof: Let f as in the statement of the proposition. The assumption a_{1} + ⋯ + a_{4} = 3m implies that dim_{k} L_{χ} = 2. Since dim_{k} L_{χ} + dim_{k} L_{χ^{−1}} = r − 2, we conclude that dim_{k} L_{χ^{−1}} = 0. Therefore Proposition 2.9 implies that V_{χ^{−1}} = M̄_{χ^{−1}}. By duality, we conclude that V_{χ} = V̄_{χ}. Therefore f has good reduction. □

3 Description of the Hurwitz space in characteristic zero

In this section, we define the Hurwitz space parameterizing metacyclic covers in characteristic zero.

Let G be a group whose order is strictly divisible by p. Let R_{0} = W (𝔽̄_{p}) and K_{0} its quotient field. Let N ≃ ℤ/p ⋊ ℤ/m be as in Section 2. Recall that we have fixed generators ϕ and ψ of N which satisfy

ϕ^{p} = ψ^{m} = 1, ψϕψ^{−1} = ϕ^{χ(1)},

where χ : ℤ/m → 𝔽_{p}^{×} is an injective character. We represent the element ϕ^{i}ψ^{j} of N as [i, χ(1)^{j}], with i ∈ 𝔽_{p} and χ(1)^{j} ∈ Im(χ) ⊂ 𝔽_{p}^{×}.

Fix a type a with gcd(m, a_{1}, …, a_{4}) = 1. The inverse image of b ∈ ℤ/m is a conjugacy class of N which we denote by C(b). Let C = (C(a_{1}), C(a_{2}), C(a_{3}), C(a_{4})). Put ζ_{i} := χ(1)^{a_{i}} ∈ 𝔽_{p}^{×}. Let ℚ_{p}(a) be the smallest extension of ℚ_{p} over which C is rational, [24, Section 7.1].

Definition 3.1 Define H(a) = H4^{in}(a)/ℚ_{p}(a) as the (inner) Hurwitz space parameterizing N-Galois covers Y → X = ℙ^{1} of type (x; a). Write π(a) : H(a) → ℙ_{λ}^{1} −{0, 1, ∞} for the cover defined by [f] ↦ λ. We denote th Galois closure of π(a) by ϖ(a) : ℍ(a) → ℙ_{λ}^{1}.

If a is understood, we sometimes drop the index a from the notation. For generalities on Hurwitz spaces, we refer to [31]. This Hurwitz space has also been considered in [3]. Berger considers types of the form a = (1, 1, 1, m−3). He calls the Hurwitz space H(a) a “fake modular curve” and shows that it is a quotient of the complex upper half-plane by a non-congruence subgroup.

Lemma 3.2 The degree of π ⊗ ℚ̄_{p} is (p^{2} − 1)/m.

Proof: Let f : Y → X be an N-cover of type (x; a) and Z → X the intermediate m-cyclic cover. The étale p-cyclic cover Y → Z corresponds to a nonzero element ξ in V_{χ} ≃ (ℤ/p)^{2}, Lemma 2.3. Two elements ξ_{1} and ξ_{2} define the same cover if ψ^{i}(ξ_{1}) = ξ_{2}, for some i. □

One can describe describe H̄ ⊗ ℚ̄_{p} using Nielsen tuples, see for example [28, Section 10.1.7]. Define

E(a) = {g = (g_{1}, g_{2}, g_{3}, g_{4})| g_{i} ∈ C(a_{i}), N = ⟨g_{i}⟩, ∏_{i} g_{i} = 1 }.

The Nielsen class is equal to

Ni_{4}^{in}(a) = E(a)/N,

where the group N acts via uniform conjugation. The elements of this set are called the Nielsen tuples. Choose a presentation of the fundamental group

π_{1}(ℙ^{1} − x, ∗) = ⟨γ_{1}, …, γ_{4}| ∏_{i} γ_{i} = 1⟩,

where γ_{i} corresponds to a “loop” around the point x_{i}, as usual. One can identify the Nielsen tuples with isomorphism classes of surjective homomorphisms

π_{1}(ℙ^{1} − x, ∗) → N, Im(γ_{i}) ∈ C(a_{i}).

We also consider E^{∗}(a) := E(a)/𝔽_{p}, where 𝔽_{p} ⊂ N acts via uniform conjugacy.

Choose a base point λ_{0} ∈ ℙ^{1} − {0, 1, ∞}. The cover π : H → ℙ_{λ}^{1} defines an action of Π := π_{1}(ℙ^{1} − {0, 1, ∞}, λ_{0}) on π^{−1}(λ_{0}). Let t be a transcendental and K = ℚ̄_{p}(t). Choose a K-model Z → ℙ_{K}^{1} of the m-cyclic cover of type (x; a), where x_{4} = t. Define W := J_{Z}[p]_{χ}(K̄) ≃ (𝔽_{p})^{2}. This defines the monodromy representation

ρ : Gal(K̄, K) → GL(W) ≃ GL_{2}(p).  (2)

Let Γ be the image of the monodromy representation. The first goal of this section is to describe Γ. These results are due to Völklein [26], [27]. We repeat the proofs as our assumptions are somewhat different.

The action of Π on π(a)^{−1}(λ_{0}) can be described via the action of the Artin braid group B_{4}. Recall that B_{4} is generated by the standard generators Q_{1}, Q_{2}, Q_{3} which act on Nielsen tuples via:

gQ_{i} = (g_{1}, …, g_{i−1}, g_{i+1}, g_{i+1}^{−1}g_{i}g_{i+1}, g_{i+2}, …, g_{r}).

We follow here the convention of [27]. In some other texts the inverses of the Q_{i} are used; this does not make much difference, [28, p. 167].

The pure Artin braid group B^{(4)} is the kernel of the map B_{4} → S_{4}, sending Q_{i} to the transposition (i, i + 1). The pure Artin braid group B^{(4)} acts naturally on the set E^{∗}(a). The fundamental group Π can be embedded into the Hurwitz braid group which is a quotient of the Artin braid group. The action of Π on the Nielsen class Ni_{4}^{in}(a) is induced from the action of B^{(4)} on E^{∗}(a). We refer to [27] for details on braid groups.

The pure Artin braid group is generated by the braids

b_{1} = Q_{3}Q_{2}Q_{1}^{2}Q_{2}^{−1}Q_{3}^{−1}, b_{2} = Q_{3}Q_{2}^{2}Q_{3}^{−1}, b_{3} = Q_{3}^{2}.  (3)

Write g^{h} = h^{−1}gh and [g, h] = g^{−1}h^{−1}gh. One checks that

(g_{1}, g_{2}, g_{3}, g_{4})b_{i} = \{ \begin{matrix}(g_{1}^{g_{4}}, g_{2}^{[g_{1},g_{4}]}, g_{3}^{[g_{1},g_{4}]}, g_{4}^{g_{1}g_{4}}) & if i = 1, \\ (g_{1}, g_{2}^{g_{4}}, g_{3}^{[g_{2},g_{4}]}, g_{4}^{g_{2}g_{4}}) & if i = 2, \\ (g_{1}, g_{2}, g_{3}^{g_{4}}, g_{4}^{g_{3}g_{4}}) & if i = 3. \end{matrix}  (4)

We start by studying the action of B^{(4)} on the set E^{∗}(a). The elements of E^{∗}(a) may be represented as

([0, ζ_{1}], [v_{1}, ζ_{2}], [v_{2}, ζ_{3}], [v_{3}, ζ_{4}]), v_{i} ∈ 𝔽_{p},  (5)

where v_{3} = −(ζ_{2}ζ_{3})^{−1}v_{1} − ζ_{3}^{−1}v_{2}. Recall that ζ_{i} = χ(1)^{a_{i}} and therefore ζ_{1}ζ_{2}ζ_{3}ζ_{4} = 1. We may identify a tuple (5) with a vector v = (v_{1}, v_{2}) ∈ W := J(Z)[p]_{χ} ≃ (𝔽_{p})^{2}. Here Z = Z_{λ} is the m-cyclic cover of type (x; a). Note that v ∈ W corresponds to a tuple in E^{∗}(a) if and only if v ≠ 0. The action of B^{(4)} on E^{∗}(a) induces an action on W.

Proposition 3.3 The image in GL(W) of the action of B^{(4)} on W is equal to Γ.

Proof: [27, Theorem A] □

This proposition gives us a concrete way to compute the image of the monodromy action. For i = 1, 2, 3, define B_{i} ∈ GL_{2}(p) as the matrix corresponding to the braid b_{i}. As in [26,

Lemma 3], one computes that

B_{1} = ( \begin{matrix}ζ_{2} & −1 + ζ_{3} \\ ζ_{2}(−1 + ζ_{2}) & 1 − ζ_{2} + ζ_{2}ζ_{3} \end{matrix} ) ,  (6)
B_{2} = ( \begin{matrix}ζ_{1}(1 − ζ_{2} + ζ_{2}ζ_{3}) & ζ_{2}^{−1}(−1 + ζ_{3})(−1 + ζ_{1} − ζ_{1}ζ_{2} + ζ_{1}ζ_{2}ζ_{3}) \\ ζ_{1}ζ_{2}(1 − ζ_{2}) & 1 − ζ_{1} + ζ_{1}ζ_{2} + ζ_{1}ζ_{3} − ζ_{1}ζ_{2}ζ_{3} \end{matrix} ) ,  (7)
B_{3} = ( \begin{matrix}1 & ζ_{1}(1 − ζ_{3}) \\ 0 & ζ_{1}ζ_{2} \end{matrix} ) .  (8)

Let Γ̄ be the image of Γ under the projection GL_{2}(p) → PGL_{2}(p). We show that Γ̄ is either a group of order prime-to-p or PSL_{2}(p) or PGL_{2}(p).

Definition 3.4 If Γ̄ ≠ PSL_{2}(p), PGL_{2}(p), we say that a is exceptional.

Proposition 3.5 Suppose that p > 5.

(a) Suppose that m is even and, after permuting the branch points, we have a = (a, a, −a + m/2, −a + m/2). Then Γ is a dihedral group of order prime-to-p.

(b) Suppose that a is exceptional, but not as in (a). The order of ζ_{i}ζ_{j} in 𝔽_{p}^{×} is either 2,3,4,5, for all i ≠ j. If a is non-exceptional, Γ contains SL_{2}(p).

(c) Suppose a is non-exceptional. The image of det : Γ → 𝔽_{p}^{×} is generated by ζ_{1}ζ_{2}, ζ_{1}ζ_{3}, ζ_{2}ζ_{3}.

(d) Suppose a is non-exceptional. The Hurwitz space H is connected.

Proof: Part (a) follows easily from the expression for the matrices B_{i}, cf. [26, Section 1.7].

Suppose that no permutation of a is equal to (a, a, −a+m/2, −a+m/2). Recall that a subgroup of GL_{2}(W) is called primitive if it is irreducible and does not permute the summands in any nontrivial direct sum decomposition of W. It is proved in [26, Section 1.7] that Γ acts primitively on W. We may regard Γ as a subgroup of PSL_{2}(p^{2}). Dickson’s classification of the subgroups of PSL_{2}(p^{2}), [11, Theorem 8.27], implies that Γ̄ is isomorphic to either A_{4}, S_{4}, A_{5}, PSL_{2}(p) or PGL_{2}(p). Therefore, if a is non-exceptional, Γ contains SL_{2}(p).

Suppose that a is exceptional. Then the matrices B_{1}, B_{2}, B_{3} have order less than or equal to 5. One checks that there exists vectors w_{i}, w′_{i} ∈ W which are eigenvectors of B_{i} with eigenvalue 1 and (ζ_{i}ζ_{4})^{−1}. If ζ_{i}ζ_{j} = 1 for some i ≠ j, then Γ̄ contains an element of order p > 5 which is impossible. This proves (b).

Suppose that a is non-exceptional. To prove (c), note that the index of SL_{2}(p) in Γ is equal to the order of the image of det : Γ → 𝔽_{p}^{×}. The subgroup det(Γ) of 𝔽_{p}^{×} is generated by the determinant of the matrices B_{i} for i = 1, 2, 3. We have

det(B_{i}) = \{ \begin{matrix}ζ_{2}ζ_{3} & if i = 1, \\ ζ_{1}ζ_{3} & if i = 2, \\ ζ_{1}ζ_{2} & if i = 3. \end{matrix}

Recall that we have identified E^{∗}(a) with the set of v ∈ W − {0}. Since Γ contains SL_{2}(p), it follows that Γ acts transitively on E^{∗}(a). But this implies that G acts transitively on Ni_{4}^{in}(C) = E^{∗}(a)/(ℤ/m) and therefore that the Hurwitz space is connected. □

It is not clear for which types the group Γ̄ is either A_{4}, S_{4} or A_{5}. The necessary condition of Proposition 3.5 is certainly not sufficient. However, for us this is not important since in the exceptional case p does not divide the order of the Galois group G, since p > 5. Therefore, the Galois closure of H(a) → ℙ_{λ}^{1} has good reduction to characteristic p. If ∑ a_{i} = 2m we show in Section 4 that the Galois closure of H(a) → ℙ_{λ}^{1} has bad reduction and we are not in the exceptional case.

Write M := ⟨χ(1)I⟩ for the m-cyclic subgroup of the center of GL_{2}(p). If a is non-exceptional, then SL_{2}(p) ⊂ Γ ⊂ GL_{2}(p). Therefore the index of SL_{2}(p) in Γ is equal to the order of det(Γ) . It follows from Proposition 3.5.(c) that det(Γ) is a subgroup of Im(χ) ≃ ℤ/m ⊂ 𝔽_{p}^{×}.

Corollary 3.6 Suppose that a is non-exceptional. Let G be the Galois group of the Galois closure of π(a). Then

G ≃ \{ \begin{matrix}SL_{2}(p) & if m is odd, \\ PSL_{2}(p) & if m if even and det(Γ) ⊂ det(M), \\ PGL_{2}(p) & if det(Γ) ⊄ det(M). \end{matrix}

Proof: Note that the action of ψ^{i} ∈ ℤ/m on E^{∗}(a) corresponds to multiplication by the matrix

( \begin{matrix}χ(1)^{i} & 0 \\ 0 & χ(1)^{i} \end{matrix} )

on W. Therefore G is the image of Γ under the projection ΓM → ΓM/M.

It follows from Proposition 3.5(b) that det(Γ) ⊂ Im(χ) ≃ ℤ/m ⊂ 𝔽_{p}^{×}. We first suppose that m is odd. Then ℤ/m does not contain an element of order 2 and −I ∉ M. This implies that det : M → 𝔽_{p}^{×} is injective and det(M) ≃ ℤ/m. Therefore det(Γ) ⊂ det(M). We conclude that the image G of Γ in ΓM → ΓM/M is equal to the image of SL_{2}(p) in ΓM → ΓM/M. Since −I ∉ Γ, we have G ≃ SL_{2}(p).

Suppose that m is even. Then −I ∈ M. If det(Γ) ⊂ det(M), then G is isomorphic to the image of SL_{2}(p) in ΓM/M. Since −I ∈ M, we conclude that G ≃ PSL_{2}(p).

Suppose that det(Γ) is not contained in det(M). Then M contains −I. The image of Γ in ΓM/M is strictly larger than the image of SL_{2}(p) in ΓM/M. We conclude that G ≃ PGL_{2}(p). □

Remark 3.7 We denote the Galois closure of π(a) by ϖ(a) : ℍ(a) → ℙ^{1}. Following [27], we can give a modular interpretation for ℍ = ℍ(a). We explain the definition and refer to [27] for details.

Let N := W ⋊ ℤ/m, where ℤ/m acts on W via χ. Let H be the Hurwitz space over ℚ̄_{p} parameterizing tuples (f, h, η), up to isomorphism. Here f : Y → ℙ^{1} is an N -Galois cover of type (x; a) with x_{1} = 0, x_{2} = 1, x_{3} = ∞, x_{4} = λ and λ different from the base point λ_{0} and h : Aut(Y, ℙ^{1}) → N is an isomorphism. Moreover, η is an point of Y above λ_{0}. We have a natural map H → ℙ_{λ}^{1}, which sends (f, h, η) to λ.

There is a natural action of W on H. Therefore we can define H^{(W)} as the quotient of H by W. Let ℍ^{(W)} be a connected component of H^{(W)}. It follows from the results of [27] that ϖ^{(W)} : ℍ^{(W)} → ℙ_{λ}^{1} is a Γ-Galois cover. The Galois closure ϖ of π is the quotient of ϖ^{(W)} by M ∩ Γ. We do not need this in the rest of the paper.

Our next goal is to describe the ramification of π(a) : H(a) → ℙ_{λ}^{1}. Write x_{1} = 0, x_{2} = 1, x_{3} = ∞ ∈ ℙ_{λ}^{1}. Let i ∈ {1, 2, 3}. The points π(a)^{−1}({x_{i}}) are called the cusps of π(a). By [6, Section 4.2.2], there is a 1-1 correspondence between cusps above x_{i} for i ∈ {1, 2, 3} and elements of

Ni_{4}^{in}(a)/⟨b_{i}⟩,

where the braids b_{i} defined in (3) act as in (4). Define d_{i} = m/ gcd(a_{i} +a_{4}, m) if a_{i} +a_{4} ≢ 0 mod m and d_{i} = p otherwise.

Proposition 3.8 The signature of the Galois closure ϖ(a) : ℍ(a) → ℙ^{1} of π(a) is (d_{1}, d_{2}, d_{3}).

Proof: Choose a cusp c of H(a) above x_{i} corresponding to g = (g_{1}, g_{2}, g_{3}, g_{4}). For simplicity we suppose that i = 3, i.e. x_{i} = ∞. The ramification index of the cusp c in π(a) : H(a) → ℙ^{1} is equal to the length of the orbit of g under b_{3}. It follows from the definition of the action of b_{i} that the ramification index of c divides the order of g_{3}g_{4}. Let g′ = (ψ^{a_{1}}, ψ^{a_{2}}, ϕψ^{a_{3}}, g′_{4}), where g′_{4} is chosen such that the product of the g′_{i} is one.

We distinguish two cases. First suppose that a_{3} + a_{4} = m. Then for every Nielsen tuple g, the order of g_{3}g_{4} divides p. One easily checks that the orbit of g′ has length p. This implies that the ramification index of ∞ in ϖ(a) is p.

Now suppose that a_{3} + a_{4} < m. Then the order of g_{3}g_{4} divides d_{3}m/ gcd(a_{3} + a_{4}, m). Since the orbit of g′ has length d_{3}, we conclude that ∞ is ramified in ϖ(a) of order d_{3}. □

4 A good reduction theorem

Let a = (a_{1}, a_{2}, a_{3}, a_{4}) be a type. To a we associated in Section 3 a Galois cover ϖ(a) : ℍ(a) → ℙ^{1}. In this section we determine when ϖ(a) has good reduction.

Let K = ℚ̄_{p}(t), where t is a transcendental element. Denote by G_{K} the absolute Galois group of K. Recall from Proposition 3.3 that the monodromy representation ρ : G_{K} → GL_{2}(p) has image Γ. Let L′/K be the corresponding Γ-Galois extension. Recall that the Galois group G of ϖ(a) is the quotient of Γ by the (central) subgroup Γ ∩ M of Γ. Let L be the sub field of L of invariants under Γ ∩ M. Denote the Gauß-valuation of K by ν_{0}; this valuation corresponds to the original component of the stable model of ϖ(a).

Lemma 4.1 The cover ϖ(a) has good reduction if and only if ν_{0} is unramified in L/K.

Proof: The forward implication is obvious. Suppose that ν_{0} is unramified in L/K. Then the restriction of the stable model of ϖ(a) to the original component is separable. It is well known that this implies that ϖ(a) has good reduction, [6, Proposition 1.1.4]. □

Theorem 4.2 The cover ϖ(a) has good reduction if and only if a_{1} + a_{2} + a_{3} + a_{4} ≠ 2m.

Proof: Let R be the ring of integers of the completion of K at ν_{0}. Let k be the residue field of R. Denote by Z_{0,R} the m-cyclic cover of ℙ_{R}^{1} of type a branched at 0, 1, ∞, t. Let Z̄_{0} be the special fiber of Z_{0,R}. There is an exact sequence of group schemes

1 → JZ̄_{0}[p]χ^{loc} → JZ̄_{0}[p]_{χ} → JZ̄_{0}[p]χ^{et} → 1.  (9)

Suppose that ∑_{i} a_{i} = m, i.e. we are in the multiplicative case of Definition 2.8. It follows from Proposition 2.9 that J_{Z_{0,R}}[p]_{χ} is étale. This implies that the action of Γ on J_{Z_{0,R}}[p]_{χ} is unramified, since R is a Henselian local ring. Lemma 4.1 implies therefore that ϖ(a) has good reduction.

Suppose that ∑_{i} a_{i} = 3m, i.e. we are in the étale case. The group scheme J_{Z_{0,R}}[p]_{χ} is dual to J_{Z_{0,R}}[p]_{χ^{−1}}. But J_{Z_{0,R}}[p]_{χ^{−1}} ≃ J_{Z_{0,R}^{∗}}[p]_{χ}, where Z_{0,R}^{∗} is the m-cyclic cover of type a^{∗} = (m − a_{1}, m − a_{2}, m − a_{3}, m − a_{4}) branched at 0, 1, ∞, t. Since ∑_{i}(m − a_{i}) = m, we may apply the theorem in the multiplicative case to a^{∗}. The theorem in the étale case follows by duality.

Suppose that ∑_{i} a_{i} = 2m, i.e. we are in the mixed case. Since the fourth branch point x_{4} = t is generic, it follows from Proposition 2.10 that both JZ̄_{0}[p]χ^{loc} and JZ̄_{0}[p]χ^{et} have rank p. Choose a valuation ν′ of L′ above ν_{0} and write Γ_{ν′} for its decomposition group. Suppose that ϖ(a) has good reduction. The Γ-Galois cover ϖ′(a) corresponding to L′/K has good reduction also. (Here we use that the index of G in Γ is prime-to-p.) We conclude that Γ is equal to Γ_{ν′} and that Γ acts faithfully on the stable reduction of ϖ′(a). In particular, Γ acts transitively on the subgroups of JZ̄_{0}[p]_{χ}. But the action of Γ on JZ̄_{0}[p]_{χ} respects the splitting (9) of JZ̄_{0}[p]_{χ} into its local and étale parts. This gives a contradiction. □

5 The a-Hasse invariant

In Sections 5 and 6, we suppose that ∑_{i} a_{i} = 2m, i.e. ϖ(a) has bad reduction. Our goal is to compute the stable reduction of ϖ(a). In this section we define the a-Hasse invariant Φ_{a}. This is a generalization if the classical Hasse invariant for elliptic curves, cf. [14, Section 12.4]. The a-Hasse invariant satisfies a hypergeometric differential equation, just as the classical Hasse invariant. In Section 6 we show that Φ_{a} essentially determines the stable reduction.

Let a be a type such that a_{1} + a_{2} + a_{3} + a_{4} = 2m. Let Z̄_{0} be the m-cyclic cover of ℙ_{k}^{1} branched at 0, 1, ∞, λ of type a. Since we are in the mixed case, Lemma 2.5 implies that the χ-isotypical part H^{0}(Z̄_{0}, Ω)_{χ} has k-dimension one. Recall that C : H^{0}(Z̄_{0}, Ω)_{χ} → H^{0}(Z̄_{0}, Ω)_{χ} is a bijection for all but finitely many λ, Proposition 2.10.

Definition 5.1 The λ̄ for which C : H^{0}(Z̄_{0}, Ω)_{χ} → H^{0}(Z̄_{0}, Ω)_{χ} is a bijection are called a-ordinary. If λ̄ is not a-ordinary it is called a-supersingular. Write Λ(a) for the set of a-supersingular λ-values.

For m = 2, the only possible type is (1, 1, 1, 1). In this case Z̄_{0} is an elliptic curve and λ is a-supersingular if and only if Z̄_{0} is supersingular.

Now suppose that λ = t corresponds to the generic point of ℙ_{k}^{1}. Then Z̄_{0} is given by an equation z^{m} = x^{a_{1}}(x − 1)^{a_{2}}(x − λ)^{a_{4}}. Define

ω = \frac{z dx}{x(x − 1)(x − λ)} ∈ H^{0}(Z̄_{0}, Ω^{1})_{χ}.

Lemma 2.5 implies that dim_{k} H^{0}(Z̄_{0}, Ω^{1})_{χ} = dim_{k} H^{1}(Z̄_{0}, OZ̄_{0})_{χ^{−1}} = 1. Therefore ω is a basis of H^{0}(Z̄_{0}, Ω^{1})_{χ}.

Definition 5.2 Suppose that a_{1} + a_{2} + a_{3} + a_{4} = 2m. Define the a-Hasse invariant as

Cω = Φ_{a}(λ)^{(1/p)}ω.

Lemma 5.3 Write a_{i}^{∗} = m − a_{i} and α = (p − 1)/m. Then

Φ_{a}(λ) = (−1)^{αa_{3}} ∑_{i+j=αa_{3}} ( \begin{matrix}αa_{2}^{∗} \\ i \end{matrix} ) ( \begin{matrix}αa_{4}^{∗} \\ j \end{matrix} ) λ^{j}.  (10)

Proof: Write

ω = \frac{z^{p}}{x^{p}(x − 1)^{p}(x − λ)^{p}} x^{p−αa_{1}}(x − 1)^{p−1−αa_{2}}(x − λ)^{p−1−αa_{4}} \frac{dx}{x}

and

e := x^{p−αa_{1}}(x − 1)^{p−1−αa_{2}}(x − λ)^{p−1−αa_{4}} = x^{1+αa_{1}^{∗}}(x − 1)^{αa_{2}^{∗}}(x − λ)^{αa_{4}^{∗}} = ∑_{i} e_{i}x^{i}.

It follows from standard properties of the Cartier operator that

Cω = \frac{z}{x(x − 1)(x − λ)} (∑_{i} e_{pi}^{1/p}x^{i}) \frac{dx}{x},

[23]. Note that deg e = 1 + α(a_{1}^{∗} + a_{2}^{∗} + a_{4}^{∗}) < 1 + α(a_{1}^{∗} + a_{2}^{∗} + a_{3}^{∗} + a_{4}^{∗}) = 1 + 2αm = 2p − 1 and deg(e) ≥ 1 + α(a_{1}^{∗} + a_{2}^{∗} + a_{3}^{∗} + a_{4}^{∗}) − αm = 1 + αm = p. Therefore

Cω = \frac{z(e_{p}^{1/p}x)}{x(x − 1)(x − λ)} \frac{dx}{x} = e_{p}^{1/p}ω.

One easily checks that e_{p} is equal to the right hand side of (10). □

We find back the classical Hasse invariant Φ for m = 2 and a = (1, 1, 1, 1). It is well known that Φ is the solution to a hypergeometric differential equation. The same holds for the a-Hasse invariant.

Proposition 5.4 The polynomial Φ_{a}(λ) is a solution to the hypergeometric differential equation

λ(1 − λ)u′′ + [C − (A + B + 1)λ]u′ − ABu = 0,  (11)

where A ≡ −αa_{3} mod p and B ≡ α(a_{4} − m) mod p and C ≡ −α(a_{2} + a_{3}) mod p.

Proof: Write Φ_{a}(λ) = ∑_{j} φ_{j}λ^{j}. To prove the proposition, we have to show that the coefficients φ_{j} satisfy the recursion relation φ_{j+1}/φ_{j} = (A + j)(B + j)/(C + j)(1 + j), [32, Section III.2]. Here j and j + 1 are in the range where the coefficients are not congruent to zero. Obviously,

\frac{φ_{j+1}}{φ_{j}} = \frac{(−αa_{3} + j)(−αa_{4}^{∗} + j)}{(−α(a_{2} + a_{3}) + j)(j + 1)} ≡ \frac{(A + j)(B + j)}{(C + j)(1 + j)} (mod p),

with A, B, C as in the statement of the proposition. □

Corollary 5.5 (a) The zeros of Φ_{a}(λ) different from 0 and 1 are simple.

(b) At λ = 0, the polynomial Φ_{a}(λ) has a zero of order max(α(a_{2} + a_{3} − m), 0).

(c) At λ = 1, the polynomial Φ_{a}(λ) has a zero of order max(α(a_{1} + a_{3} − m), 0).

Proof: Part (a) follows immediately from Proposition 5.4 as the hypergeometric differential equation (11) has essential singularities only at 0, 1, ∞. Part (b) is obvious. Part (c) follows from Part(b) by interchanging the branch points x_{1} and x_{2}. □

Corollary 5.6 The number of a-supersingular λ-values is

min(αa_{4}^{∗}, αa_{3}) − max(α(a_{2} + a_{3} − m), 0) − max(α(a_{1} + a_{3} − m), 0).

Proof: The degree of Φ_{a} is equal to min(αa_{4}^{∗}, αa_{3}). Therefore the corollary follows from Corollary 5.5. One easily checks from the formula that the number of a-supersingular λ-values is non-zero.

□

6 Computation of the deformation datum

The goal of this section is to describe the stable reduction of ϖ(a) : ℍ(a) → ℙ^{1} By Theorem 1.5, this amounts to computing the deformation datum associated to ϖ(a).

The notation is as in Section 4. In particular, we suppose that ∑_{i} a_{i} = 2m. Let k be an algebraically closed field of characteristic p. We denote by P the Sylow p-subgroup of Γ consisting of upper triangular matrices with ones on the diagonal. Choose a valuation ν of L′ above ν_{0} such that the inertia group I_{ν} of ν is P. This is possible as ϖ(a) has bad reduction by Theorem 4.2. Let Γ_{ν} be the decomposition group of ν. By our assumption on ν, the group Γ_{ν} is a subgroup of the Borel group B of GL_{2}(p) consisting of upper triangular matrices. Define

Γν^{res} = Γ_{ν}/I_{ν}.

The monodromy representation (2) induces a representation

ρ^{res} : Γν^{res} → B/P = 𝔽_{p}^{×} × 𝔽_{p}^{×}.

We may identify ρ^{res} with two characters ξ_{i} : Γν^{res} → 𝔽_{p}^{×}, where ξ_{i} is the composition of ρ^{res} with the ith projection.

Proposition 6.1 Denote by G_{k(t)} the absolute Galois group of k(t). Under the natural identification

k(t)^{×}/(k(t)^{×})^{p−1} ≃ Hom(G_{k(t)}, μ_{p−1})
u ↦ [σ ↦ \frac{^{σ}( \sqrt[p−1]{u})}{\sqrt[p−1]{u}}]

the character ξ_{2} (resp. ξ_{1}) corresponds to Φ_{a}(t) (resp. Φ_{a^{∗}}(t)^{−1}). Here Φ_{a} is the a-Hasse invariant and a^{∗} := (m − a_{1}, m − a_{2}, m − a_{3}, m − a_{4}) is the dual type.

Proof: Let x = (0, 1, ∞, t) and write Z̄_{0} for the m-cyclic cover of type (x; a) over k(t). It follows from (9) that the character ξ_{2} (resp. ξ_{1}) corresponds to the action of G_{k(t)} on JZ̄_{0}[p]χ^{et} (resp.

JZ̄_{0}[p]χ^{loc}).

We start by computing ξ_{2}. For this we use the canonical identification

JZ̄_{0}[p]χ^{et} ≃ H^{0}(Z̄_{0}, Ω)_{χ}^{C},

as in Section 2. The differential

ω := \frac{z dx}{x(x − 1)(x − t)}

is a basis of H^{0}(Z̄_{0}, Ω)_{χ}, cf. Section 5. It follows from Definition 5.2 that Cω = Φ_{a}(t)^{1/p}ω. Since Ccω = c^{1/p}Cω, we see that

CΦ_{a}(t)^{\frac{1}{p−1}} ω = Φ_{a}(t)^{\frac{1}{p−1}} ω.

We conclude that ξ_{2} corresponds to the (p − 1)-cyclic extension of k(t) obtained by adjoining a (p − 1)th root of the a-Hasse invariant Φ_{a}(t).

To prove the statement of the proposition for ξ_{1}, we remark that J_{Z}[p]χ^{loc} is dual to J_{Z}[p]χ^{−1}^{et}. Therefore the above argument applied to the character χ^{−1} shows that ξ_{1} corresponds to the inverse of the Hasse invariant Φ_{a^{∗}}(t) corresponding to the dual type a^{∗} = (m − a_{1}, m − a_{2}, m − a_{3}, m − a_{4}).

□

Let W̄_{0} be the original component of the stable reduction of the G-Galois cover ϖ(a) : ℍ(a) → ℙ_{λ}^{1}. We choose a component Ū_{0} of ℍ̄(a) above W̄_{0}. The map Ū_{0} → W̄_{0} factors through a curve V̄_{0} such that Ū_{0} → V̄_{0} is a μ_{p}-torsor and V̄_{0} → W̄_{0} is a tame cover, cf. Section 1. The following theorem describes the cover V̄_{0} → W̄_{0}.

Theorem 6.2 Suppose that a is non-exceptional, Definition 3.4. Write d := gcd(m, a_{1} + a_{3}, a_{2} + a_{3}). If m ≠ 2, we assume that d ≠ m. This is no restriction.

(a) Suppose that m = 2. Then G = PSL_{2}(p) and the cover V̄_{0} → W̄_{0} is given by

θ^{(p−1)/2} = Φ_{a}.

(b) Suppose that G = SL_{2}(p). Write m/d = 1 + 2j. The cover V̄_{0} → W̄_{0} is given by

θ^{p−1} = Φ_{a}^{1+j}Φ_{a^{∗}}^{−j}.

(c) Suppose G = PSL_{2}(p) or G = PGL_{2}(p) and d is even. Then there exists a function g ∈ k(t) such that

Φ_{a}^{1+m/d}(t)Φ_{a^{∗}}^{1−m/d}(t) = g^{2}.

The cover V̄_{0} → W̄_{0} is given by

θ^{(p−1)/2} = g.

(d) Suppose that G = PGL_{2}(p) and d is odd. The cover V̄_{0} → W̄_{0} is given by

θ^{p−1} = Φ_{a}^{1+m/(2d)}Φ_{a^{∗}}^{1−m/(2d)}.

Proof: We first note that

Φ_{a^{∗}}(t) = (−1)^{αa_{3}^{∗}} ∑_{i+j=αa_{3}^{∗}} ( \begin{matrix}αa_{2} \\ i \end{matrix} ) ( \begin{matrix}αa_{4} \\ j \end{matrix} ) λ^{j}.

The polynomial Φ_{a^{∗}}(t) has a zero at t = 0 of order max(α(m − a_{2} − a_{3}), 0) and a zero at t = 1 of order max(α(m − a_{1} − a_{3}), 0).

Since the group scheme JZ̄_{0}[p]_{χ} is dual to JZ̄_{0}[p]_{χ^{−1}}, it is obvious that JZ̄_{0}[p]_{χ} is local-local if and only if JZ̄_{0}[p]_{χ^{−1}} is local-local. This implies that t ≠ 0, 1 is a zero of Φ_{a}(t) if and only if it is a zero of Φ_{a^{∗}}(t). Recall from Corollary 5.5 that all zeros of Φ_{a} and Φ_{a^{∗}}, except 0 and 1, are simple. Therefore

\frac{Φ_{a}(t)}{Φ_{a^{∗}}(t)} = t^{±α(a_{2}+a_{3}−m)}(t − 1)^{±α(a_{1}+a_{3}−m)}.

Case (a). Suppose that m = 2. The only possibility for the type is a = (1, 1, 1, 1). Since a_{i} + a_{j} = m, for all i and j, we conclude that Φ_{a} = Φ_{a}^{∗}. This implies the theorem in this case. This is well known, see for example [14].

Suppose now that m ≠ 2. Then the exist i ≠ j such that a_{i} + a_{j} ≠ m. Therefore, after permuting the branch points if necessary, we may assume that d ≠ m. Let L_{2} (resp. L_{1}) be the extension of k(t) corresponding to adjoining a (p − 1)th root θ_{i} of Φ_{a}(t) (resp. 1/Φ_{a^{∗}}(t)). Let L̃ = L_{1} ⊗_{k(t)} L_{2}. Choose a primitive (p − 1)th root of unity ξ ∈ k. Choose generators σ_{i} for the Galois group of F̃ over k(t) such that

σ_{1}θ_{1} = ξθ_{1},  σ_{1}θ_{2} = θ_{2},  σ_{2}θ_{1} = θ_{1},  σ_{2}θ_{2} = ξθ_{2}.

For ℓ|(p − 1), denote by L_{i}^{(ℓ)} the (unique) subfield of L_{i} containing k(t) whose degree over k(t) is ℓ. Note that L_{1}^{(ℓ)} ≃ L_{2}^{(ℓ)} if and only if Φ_{a}Φ_{a^{∗}}^{i} = δ^{ℓ} for some δ ∈ k(t) and some i prime to (p − 1). Let d be as in the statement of the theorem and α = (p − 1)/m. Then [L_{1} ∩ L_{2} : k(t)] = αd. Therefore the algebra L̃ is the product of αd fields L(η) with

L(η) = k(t)[θ_{1}, θ_{2}| θ_{2}^{p−1} = Φ_{a}, θ_{1}^{p−1} = 1/Φ_{a^{∗}}, (θ_{1}θ_{2})^{m/d} = ηδ],

where η is a αdth root of unity and δ^{αd} = Φ_{a}/Φ_{a^{∗}}. The group G(η) = Gal(L(η), k(t)) is generated by σ_{1}σ_{2}^{−1} and σ_{1}^{αd}.

Recall that M is the subgroup of GL_{2}(p) consisting of the scalar matrices xI with x ∈ ℤ/m ⊂ 𝔽_{p}^{×}. To find the subfield of L(η) which corresponds to V̄_{0}, we have to take invariants under the restriction of M ∩ Γ to L(η), cf. Section 3. From now on, we suppose η = 1. This is no restriction.

Define J as the intersection of M ∩ Γ with G(η). The group M ∩ Γ is generated by (σ_{1}σ_{2})^{α}. Therefore I is generated by (σ_{1}σ_{2})^{αd/2} if d is even and by (σ_{1}σ_{2})^{αd} if d is odd.

Case (b) Suppose that G = SL_{2}(p). Corollary 3.6 implies that m is odd. Write m/d = 1 + 2j. The degree of L(η)^{J} over k(t) is p − 1. Define θ := θ_{1}^{j}θ_{2}^{1+j}. We claim that θ generates L(η)^{J} over k(t). It is clear that θ ∈ L(η)^{J}. Since we assumed that m ≠ d, the integer j does not divide p − 1. Therefore θ^{i} ∉ k(t), for every 0 < i < p − 1 and θ generates L(η)^{J} over k(t). We have

θ^{p−1} = Φ_{a}^{1+j}Φ_{a^{∗}}^{−j} = x^{b_{1}}(x − 1)^{b_{2}} ∏_{λ∈Λ(a)}(x − λ).  (12)

The b_{i} are expressions in terms of the a_{j} which we leave to the reader to compute.

Case (c). Suppose that G = PSL_{2}(p). Corollary 3.6 implies that m is even and that det(Γ) is a subgroup of det(M). Therefore the order of det(Γ) divides m/2. This implies that gcd(m, a_{1} + a_{3}, a_{1} + a_{3}, a_{3} + a_{4}) is even and so d is even. We conclude that J is generated by (σ_{1}σ_{2})^{αd/2} and that the degree of L(η)^{J} over k(t) is (p − 1)/2.

One checks that L(η)^{J} is generated over k(t) by θ := θ_{1}^{−1−m/d}θ_{2}^{+1−m/d} which satisfies

θ^{p−1} = Φ_{a}^{1+m/d}Φ_{a^{∗}}^{1−m/d} = x^{b_{1}}(x − 1)^{b_{2}} ∏_{λ∈Λ(a)}(x − λ)^{2}.

Then

b_{i} = \{ \begin{matrix}α(a_{i} + a_{4} − m)(1 − \frac{m}{d}) & if a_{i} + a_{4} > m, \\ α(m − a_{i} − a_{4})(1 + \frac{m}{d}) & if a_{i} + a_{4} < m. \end{matrix}

In case a_{i} + a_{4} = m the branch point x_{i} is unramified in V̄_{0} → W̄_{0}.

Since d is even, also b_{1} and b_{2} are even. We conclude that the (p − 1)/2-cyclic cover V̄_{0} → W̄_{0} is given by the equation

(θ)^{(p−1)/2} = x^{b_{0}/2}(x − 1)^{b_{1}/2} ∏_{λ}(x − λ).  (13)

We leave it for the reader to verify that if G = PGL_{2}(p) and d is even the cover V̄_{0} → W̄_{0} is given by the same equation.

Case (d) Suppose that G = PGL_{2}(p) and that d is odd. It follows from the proof of Corollary 3.6 that in this case m is even. Define j = m/(2d). Since d is odd, we know that J is generated by (σ_{1}σ_{2})^{αd} and that the degree of L(η)^{J} over k(t) is p − 1. One checks that L(η)^{J}/k(t) is generated by θ := θ_{1}^{−1+j}θ_{2}^{1+j} which satisfies

θ^{p−1} = Φ_{a}^{1+j}Φ_{a^{∗}}^{1−j} = x^{b_{1}}(x − 1)^{b_{2}} ∏_{λ∈Λ(a)}(x − λ)^{2}.  (14)

□

Recall the following from Section 1. The stable reduction ϖ̄(a) : ℍ̄(a) → ℙ̄ of ϖ(a) has a very simple structure. The curve ℙ̄ is a comb. It consist of tails W_{b} for b ∈ 𝔹 which intersect the original component in one point which we denote by τ_{b}. There are two types of tails: new tails and primitive tails. A tail W_{b} is primitive if W_{b} contains the specialization of one of the branch points of ϖ(a) whose ramification index is prime-to-p. Therefore x_{i} for i ∈ {1, 2, 3} specializes to a primitive tail if a_{i} + a_{4} ≠ m, Proposition 3.8. All other tails are called the new tails. They intersect the original component W̄_{0} in an a-supersingular λ ∈ Λ(a). We may regard the set 𝔹 indexing the tails as a subset of {1, 2, 3} ∪ Λ(a).

To a tail W_{b} of ℙ̄ we associated a ramification invariant σ_{b} = h_{b}/m_{b} which describes the ramification above the intersection point of W_{b} with W̄_{0}. As a consequence of Theorem 6.2, we can now describe the ramification invariants of the tails of ℙ̄.

Recall that we have chosen an irreducible component Ū_{0} of ℍ̄(a) such that the inertia group of Ū_{0} is the Sylow p-subgroup P of G. The map Ū_{0} → W̄_{0} factors through V̄_{0} and Ū_{0} → V̄_{0} is a μ_{p}-torsor. Write Ū′_{0} for the quotient of Ū_{0} by the prime-to-p part of the center of its decomposition group and write V̄′_{0} for the quotient of Ū′_{0} by μ_{p}. Then V̄′_{0} → W̄_{0} is a cyclic cover, which is branched at 0, 1, ∞, λ for λ ∈ Λ(a). Let n be the order of this cover.

Remark 1.6 implies that 0 < σ_{i} < 1 for i ∈ {1, 2, 3} and 1 < σ_{λ} < 2 for λ ∈ Λ(a). Moreover, the ramification invariants are related to the type of the cover g′_{0} : V̄′_{0} → W̄_{0}, [30]. The relation is as follows. Write σ_{b} = h_{b}/n_{b}, where n_{b} is the ramification index of τ_{b} in g′_{0}. Let (β_{b}, τ_{b}) be the type of g′_{0}, as defined in Definition 2.1. Then h_{b}n/n_{b} ≡ β_{b} mod n. The invariants β_{b} are easily computed from the explicit equation for g′_{0}. This determines the ramification invariants.

We will make this explicit in case λ ∈ Λ(a). Suppose first that we are in Case (b) of Theorem 6.2, i.e. G = SL_{2}(p). Then V̄_{0} → W̄_{0} has order p − 1. The normalizer N_{G}(P) of the Sylow p-subgroup P in SL_{2}(p) has order p(p − 1), therefore the decomposition group of Ū_{0} is the full normalizer N_{G}(P). The order of the prime-to-p centralizer of N_{G}(P) is two. Therefore the degree of V̄′_{0} → W̄_{0} is (p − 1)/2. It follows from (12) that the integer β_{λ} = 1, for every λ ∈ Λ(a). The ramification invariant is now determined by the fact that (p − 1)/2 = n_{λ} < h_{λ} < 2n_{λ} = (p − 1) and 2h_{λ} ≡ 1 mod (p − 1). We conclude that σ_{λ} = (p + 1)/(p − 1). One checks that the same holds in the other cases.

We note that the fact that all new tails have ramification invariant (p+1)/(p−1) is a consequence of the fact that the a-Hasse invariant is a solution of a hypergeometric differential equation. The decomposition group of a new tail is either PSL_{2}(p) or SL_{2}(p). One can show the following. Let f be a Galois cover of ℙ_{k}^{1} with Galois group either PSL_{2}(p) or SL_{2}(p). Suppose that f is branched only at ∞ and that the ramification invariant σ of f satisfies 1 < σ < 2. Then σ = (p + 1)/(p − 1). The proof uses ideas from [5, Section 3.3].

Corollary 6.3 The differential ω is given by

ω = u\frac{θ dx}{x(x − 1)},  (15)

where θ and x are as in Theorem 6.2 and u ∈ k^{×}.

Proof: This is immediate. □

An important property of ω is that Cω = ω. This determines the constant u in (15), up to an element of 𝔽_{p}^{×}. (Actually, u ∈ 𝔽_{p}^{×}.) One can also check directly that the differential defined by (15) with u ∈ 𝔽_{p}^{×} is fixed by the Cartier operator, for example by using the method of [12, Section 4.3]. It turns out that Cω = ω is equivalent to the fact that Φ_{a} satisfies the hypergeometric differential equation of Proposition 5.4.

7 Reduction of the Hurwitz spaces

In the previous sections, we computed the stable reduction of ϖ(a) : ℍ(a) → ℙ_{λ}^{1} as Galois cover. We did not give an interpretation of the stable model ℍ̄(a) as moduli space; it is not clear whether the model ℍ̄(a) has a “reasonable” interpretation as a moduli space. We merely used the interpretation of ℍ(a) (or rather of its quotient H(a)) as Hurwitz space to determine the stable reduction of the cover ϖ(a) : ℍ(a) → ℙ_{λ}^{1}.

Let H(a)/K be the Hurwitz space parameterizing metacyclic covers of type (x; a), as defined in Section 3. Here K is some finite extension of ℚ_{p}. Write π(a) : H(a) → ℙ_{λ}^{1} for the projection to the λ-line. In [6], we find a definition of a compactification H̄(a) of H(a) over the ring of integers R of K. Roughly speaking, H̄(a)⊗_{R} 𝔽_{p} parameterizes the stable reduction of the metacyclic covers of type (x; a). In this section, we want to describe the structure of H̄(a) ⊗_{R} 𝔽_{p} without proofs. To make the proofs rigorous one needs to carefully analyze the deformation theory. Details will appear elsewhere.

The étale case. In this case all metacyclic covers of type (x; a) have good reduction, Proposition 2.11. Therefore the complete model H̄(a) parameterizes so called admissible covers. It is well known that the moduli space of admissible covers is smooth, [31]. We conclude that π(a) : H(a) → ℙ_{λ}^{1} has good reduction in this case, confirming Theorem 4.2.

The mixed case. Let λ ∈ ℙ_{K}^{1} be such that its reduction λ̄ ∈ ℙ_{k}^{1} is a-ordinary and different from 0, 1, ∞. (See Definition 5.1 for the definition of a-ordinary and a-supersingular.) Let x_{1} = 0, x_{2} = 1, x_{3} = ∞, x_{4} = λ. It follows from Proposition 2.10 that there are (p − 1)/m metacyclic covers of type (x; a) with good reduction. The other p(p − 1)/m such covers have multiplicative bad reduction. If λ̄ is a-supersingular, then all covers have additive bad reduction.

Suppose f : Y → X is a metacyclic cover of type (x; a) with bad reduction. Lemma 1.4 implies that X̄ has 5 irreducible components: the original component X̄_{0} and the four primitive tails X_{1}, X_{2}, X_{3}, X_{4}. Lemma 1.4.(b) implies that the ramification invariant σ_{i} corresponding to the tail X_{i} is smaller than 1. Therefore it follows from [18, Lemma 2.3.3] that there are only finitely many possibilities for f̄|_{X_{i}}. This implies that π̄(a) : H̄(a) → ℙ_{R}^{1} is finite, outside the cusps.

It follows from the description of the monodromy given in Section 4 that H̄(a) ⊗_{R} 𝔽_{p} has two irreducible components H^{good} and H^{bad}. Over the a-ordinary locus, H^{good} parameterizes metacyclic covers and H^{bad} parameterizes the stable reduction of the covers with multiplicative bad reduction. The two components intersect above λ ∈ Λ(a) (the a-supersingular λ’s).

For m = 2, the Hurwitz space H(a) is a version of the modular curve X_{1}(p) with ordered branch points. (In other words, one adds a full level 2-structure, cf. [5, Section 2.2].) In this case, the reduction of H is of course well known.

The multiplicative case. All metacyclic covers of type (x; a) have multiplicative bad reduction, Proposition 2.9. To describe the reduction of π(a) : H(a) → ℙ_{λ}^{1}, we first need some results on the stable reduction of metacyclic covers.

Let a = (a_{1}, a_{2}, a_{3}, a_{4}) be a type such that a_{1}+a_{2}+a_{3}+a_{4} = m. Let f : Y → X be a metacyclic cover of type a, branched at x_{1} = 0, x_{2} = 1, x_{3} = ∞, x_{4} = λ and suppose that λ ≢ 0, 1, ∞ mod p. Write (Z̄_{0}, ω) for the deformation datum of f, as defined in Definition 1.1. As in Remark 1.6, one checks that for z_{i} ∈ Z̄_{0} above x_{i} we have

gcd(a_{i}, m)ord_{z_{i}}(ω) ≡ a_{i} (mod m).

Let e ∈ 𝔼 be an edge with source v_{0}. Let x_{e} be the corresponding point of X̄_{0}. Choose a point z_{e} ∈ Z̄_{0} above x_{e} and write m_{e} for the ramification index of this point in Z̄_{0} → X̄_{0}. Analogous to the notation introduced in Section 1, we define ν_{e} = [h_{e}/m_{e}] and a_{e} = m(h_{e}/m_{e} − ν_{e}).

Suppose that the subtree of T with root e contains the primitive tail X_{i}. As in Proposition 1.6, one checks that a_{e} = a_{i}. The analog of Remark 1.6 becomes in this case ∑_{s(e)=v_{0}} ν_{e} = 1. It follows that there are two cases.

• The first possibility is that ν_{i} = 0, for all i ∈ 𝔹_{prim}. In this case, there is one new tail X_{5} which intersects the original component.

• The second possibility is that ν_{i} = 1 for a unique i ∈ 𝔹_{prim}.

Definition 7.1 For λ ∈ k − {0, 1}, let Z̄_{0} → ℙ_{k}^{1} be the m-cyclic cover of type (x; a) branched at x_{1} = 0, x_{2} = 1, x_{3} = ∞, x_{4} = λ. We say that λ ∈ k is a-special if there exists an ω ∈ H^{0}(Z̄_{0}, Ω^{1})_{χ} and 1 ≤ i ≤ 4 such that ν_{i}(ω) = 1. If λ is not a-special, we call it is a-general.

This terminology is inspired by [29], but the notation of a special cover in that paper is more restrictive than the way we use this concept here.

Lemma 7.2 Suppose that a_{1} + ⋯ + a_{4} = m. There are at most finitely many a-special λ’s.

Proof: Similar to [29, Section 3.5] □

Suppose that λ̄ ∈ ℙ_{k}^{1} − {0, 1, ∞} is a-general. One checks that X̄ has 6 irreducible components: the original component X̄_{0}, the primitive tails X_{1}, X_{2}, X_{3}, X_{4} and the new tail X_{5}.

We have σ_{i} < 1 if i ≤ 4 and σ_{5} = 2. We claim that this implies that, for given λ̄, that there are only finitely many possibilities for f̄. It follows from [18, Lemma 2.3.3] that there are only finitely

X̄_{0} X_{1} X_{2} X_{3} X_{4} X_{5} x_{1} x_{2} x_{3} x_{4}

Figure 1: Reduction in the multiplicative case for general λ

many possibilities for f̄|_{X_{i}}. The intersection point τ of X_{5} with X̄_{0} is the image of a zero of ω. Therefore it is obvious that there are at most finitely many possibilities for τ.

Suppose now that λ̄ ∈ ℙ_{k}^{1} is a-special. For simplicity, we suppose that ν_{1} = 1. There are two possibilities for the stable reduction f̄.

(a) The curve X̄ consists of 5 irreducible components: the original component X̄_{0} and the primitive tails X_{1}, X_{2}, X_{3}, X_{4}, see Figure 2. The ramification invariants are: σ_{1} = (m+a_{1})/m and σ_{i} = a_{i}/m for i > 1.

(b) There is one new tail which does not intersect the original component, see Figure 3.

X̄_{0} X_{1} X_{2} X_{3} X_{4} x_{1} x_{2} x_{3} x_{4}

Figure 2: Reduction in the multiplicative case for a-special λ (Possibility a)

X̄_{0} X_{1} X_{5}  W X_{2} X_{3} X_{4} x_{1} x_{2} x_{3} x_{4}

Figure 3: Reduction in the multiplicative case for a-special λ (Possibility b)

□

One can show using a patching argument as in [10] that both possibilities occur. It follows from [18, Proposition 2.2.6] that there is a one dimensional family of covers of type (a), which all occur as the reduction of a metacyclic cover. Therefore the map π̄ : H̄ → ℙ_{R}^{1} is not finite: there are vertical components whose image under π̄ is a point. The occurrence of vertical components in the stable reduction is not unexpected. The same phenomenon occurs for the reduction of the Hurwitz space of 2-cyclic covers branched at four points to characteristic 2, [1, Section 4].

Theorem 4.2 implies that there exists a smooth model of H over R, but the model H̄ which has a modular interpretation is not smooth.

8 Examples

Suppose that p > 5 and choose 2 ≠ d|(p − 1)/2. Let f : Y → ℙ_{ℂ}^{1} be a cover with Galois group PSL_{2}(p) branched at three points of order p, p, d. In this section we show that there exists an m|(p−1) and a type a such that f is isomorphic to the quotient of ϖ(a) : ℍ(a) → ℙ^{1} by the center of its Galois group. A similar result is shown in [5, Section 2.2] for PSL_{2}(p)-covers of ℙ_{ℂ}^{1} branched at three points of order p. In that case one may take m = 2 and a = (1, 1, 1, 1). The cover ϖ(a) is in this case isomorphic to X(2p) → X(2), where X(N) is the modular curve with full level-N structure.

Recall that SL_{2}(p) has two conjugacy classes of elements of order p, which we denote by pA and pB, [27, Lemma 3.27]. We make the convention that

( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ∈ pA.

Choose a primitive (p − 1)th root of unity ξ ∈ 𝔽_{p}^{×}. For 0 < i < (p − 1)/2, let C(i) = {A ∈ SL_{2}(p)| Tr(A) = ξ^{i} + ξ^{−i}}. Then C(i) is a conjugacy class of SL_{2}(p) of elements whose order is equal to the order of ξ^{i} in 𝔽_{p}^{×}. Moreover, every element of SL_{2}(p) of order n dividing p − 1 with n > 2 is contained in some C(i). For a triple of conjugacy classes C = (C_{1}, C_{2}, C_{3}) of SL_{2}(p), we write Ni_{3}^{in}(C) for the set of isomorphism classes of SL_{2}(p)-cover Y → ℙ_{ℂ}^{1} with class vector C. (This means that the canonical generator of some point of Y above x_{i} is contained in the conjugacy class C_{i}, with respect to some fixed system of roots of unity.)

Lemma 8.1 Let 0 < i < (p−1)/2. Then either | Ni_{3}^{in}(pA, pA, C(i))| = 0 and | Ni_{3}^{in}(pA, pB, C(i))| = 1, or | Ni_{3}^{in}(pA, pA, C(i))| = 1 and | Ni_{3}^{in}(pA, pB, C(i))| = 0.

Proof: The proof is similar to [28, Lemma 3.29]. □

The lemma says that the triples (pA, pA, C(i)) and (pA, pB, C(i)) are rigid. Since there is an outer automorphism of SL_{2}(p) which interchanges the conjugacy classes pA and pB and fixes C(i), the lemma may be rephrased as follows. For every 0 < i < (p − 1)/2, there exists a unique SL_{2}(p)-cover of ℙ_{ℂ}^{1} with class vector (p∗, p ∗ ∗, C(i)) up to isomorphism.

We will now construct types a such that ϖ(a) is ramified of order p, p, d, for some d|(p−1). The cover ϖ(a) is branched at 0 and 1 of order p if and only if a_{1} + a_{4} = a_{2} + a_{4} = m, by Proposition 3.8. Therefore the only types we need to consider are types of the form (a, a, m − a, m − a), for some m|(p − 1) with gcd(a, m) = 1.

Lemma 8.2 Suppose m ≠ 2 divides p − 1. Choose 0 < a < m such that gcd(m, a) = 1 and write a = (a, a, m − a, m − a). Then the Galois group of ϖ(a) is SL_{2}(p) if m is odd and PSL_{2}(p) if m is even.

Proof: We first note that p divides the order of the Galois group of ϖ(a), since the ramification index at 0 and 1 is p. Therefore we are not in the exceptional case, Definition 3.4.

We denote by B_{1}, B_{2}, B_{3} the matrices defined in (6)–(8). Then det(B_{1}) = det(B_{2}) = 1 and det(B_{3}) = (ξ^{αa})^{2}. Corollary 3.6 implies that the Galois group of ϖ(a) is SL_{2}(p) if m is odd and PSL_{2}(p) if m is even. □

For every type a, we can define an SL_{2}(p)-cover ϖ′(a). If m is odd, this is just ϖ(a). If m is even, we choose a lift of ϖ(a) to an SL_{2}(p)-cover of ℙ^{1} branched at three points. We may suppose that ϖ′(a) is branched of order p at 0 and 1. This uniquely characterizes the lift.

The following proposition states that there exists a type a such that ϖ′(a) has class vector (C_{1}, C_{2}, C(i)), for some choice of C_{1}, C_{2}. In other words, every SL_{2}(p)-cover with class vector (p∗, p∗, C(i)) is isomorphic to one of the covers ϖ′(a).

Proposition 8.3 Let 2 ≠ m|(p − 1) and a = (a, a, m − a, m − a). Write α = (p − 1)/m. Suppose a < m/2 and gcd(m, a) = 1. Then the class vector of ϖ′(a) : ℍ(a) → ℙ^{1} is (C_{1}, C_{2}, C(aα)), for some C_{1}, C_{2} ∈ {pA, pB}.

Proof: The Nielsen tuple of ϖ′(a) is (B_{1}, B_{2}, B′_{3}) with B′_{3} = B_{3}/ξ^{αa} ∈ SL_{2}(p). Therefore Tr(B′_{3}) = ξ^{αa} + ξ^{−αa} and B′_{3} ∈ C(αa). □

Corollary 8.4 Suppose d ≠ 2 divides (p − 1)/2. Let f : Y → ℙ_{ℂ}^{1} be a cover with Galois group PSL_{2}(p) which is branched only at 0, 1, ∞ with ramification index p, p, d, respectively. Then there exists a type a such that f is isomorphic to the quotient of ϖ(a) ⊗_{K} ℂ by the center of the Galois group G of ϖ(a).

Proof: This follows from Proposition 8.3. □

One may expect Corollary 8.4 to hold more generally. Let G = SL_{2}(p) and suppose that C = (C_{1}, C_{2}, C_{3}) is a triple of conjugacy classes of G. In general, C is not rigid, but it is still linearly rigid, [25, Example 2.4]. This means the following. Let (M_{1}, M_{2}, M_{3}) be a triple of matrices in GL_{2}(p) such that M_{i} ∈ C_{i} and M_{1}M_{2}M_{3} = 1. Then if (M′_{1}, M′_{2}, M′_{3}) is another such triple, there exists an N ∈ GL_{2}(p) such that NM′_{i}N^{−1} = M_{i} for all i. This was essentially already known to Riemann, see [13, Introduction].

It follows from Proposition 3.8 that the ramification indices of ϖ(a) are either p or divide p−1. Therefore there exist covers of ℙ^{1} branched at three points with Galois group PSL_{2}(p), SL_{2}(p) or PGL_{2}(p) which are not isomorphic to some ϖ(a). To obtain covers with ramification indices dividing p + 1 one should omit the condition p ≡ 1 mod m.

Comparison to Raynaud’s criterion for good reduction. In the rest of this paper, we want to compare Theorem 4.2 to Raynaud’s criterion for good reduction, [21]. We illustrate that our result does not follow from Raynaud’s Criterion.

Suppose that m = 5 and p ≡ 1 mod 5. Choose a primitive 5th root of unity ζ_{5} ∈ 𝔽_{p}^{×}. For 1 ≤ i ≤ 4, we write a_{i} = (i, i, i, −3i). One checks that a_{i} is non-exceptional. Therefore ϖ_{i} := ϖ(a_{i}) is a Galois cover with Galois group SL_{2}(p), Corollary 3.6. Proposition 3.8 implies that ϖ_{i} is branched at three points of order 5.

Let 5A (resp. 5B) be the conjugacy class of SL_{2}(p) of matrices with trace ζ_{5} +ζ_{5}^{4} (resp. ζ_{5}^{2} +ζ_{5}^{3}). Write C_{A} = (5A, 5A, 5A) and C_{B} = (5B, 5B, 5B). As before, we denote by Ni_{3}^{in}(C) the Nielsen class.

Lemma 8.5 (a) We have ϖ_{1}, ϖ_{4} ∈ Ni_{3}^{in}(C_{B}) and ϖ_{2}, ϖ_{3} ∈ Ni_{3}^{in}(C_{A}).

(b) The sets Ni_{3}^{in}(C_{A}) and Ni_{3}^{in}(C_{B}) both contain exactly two elements.

Proof: For each type, we defined matrices B_{1}, B_{2}, B_{3} in (6)–(8). We denote these matrices corresponding to the type a_{i} by B_{1}(i), B_{2}(i), B_{3}(i).

The class vector of ϖ_{i} is described by matrices B′_{1}(i), B′_{2}(i), B′_{3}(i) which are characterized by the following two properties:

• B′_{j}(i) = B_{j}(i) · γ_{j}(i)I, for some scalar matrix γ_{j}(i)I ∈ M, such that det(B′_{j}(i)) = 1,

• the order of B′_{j}(i) is 5.

It follows that γ_{j}(i) = ζ_{5}^{−i}, for every j. Part (a) follows from this. The triples C_{A} and C_{B} are linearly rigid, [25, Example 2.4]. The outer automorphism group of SL_{2}(p) has order two and fixes the conjugacy classes of order five. Therefore, the Nielsen class has either zero or two elements and (b) follows from (a). □

In SL_{2}(p) there are two conjugacy classes of elements of order 5. Therefore the triples C_{A} and C_{B} are rational over ℚ(\sqrt{5}), [24, Section 7.1]. Lemma 8.5.(b) implies that the SL_{2}(p)-covers ϖ_{i} are defined over some extension K_{i} of ℚ(\sqrt{5}) of degree at most two. Let ℘ be a prime of K_{i} above p and write e_{i} for the ramification index of ℘. Since the degree of K_{i} over ℚ(\sqrt{5}) is at most two, we have e_{i} ≤ 2. Raynaud’s Criterion for good reduction states in this case that if ϖ_{i} has bad reduction then e_{i} is greater than or equal to the number of Sylow p-subgroups in SL_{2}(p), i.e. e_{i} ≥ 2. Therefore we cannot conclude directly whether ϖ_{i} has good reduction or not, unless we have more information on the field of definition K_{i} of ϖ_{i}. But such information is hard to obtain for triples which are not rigid. In what follows we take an alternative approach. We use Theorem 4.2 to decide whether ϖ_{i} has good reduction and deduce from this information on the field of definition K_{i}.

The following proposition is essentially the statement of Raynaud’s criterion for good reduction, in our special case.

Proposition 8.6 The cover ϖ_{i} has good reduction at ℘ if and only if e_{i} = 1.

Proof: If ϖ_{i} has good reduction at ℘ it follows from Beckmann’s Theorem that e_{i} = 1, [2]. Suppose that ϖ_{i} has bad reduction. Raynaud’s Criterion [21] implies that e_{i} ≥ 2. In [21] the assumption is made that the Galois group G has trivial center, but this is not essential. □

Proposition 8.7 (a) Let f : Y → ℙ^{1} be an SL_{2}(p)-cover with class vector C_{A}. Then f has bad reduction.

(b) Let f : Y → ℙ^{1} be an SL_{2}(p)-cover with class vector C_{B}. Then f has good reduction.

Proof: Lemma 8.5 implies that, over ℂ, there are two covers with class vector C_{A} and two with class vector C_{B}. Write Ni_{4}^{in}(C_{B}) = {f_{1}, f_{4}} and Ni_{4}^{in}(C_{A}) = {f_{2}, f_{3}}. Let K/ℚ be a field over which all f_{i} are defined. The definition of the type a_{i} depends on a choice of primitive 5th root of unity ξ_{5} ∈ K. Recall that we have also fixed a primitive 5th root of unity ζ_{5} ∈ 𝔽_{p}^{×}. Choose a prime ideal ℘ of K above p such that ξ_{5} ≡ ζ_{5} mod ℘.

We may assume that f_{1} = ϖ_{1} and f_{2} = ϖ_{2}. Theorem 4.2 implies that f_{1} has good reduction and f_{2} has bad reduction at ℘. It follows from Proposition 8.6 that f_{2} is defined over a field K_{2} such that ℚ(\sqrt{5}) ⊂ K_{2} ⊂ K in which the ramification index of ℘∩K_{2} is two. Let σ_{2} be a generator of Gal(K_{2}, ℚ(\sqrt{5})). Then f_{3} = f_{2}^{σ}. we conclude that f_{3} has bad reduction at ℘ also.

The Galois group of ℚ(\sqrt{5})/ℚ permutes the class vectors C_{A} and C_{B}, therefore the covers f_{1}, f_{2}, f_{3}, f_{4} are conjugated over K by an automorphism of order four. It follows that we may take K to be the Galois closure of K_{2}/ℚ. The Galois group of K/ℚ is a transitive group on four letters which contains an element of order four and has a quotient of order two, i.e. Gal(K, ℚ) is either cyclic of order four or a dihedral group of order eight.

Suppose Gal(K, ℚ) ≃ ℤ/4. Then K_{2}/ℚ is Galois and f_{1} is defined over K_{2}. Moreover, f_{1}^{σ} = f_{4}. This implies that K_{2} is a minimal field of definition for f_{1}. Since ℘ is ramified in K_{2}, Proposition 8.6 implies that f_{1} has bad reduction at ℘ contradicting Theorem 4.2. We conclude that Gal(K, ℚ) is a dihedral group of order eight. We represent the elements of Gal(K, ℚ) as permutations on {1, 2, 3, 4}. It follows that K_{2} is the subfield of K of invariants under (1 4).

The covers f_{1} and f_{4} may be defined over the subfield K_{1} of K of invariants of (2 3). The permutation (1 4)(2 3) restricts to an automorphism of K_{1} which permutes f_{1} and f_{4}, therefore K_{1} is a minimal field of definition of f_{1}. Since f_{1} has good reduction at ℘, we conclude that ℘ ∩ K_{1} is unramified in K_{1}. Therefore f_{4} has good reduction at ℘ also.

We remark that the choice of ℘ depends on the choice of the 5th root of unity ξ_{5} ∈ K. The permutation (1 2 4 3) ∈ Gal(K, ℚ) sends ℘ to a different prime ℘′ and also changes the class vector of a cover f_{i}. If f_{i} has good reduction at ℘ then f_{i} has bad reduction at ℘′, and conversely. □

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