Galois covers of degree p : semi-stable reduction and Galois action

Mohamed Saïdi

Abstract

In this paper we study the semi-stable reduction of Galois covers of degree p above semistable curves over a complete discrete valuation ring of inequal chracteristics (0, p). We are also able to describe the Galois action on these covers in terms of some geometric and combinatorial data in characteristic p endowed with the action of the Galois group of the residue field.

0. Introduction. In this paper we study the semi-stable reduction of Galois covers of degree p above semi-stable curves over a complete discrete valuation ring of inequal chracteristics (0, p), as well as the Galois action on these data.

More precisely, let R be a complete discrete valuation ring of inequal chracteristics (0, p), with fraction field K, and residue field k. In the first part of the paper we consider the following local situation. Let f : Y → X be a Galois cover of degree p between formal germs of R-curves at the closed points y and x respectively, with Y normal, and X semistable i.e. X is the formal germ of an R-curve at a closed point x which is either a smooth point or an ordinary double point. It follows from the theorem of semi-stable reduction for curves that the germ Y admits potentially a semi-stable reduction i.e. there exists, after eventually a finite extension of R, a semi-stable model Ỹ → Y which is “essencially” unique. In particular, the Galois group of the cover f acts on Ỹ and the quotient X̃ of Ỹ by this action is a semi-stable blow-up of X. We have a canonical morphism f̃: Ỹ → X̃ which is Galois of degree p. To the above cover f : Y → X we associate canonically some degeneration data which determine completely the special fibre Ỹ_{k} := Ỹ ×_{R} k of Ỹ. These degeneration data consist of the (canonically marked) tree associated to the special fibre X̃_{k} := X̃ ×_{R} k of X̃, plus some geometric data which consists of a torsor f_{i} : V_{i} → U_{i} under a finite and flat group scheme of rank p above each irreducible marked component U_{i} of X̃_{k} such that the “conductors” of these torsors at the marked points satisfy certain “compatibility” conditions (these are what we called in [Sa] Kummerian mixed torsors) (cf.

1.2, 1.3 for a more precise definition). For an illustration of this situation we refeer to the example 1.2.1. Let’s denote by Deg_{p} the set of isomorphism classes of such degeneration data which are defined over an algebraic closure \overline{k} of k (cf. 1.2.2, 1.3.1 for a precise definition). We show that the set Deg_{p} is a G_{k}-set, where G_{k} is the Galois group of the separable closure of k contained in \overline{k} (cf. 1.2.3). Let \overline{K} be an algebraic closure of K, and let Spec L be the geometric generic point of X. The étale cohomology group H_{et}^{1}(Spec L, μ_{p}) classifies (pointed)-geometric Galois covers of X of degree p. Moreover our result which exhibit the above degeneration data can be translated as the existence of a canonical specialisation map Sp : H_{et}^{1}(Spec L, μ_{p}) → Deg_{p}. Note that both H_{et}^{1}(Spec L, μ_{p}) and Deg_{p} are G_{K}-sets, where G_{K} is the Galois group of \overline{K} over K, and G_{K} acts on Deg_{p} via its canonical quotient G_{k}. Our main result in this part of the paper is the following:

Theorem (1.2.7, 1.3.2). The above specialisation map Sp : H_{et}^{1}(Spec L, μ_{p}) → Deg_{p} is surjective and G_{K}-equivariant.

In other words the association of degeneration data to a cover f : Y → X as above is compatible with the action of the Galois group on covers and on degeneration data, Moreover one has a realisation result for such degeneration data (cf. example 1.2.5 which illustrates the realistaion of degeneration data). The proof of the above result relies heavily on the results in [Sa] and [Sa-1]. We also use formal patching techniques à la Harbater. As an application of the above result we construct in 2.1.8 an example of a covers f : Y → X as above where X is smooth at the closed point x, and where the special fibre Y_{k} := Y ×_{R} k of Y is singular and unibranche at the closed point y, and such that the configuration of the special fibre of the semi-stable model Ỹ of Y is not a tree-like. This indeed answers a question raised by Lorenzini whether such a situation can occur in the inequal characteristic case.

In the second part of the paper we study the global situation of a Galois cover f : Y → X of degree p above a proper and semi-stable R-curve X with Y normal. As above, and using the theorem of semi-stable reduction for curves (cf. [De-Mu]), one obtains after eventually a finite extension of R a semi-stable model Ỹ → Y of Y and a Galois cover f : Ỹ → X̃ of degree p, where X̃ is a semi-stable blow-up of X. We associate then canonically to the cover f : Y → X some “degeneration data” which determine completely the special fibre Ỹ_{k} := Ỹ ×_{R} k of the semi-stable model Ỹ of Y (cf. 2.2). This consists of the graph associated to the semi-stable k-curve X_{k}, and a given “mixed torsor” above X_{k}, plus given local degeneration data at the critical points of this mixed torsor (cf. 2.2.1 for a more precise definition). For an illustration of this we refeer to the example 2.2.3. Let’s denote by Deg_{p}(X_{k}) the set of isomorphism classes of such degeneration data which are defined over \overline{k}. We show that this set is a G_{k}-set in a canonical way. Let \overline{η} be the geometric generic point of X. The étale cohomology group H_{et}^{1}(\overline{η}, μ_{p}) classifies (pointed)- geometric Galois covers of X of degree p. Moreover our result which exhibit the above degeneration data can be translated as the existence of a canonical specialisation map Sp : H_{et}^{1}(\overline{η}, μ_{p}) → Deg_{p}(X_{k}). Note that both H_{et}^{1}(\overline{η}, μ_{p}) and Deg_{p}(X_{k}) are G_{K}-sets, where G_{K} is the Galois group of \overline{K} over K, and G_{K} acts on Deg_{p} via its canonical quotient G_{k}. Our main result in the second part of the paper is the following:

Theorem (2.3.2). The above specialisation map Sp : H_{et}^{1}(\overline{η}, μ_{p}) → Deg_{p}(X_{k}) is G_{K}- equivariant.

The above specialisation map Sp can not be surjective in this global case. However we prove the following result of realisation of degeneration data:

Theorem (2.3.1). Let X_{k} be a proper and semi-stable k-curve. Let R be a complete discrete valuation ring of inequal characteristics with residue field k, and fractions field K. Suppose given a \overline{k}-degeneration data deg(X_{k}) of rank p associated to X_{k} (in other words suppose given an element of Deg_{p}(X_{k})). Then there exists, after eventually a finite extension of R, a proper and semi-stable R-curve X̃ with smooth generic fibre and a special fibre X̃_{k} := X̃ ×_{R} k isomorphic to X_{k}, and a Galois cover f̃: Ỹ → X̃, such that the degeneration data associated to f is isomorphic to the given data deg(X_{k}).

We refeer to the example 2.3.4 which illustrate the realisation of global degeneration data. Finally, I believe the ideas presented in this paper provide the framework to construct Hurwitz spaces over ℤ whose fibre in characteristic zero classifies Galois covers of degree p plus extra data. Also in the case of Galois covers of degree p of the projective line over a p-adic field these results should lead to an algorithm which compute the semi-stable reduction of these covers at least in the case where the number of branched points is smaller than p.

I. Semi-stable reduction of Galois covers of degree p above formal germs of curves with Galois action.

1.0. In what follows we use the following notations: R is a complete discrete valuation ring of inequal characteristics, with residue characteristic p > 0, and which contains a primitive p-th root of unity ζ. We denote by K the fraction field of R, π a uniformising parameter, and k the residue field of R. Let \overline{K} be a fixed algebraic closure of K, and let G_{K} be the Galois group of \overline{K} of K. Let \overline{R} be the integral closure of R in \overline{K} which is a valuation ring, and let \overline{k} be the residue field of \overline{R} which is an algebraic closure of k. Let G_{k} be the Galois group of a separable closure of k contained in \overline{k}. We have the following canonical exact sequence, where I_{K} denotes the inertia subgroup:

0 → I_{K} → G_{K} → G_{k} → 0

In this section we will consider a formal germ X of a semi-stable R-curve at a closed point, a Galois cover f : Y → X with group G = ℤ/pℤ, and we will study the semi-stable reduction of Y, as well as the action of G_{K} on these data.

1.1. Let X := Spec Ô_{X,x} be the formal germ of an R-curve X at a closed point x, and let f : Y → X be a Galois cover with group G ≃ ℤ/pℤ, such that Y is normal and local. It follows then easily from the theorem of semi-stable reduction for curves (cf. [De-Mu], as well as the compactification process in [Sa-1] 2.3), that after eventually a finite extension R′ of R with residue field k′, and fractions field K′, the formal germ Y has a semi-stable reduction. More precisely there exists a birational and proper morphism f̃: Ỹ → Y′, where Y′ is the normalisation of Y ×_{R} R′, such that Ỹ_{K′} ≃ Y′_{K′}, and the following conditions hold:

(i) The special fibre Ỹ_{k′} := Ỹ ×_{Spec R′} Spec k′ of Ỹ is reduced. (ii) Ỹ_{k} has only ordinary double points as singularities.

Moreover there exists such a semi-stable model f̃ : Ỹ → Y′ which is minimal for the above properties. In particular the action of G on Y′ extends to an action on Ỹ. Let X̃ be the quotient of Ỹ by G, which is a semi-stable model of X. One has the following commutative diagram:

Ỹ ⟶^{f̃} Y′
g↓  ^{f′}↓
X̃ ⟶^{g̃} X′

With the same notations as above, one can moreover choose the semi-stable models Ỹ and X̃ such that the set of points B_{K′} := {x_{i,K′}}_{1≤i≤r}, consisting of the branch locus in the morphism Y′_{K′} → X′_{K′}, which we may assume to be rational, specialise in smooth (resp. smooth distincts) points of X′_{k′}. We may also, after eventually a finite extension of K, suppose that the double points of X̃_{k} are rational. Moreover one can choose such X̃ and Ỹ which are minimal for these properties. We will denote by f̃^{nsp} : Ỹ^{nsp} → Y′ (resp. f̃^{sp} : Ỹ^{sp} → Y′) the minimal semi-stable model of f such that the points of B_{K} specialise in smooth (resp. smooth distincts) points of X̃^{nsp} := Ỹ^{nsp}/G (resp. of X̃^{sp} := Ỹ^{sp}/G). We call Ỹ^{nsp} (resp. Ỹ^{sp}) the minimal non split (resp. split) semi-stable model of Y. We have the following commutative diagrams:

Ỹ^{nsp} ⟶^{f̃^{nsp}} Y′
g^{nsp}↓  ^{f′}↓
X̃^{nsp} ⟶^{g̃^{nsp}} X′

and

Ỹ^{sp} ⟶^{f̃^{sp}} Y′
g^{sp}↓  ^{f′}↓
X̃^{sp} ⟶^{g̃^{sp}} X′

Moreover the morphism f̃^{sp} : Ỹ^{sp} → Y′ factors throught f̃^{nsp} as follows: Ỹ^{sp} → Ỹ^{nsp} → Y′, and the later commutative diagram factors through the first one. In the case where X is the formal germ of a semi-stable R-curve at a closed point x, the fibre (g̃^{nsp})^{−1}(x) (resp. (g̃^{sp})^{−1}(x)) of the closed point x in X̃^{nsp} (resp. in X̃^{sp}) is a tree Γ^{nsp} (resp. Γ^{sp}) of projective lines. This tree is canonically endowed with some “degeneration data” that we will exhibit below, and which follow mainly from the results in [Sa] and [Sa-1].

1.2. We will use the same notations as in 1.1. We first consider the case where X ≃ Spf A is the formal germ of a semi-stable R-curve at a smooth point x. Let R′ be a finite extension of R as in 1.1, and let π′ be a uniformiser of R′. We assume that X is geometrically connected, in which case Y is also geometrically connected. Below we exhibit the degeneration data associated to the non split (resp. split) semi-stable reduction of Y, and which are consequences of the results in [Sa] and [Sa-1].

Deg.1. Let ℘ := (π′) be the ideal of A′ := A ⊗_{R} R′ generated by π′, and let Â′_{℘} be the completion of the localisation of A′ at ℘. Let X′_{η} := Spf Â′_{℘} be the boundary of X′, and let X′_{η} → X′ be the canonical morphism. Consider the following cartesian diagram:

Y_{η} ⟶^{f_{η}} X′_{η}
↓  ↓
Y′ ⟶^{f′} X′

Then f_{η} : Y_{η} → X′_{η} is a torsor under a commutative finite and flat R′-group scheme G_{R′} of rank p. Let δ be the degree of the different associated to the torsor f_{η}. One has the degeneration type (G_{k′}, m, h) of the torsor f_{η} (cf. [Sa] 3.2) which is canonically associated to f. The geometric genus g_{y} of the point y equals (r − m − 1)(p − 1)/2 (cf. [Sa-1] 3.1.1), where r is the cardinality of B_{K′}.

Deg.2. The fibre (g̃^{nsp})^{−1}(x) (resp. (g̃^{sp})^{−1}(x)) of the closed point x of X′ in X̃^{nsp} (resp. in X̃^{sp}) is a tree Γ^{nsp} (resp. Γ^{sp}) of projective lines, Γ^{nsp} is a sub-tree of Γ^{sp}. Let Vert(Γ^{nsp}) := {X_{i}}_{i∈I^{nsp}} (resp. Vert(Γ^{sp}) := {X_{i}}_{i∈I^{sp}}) be the set of irreducible components of (g̃^{nsp})^{−1}(x) (resp. of (g̃^{sp})^{−1}(x)), which are also the vertices of the tree Γ^{nsp} (resp. Γ^{sp}). The tree Γ^{sp} (hence also Γ^{nsp}) is canonically endowed with an origin vertex X_{i_{0}}, which is the unique irreducible component of (g̃^{sp})^{−1}(x) which meets the point x. We fix an orientation of the tree Γ^{sp} starting from X_{i_{0}} in the direction of the ends. Such an orientation induces of course an orientation of the subtree Γ^{nsp}.

Deg.3. For each i ∈ I^{nsp} (resp. I^{sp}), let {x_{i,j}}_{j∈S_{i}} be the set of points of X_{i} in which specialise some points of B_{K′} (S_{i} may be empty), say in each point x_{i,j} specialise r_{i,j} points of B_{K}. If i ∈ I^{sp}, and S_{i} is non empty, we have r_{i,j} = 1. Also let {z_{i,j}}_{j∈D_{i}} be the set of points of the irreducible component X_{i} where X_{i} meets the rest of the components of X̃_{k′}^{nsp} (resp. of X̃_{k′}^{sp}). These are the double points of X̃_{k′}^{nsp} (resp. X̃_{k′}^{sp}) supported by X_{i}. We denote by B_{k′}^{nsp} (resp. B_{k′}^{sp}) the set of all points ∪_{i∈I^{nsp}}{x_{i,j}}_{j∈S_{i}} (resp. ∪_{i∈I^{sp}}{x_{i,j}}_{j∈S_{i}}), which is the set of specialisation of the branch locus B_{K′}, and by D_{k′}^{nsp} (resp. D_{k′}^{sp}) the set of double points of X̃_{k′}^{nsp} (resp. X̃_{k′}^{sp}). In particular x_{i_{0},j_{0}} := x is a double point of X̃_{k′}^{nsp} (resp. of X̃_{k′}^{sp}).

Deg.4. Let U^{nsp} := X^{nsp} − {B_{k′}^{nsp} ∪ D_{k′}^{nsp}} (resp. U^{sp} := X^{sp} − {B_{k′}^{sp} ∪ D_{k′}^{sp}}). The cover f induces a mixed torsor f^{nsp} : V → U^{nsp} above U^{nsp} (resp. f^{sp} : V′ → U^{sp} above U^{sp}), whose special fibre f_{k′}^{nsp} : V_{k′} → U_{k′}^{nsp} (resp. f_{k′}^{sp} : V′_{k′} → U_{k′}^{sp}) is an element of H_{fppf}^{1}(U_{k′}^{nsp})_{p} (resp. an element of H_{fppf}^{1}(U_{k′}^{sp})_{p}) (cf. [Sa] 1.6 for the definition of H_{fppf}^{1}( )_{p}). In particular the restriction of f^{nsp} (resp. f^{sp}) to each connected component U_{i} of U^{nsp} (resp. U^{sp}) is a torsor f_{i} : V_{i} → U_{i} under a commutative finite and flat R′-group scheme G_{R′,i} of rank p, and f_{i,k′} : V_{i,k′} → U_{i,k′} := U_{i} ×_{R′} k′ is a torsor under the k′-group scheme G_{k′,i} := G_{R′,i} ×_{R′} k′. Moreover if G_{k′,i} is radicial, and if ω_{i} is the associated differential form, then the set of zeros and poles of w_{i} is necessarily contained in {x_{i,j}}_{j∈S_{i}} ∪{z_{i,j}}_{j∈D_{i}}, as V_{i,k′} is smooth (cf. [Sa], I).

Deg.5. Each smooth point x_{i,j} ∈ B_{k′}^{nsp} (resp. ∈ B_{k′}^{sp}) is endowed via f with degeneration data on the boundary of the formal fibre at x_{i,j}, as in Deg.1 above, and which satisfy certain compatibility conditions. More precisely for each smooth point x_{i,j} we have the reduction type (G_{k′,i}, m_{i,j}, h_{i,j}) on the boundary of the formal fibre at this point, induced by g^{nsp} (resp. g^{sp}), and such that r_{i,j} = m_{i,j} + 1, as a consequence of [Sa-1] 3.1.1. Also if x_{i,j} ∈ B_{k′}^{sp} (in other words if i ∈ I^{sp}, and S_{i} is non empty), then necessarily G_{k′,i} = μ_{p}, m_{i,j} = −1 and h_{i,j} = 0 (cf. [Sa-1, 3.1.2).

Deg.6. Each double point z_{i,j} = z_{i′,j′} ∈ X_{i} ∩ X_{i′} of X̃^{nsp} (resp. X̃^{sp}) is endowed with degeneration data (G_{i,k′}, m_{i,j}, h_{i,j}) and (G_{i′,k′}, m_{i′,j′}, h_{i′,j′}), induced by g^{nsp} (resp. g^{sp}), on the two boundaries of the formal fibre at this point as in Deg.1, such that m_{i,j} +m_{i′,j′} = 0, and h_{i,j} + h_{i′,j′} = 0, as a consequence of [Sa-1] 3.2.3. In particular m_{i_{0},j_{0}} + m = 0, and h_{i_{0},j_{0}} + h = 0. In other words the above induced by f element of H_{fppf}^{1}(U_{k′}^{nsp})_{p} (resp. of H_{fppf}^{1}(U_{k′}^{sp})_{p}) is indeed an element of H_{fppf}^{1}(U_{k′}^{nsp})_{p}^{kum} (resp. an element of H_{fppf}^{1}(U_{k′}^{sp})_{p}^{kum}) (cf. [Sa] 1.7 for the definition of H_{fppf}^{1}( )^{kum}).

Deg.7. For each double point z_{i,j} = z_{i′,j′} ∈ X_{i} ∩ X_{i′} of X̃^{nsp} (resp. X̃^{sp}), let e_{i,j} be the thikeness of z_{i,j}. Then e_{i,j} = pt_{i,j} is necessarily divisible by p. For each irreducible component X_{i}, i ∈ I^{nsp} (resp. i ∈ I^{sp}), let η_{i} be the generic point of X_{i}. Let δ_{i} be the degree of different above η_{i} in the cover g^{nsp} (resp. g^{sp}). Then we have δ_{i} − δ_{i′} = t_{i,j}m_{i,j}(p − 1) as follows from [Sa-1] 3.2.5. In particular δ − δ_{i_{0}} = t_{i_{0},j_{0}}m(p − 1), and |D_{i}|δ_{i} = ∑_{j∈D_{i}}(δ_{j} + t_{i,j}m_{i,j}(p − 1)), where |D_{i}| denotes the cardinality of D_{i}.

Deg.8. The contribution to the arithmetic genus g_{y} of the point y was computed in [Sa-1] 3.1.1, in terms of the degeneration data (G_{k′}, m, h) on the boundary of the formal fibre at x, and the cardinality r of B_{K}. It follows also from the above considerations and after easy calculation that: g_{y} = ∑_{i∈I_{et}}(−2 + ∑_{j∈S_{i}}(m_{i,j} + 1) + ∑_{j∈D_{i}}(m_{i,j} + 1))(p − 1)/2, where I_{et} denote the subset of I^{nsp}, or I^{sp}, which among to the same, consisting of those i for which G_{k′,i} is étale.

1.2.1. Example. In the following we give an example where one can exhibit the degeneration data associated to a Galois cover f : Y → X of degree p where X ≃ Spf R[[T]] is the formal germ of a smooth point. More precisely, for m > 0 an integer prime to p consider the cover given generically by the equation X^{p} = 1 + λ^{p}(T^{−m} + πT^{−m+1}) (this is the example 1 in [Sa-1], 3.2.4, with m′ = m + 1). Hier r = m + 2 and this cover has a reduction of type (ℤ/pℤ, m, 0) on the boundary. In particular the geometric genus g_{y} of the closed point y of Y equals (p − 1)/2. The non split degeneration data associated to the above cover consists necessarily of a treee with only one vertex and no edges, i.e. a unique projective line X_{1} with a marked \overline{k}-point x_{1}, and an étale torsor f_{1} : V_{1} → U_{1} := X_{1} −\{x_{1}\} above U_{1} with conductor 2 at the point x_{1}.

The above considerations leads naturally to the following abstract geometric and combinatorial definition of degeneration data.

1.2.2. Definition. Let k′ be an algebraic extension of k. A k′-simple non split (resp. split) degeneration data Deg(x) of type (r, (G_{k′}, m, h)) and rank p consists of the following data:

Deg.1. G_{k′} is a commutative finite and flat k′-group scheme of rank p, r ≥ 0 is an integer, m is an integer prime to p such that r − m − 1 ≥ 0, and h ∈ 𝔽_{p} equals 0 unless G_{k′} = μ_{p} and m = 0.

Deg.2. Γ := X_{k′} is an oriented tree of projective lines with vertices Vert(Γ) := {X_{i}}_{i∈I} (resp. Vert(Γ) := {X_{i}}_{i∈I^{sp}}), endowed with an origin vertex X_{i_{0}}, and a marked smooth point x := x_{i_{0},j_{0}} on X_{i_{0}} which is a k′-rational point. We denote by {z_{i,j}}_{j∈D_{i}} the set of double points, or (non oriented) edges, of Γ which are supported by X_{i}, and which we assume to be k′-rational points.

Deg.3. For each vertex X_{i} of Γ is given a set, may be empty, of smooth marked k′-rational points {x_{i,j}}_{j∈S_{i}}.

We will summarise the data Deg.2 and Deg.3 by saying that we are given a k′-rational marked tree of projective lines, or a marked semi-stable k′-curve with arithmetic genus 0.

Deg.4. For each i ∈ I (resp. i ∈ I^{sp}), is given a torsor f_{i,k′} : V_{i,k′} → U_{i,k′} := X_{i,k′} − {{x_{i,j}}_{j∈S_{i}} ∪{z_{i,j}}_{j∈D_{i}}} under a commutative finite and flat k′-group scheme G_{k′,i} of rank p, where X_{i,k′} := X_{i} ×_{Spec k} Spec k′, with V_{i,k′} smooth. Let U_{k′} be the open subscheme of X_{k′} obtained by deleting the double and marked points of X_{k′}. Then the above data determines an element of H_{fppf}^{1}(U_{k′})_{p} (cf. [Sa], I). Moreover if G_{k′,i} is radicial, and if ω_{i} is the associated differential form, then the set of zeros and poles of w_{i} is necessarily contained in {x_{i,j}}_{j∈S_{i}} ∪{z_{i,j}}_{j∈D_{i}}, as V_{i,k′} is smooth. Also if i ∈ I^{sp}, and S_{i} is non empty, we assume that G_{i,k′} = μ_{p}.

Deg.5. For each i ∈ I (resp. i ∈ I^{sp}), are given for each j ∈ S_{i} the integres (m_{i,j}, h_{i,j}), where m_{i,j} is the conductor of the torsor f_{i} at the point x_{i,j}, and h_{i,j} its residue at this point. Let r_{i,j} := m_{i,j} + 1. If i ∈ I^{sp} then we assume that r_{i,j} = 1. Also for the marked point x_{i_{0},j_{0}} is given the data (m_{i_{0},j_{0}}, h_{i_{0},j_{0}}), such that m_{i_{0},j_{0}} + m = 0, and h_{i_{0},j_{0}} + h = 0.

Deg.6. For each double point z_{i,j} = z_{i′,j′} ∈ X_{i} ∩ X_{i′} is given the pair (m_{i,j}, h_{i,j}) (resp. (m_{i′,j′}, h_{i′,j′})), where m_{i,j} (resp. m_{i′,j′}) is the conductor of the torsor f_{i,k′} (resp. f_{i′,k′}) at the point z_{i,j} (resp. z_{i′,j′}), and h_{i,j} (resp. h_{i′,j′}) its residue at this point. These data must satisfy m_{i,j} + m_{i′,j′} = 0, and h_{i,j} + h_{i′,j′} = 0. In other words the element of H_{fppf}^{1}(U_{k′})_{p} determined by the data Deg.4 must belong to the subgroup H_{fppf}^{1}(U_{k})_{p}^{kum} (cf. [Sa]. I).

Deg.7. For each irreducible component X_{i} of Γ is given an integer δ_{i} ≤ v_{K}(p) which is divisible by p − 1. For each double point z_{i,j} = z_{i′,j′} ∈ X_{i} ∩ X_{i′} of Γ is given an integer e_{i,j} = pt_{i,j} divisible by p, such that with the same notations as above we have δ_{i} − δ_{i′} = m_{i,j}t_{i,j}(p − 1).

Deg.8. Let I_{et} denote the subset of I (resp. of I^{sp}) consisting of those i for which G_{k′,i} is étale. Then the following equality should hold: (r − m − 1)(p − 1)/2 = ∑_{i∈I_{et}}(−2 + ∑_{j∈S_{i}}(m_{i,j} + 1) + ∑_{j∈D_{i}}(m_{i,j} + 1))(p − 1)/2. The integer g_{x} := (r − m − 1)(p − 1)/2 is called the genus of the degeneration data Deg(x).

There is a natural notion of isomorphism of simple split (resp. non split) degeneration data of a given type over a given algebraic extension of k. We will denote by SS-Deg_{p} (resp. SNS-Deg_{p}) the set of isomorphism classes of \overline{k}-simple split (resp. k′-simple non split) degeneration data of rank p.

1.2.3. Galois action on degeneration data.

The Galois group G_{k} of the separable closure of k contained in \overline{k} acts in a canonical way on the set SS-Deg_{p} (resp. SNS-Deg_{p}) of isomorphism classes of \overline{k}-simple split (resp. non split) degeneration data of rank p, in a way that is compatible with the group law on G_{k}. In other words SS-Deg_{p} and SNS-Deg_{p} are G_{k}-sets. More precisely, to a \overline{k}-simple split (resp. non split) degeneration data Deg(x) of type (r, (G\overline{k}, m, h)), and an element σ ∈ G_{k}, we associate the simple degeneration data Deg(x)^{σ} of the same type (r, G\overline{k}, m, h)). We explain briefly the defining data for Deg(x)^{σ}:

Deg^{σ}.1. The data Deg^{σ}.1 are the same as in Deg.1.

Deg^{σ}.2. and Deg^{σ}.3. The \overline{k}-rational marked tree Γ^{σ} is the marked tree Γ.

Deg^{σ}.4. For each i ∈ I (resp. i ∈ I^{sp}), is given the torsor f_{i}^{σ} : \overline{V} _{i}^{σ} → \overline{U}_{i} := \overline{X}_{i} − \{\{x_{i,j}\}_{j∈S_{i}} ∪ \{z_{i,j}\}_{j∈D_{i}}\} under the commutative finite and flat \overline{k}-group scheme G_{\overline{k},i} of rank p, where \overline{X}_{i} := X_{i} ×_{k} \overline{k}, and f_{i}^{σ} is the image of the torsor f_{i} above \overline{U}_{i} given by the data Deg.4, via the action of σ on torsors (cf. [Sa], I). In other words the element (f′_{t})_{t} of H_{fppf}^{1}(U\overline{k})_{p} determined by the data Deg^{σ}.4 is the transform (f_{t})_{t}^{σ} of the element (f_{t})_{t} of H_{fppf}^{1}(U\overline{k})_{p} determined by the data Deg.4, under the canonical action of σ on H_{fppf}^{1}(U\overline{k})_{p} (cf. [Sa], I).

Deg^{σ}.5. For each i ∈ I (resp. i ∈ I^{sp}), are given for each j ∈ S_{i} the integers (m_{i,j}^{σ}, h_{i,j}^{σ}), where m_{i,j}^{σ} is the conductor of the torsor f_{i}^{σ} at the point x_{i,j}^{σ} of S_{i}^{σ}, and h_{i,j}^{σ} its residue at this point. We have m_{i,j}^{σ} = m_{i,j} and h_{i,j}^{σ} = h_{i,j}.

Deg^{σ}.6. For each double point z_{i,j}^{σ} = z_{i′,j′}^{σ} ∈ X_{i} ∩ X_{i′} is given the pair (m_{i,j}^{σ}, h_{i,j}^{σ}) (resp. (m_{i′,j′}^{σ}, h_{i′,j′}^{σ})), where m_{i,j}^{σ} (resp. m_{i′,j′}^{σ}) is the conductor of the torsor f_{i}^{σ} (resp. f_{i′}^{σ}) at the point z_{i,j}^{σ} (resp. z_{i′,j′}^{σ}), and h_{i,j}^{σ} (resp. h_{i′,j′}^{σ}) its residue at this point. We have m_{i,j}^{σ} = m_{i,j} and h_{i,j}^{σ} = h_{i,j}.

Deg^{σ}.7. For each double point z_{i,j}^{σ} = z_{i′,j′}^{σ} ∈ X_{i}^{σ}∩X_{i′}^{σ} of Γ^{σ} is given the integer e_{i,j}^{σ} = pt_{i,j}^{σ} divisible by p. For each irreducible component X_{i} of Γ is given the integer δ_{i}^{σ} ≤ v_{K}(p) which is divisible by p − 1, such that with the same notations as above δ_{i}^{σ} − δ_{i′}^{σ} = m_{i,j}^{σ}t_{i,j}^{σ}(p − 1). We have e_{i,j}^{σ} = e_{i,j}, and δ_{i}^{σ} = δ_{i}.

Deg^{σ}.8. The integer g_{x} := (r − m − 1)(p − 1)/2 is the genus of the degeneration data D_{x}^{σ}.

In conclusion, the action of the group G_{k} on the set of isomorphism classes of simple degeneration data is given, and completely determined, through the canonical action of G_{k} on the data Deg.4, which is essencially given by the canonical action of G_{k} on the group H_{fppf}^{1}(U\overline{k})_{p} and which was studied in [Sa].

Let X := Spf A be the formal germ of an R-curve at a smooth closed point x, and let \overline{X} := Spf \overline{A} where \overline{A} := A ⊗_{R} \overline{R}. Let L := Fr \overline{A} be the fractions field of \overline{A}. In 1.2 above we associated to a Galois cover f : Y → X of degree p, above a formal germ of an R-curve at a smooth point x, a simple non split (resp. split) degeneration data D_{x}. This indeed can be interpreted as the existence of a canonical specialisation map:

Sp : H_{et}^{1}(Spec L, μ_{p}) → SS-Deg_{p}

resp.

Sp : H_{et}^{1}(Spec L, μ_{p}) → SNS-Deg_{p}

Reciproqually, we have the following result of realisation for degeneration data for such covers:

1.2.4. Theorem. Let k′ be a finite extention of k. Let Deg(x) be a k′-simple non split (resp. split) degeneration data of type (r, (G_{k}, m, h)) and rank p. Then there exists, after eventually a finite ramified extension of R, a non-uniquely determined Galois cover f : Y → X of degree p, where X is a formal germ of an R-curve at a smooth point x, and such that the degeneration data associated to f via 1.2. is isomorphic to Deg(x). In other words the above specialisation maps Sp : H_{et}^{1}(Spec L, μ_{p}) → SS-Deg_{p} and Sp : H_{et}^{1}(Spec L, μ_{p}) → SNS-Deg_{p} are surjective.

Proof. One uses the formal patching results as explained in [Sa-1], plus the examples given in [Sa-1] 3.1.3, 3.1.4 and 3.2.4. We may also assume that k = k′. First, for each i ∈ I^{nsp} (resp. i ∈ I^{sp}), let U_{i} be a formal affine scheme with special fibre U_{i} := U_{k,i}. The given torsor f_{i} : V_{i} → U_{i} is admissible (cf. [Sa], IV), hence can be lifted, after eventually a ramified extension of R, to a torsor f′_{i} : V_{i} → U_{i} under a finite and flat R-group scheme of rank p, which is either μ_{p} or H_{R,n} for 0 < n ≤ v_{K}(λ) where n := (v_{K}(p)−δ_{i})/(p−1). Such a lifting is non-unique in the radicial case (cf. loc. cit). Moreover, for each marked point x_{i,j} (resp. double point z_{i,j}) one can find a Galois cover f_{i,j} : Y_{i,j} → X_{i,j} of degree p where X_{i,j} is a formal germ of a smooth point x_{i,j}, and Y_{i,j} is smooth (resp. f_{i,j} : Y_{i,j} → Z_{i,j} of degree p, where Z_{i,j} is a formal germ of a double point z_{i,j} = z_{i′j′} of thikeness e_{i,j}, and Y_{i,j} is semi-stable) and with reduction type (G_{i,k}, m_{i,j}, h_{i,j}) (resp. ((G_{k,i}, m_{i,j}, h_{i,j}), (G_{k,i′}, m_{i′,j}, h_{i′,j})) on the boundaries) (cf. [Sa-1] 3.1.3, 3.1.4, 3.2.6). Also one can find a Galois cover f_{x} : Y_{x} → X_{x} of degree p, above a formal germ of a double point x, and with reduction type (G_{k}, m, h) and (G_{i_{0}}, −m, −h) on the boundaries of the formal fibre at x, as well as a Galois cover f′_{x} : Y′_{x} → X′_{x} above a formal closed disc X′_{x} := Spf R < T >, with degeneration data (G_{k}, m, h) on the boundary of the formal fibre at the closed point T = 0 (cf. [Sa-1] 2.3.1). Now using the formal patching result as in [Sa-1] I, as well as the result 3.1 in [Sa] , and after carefully adjusting the Galois action on the f_{i,j} and the f′_{i}, in order to obtain Galois patching data, one can patch the covers f′_{i}, f_{i,j}, f_{x} and f′_{x} along the points x_{i,j}, z_{i,j} and x, in order to obtain a Galois cover f′ : Ỹ → X̃ of degree p, where X̃ is a formal proper semi-stable R-curve, whose special fibre consists of the tree Γ plus a projective line linked to Γ via the double point x. A G-equivariant contraction of the vertices (X_{i})_{i} of the tree Γ in X̃_{k}, will yield to a Galois cover f : Y′ → X′ of degree p, where X′ is a formal proper and smooth projective line, with a marked point x on X′_{k}. A formal localisation now at the point x will give the desired cover f : Y_{x} → X_{x} := Spf Ô_{X′,x}, and by construction the simple degeneration data associated to f via 1.2 is isomorphic to D_{x}.

1.2.5. Example. This in particular is the realisation given by the above theorem 1.2.4 of the degeneration data which arises in the example 1.2.1. Consider the non split simple degeneration data of rank p which consists of a tree with one vertex and no edges. Hence a projective line X_{1} with one \overline{k}-marked point x = x_{1}, and a given étale torsor f_{1} : V_{1} → U_{1} := X_{1} − {x_{1}} with conductor m′ − m + 1 at the point x_{1} where m′ > m are positif integers. Then by 1.2.4 one can construct (after eventually a finite extension of R) a Galois cover f : Y → X of degree p above a formal germ X at a smooth point x, such that the singularity of Y at the closed point y is unibranche and the geometric genus of the point y equals (m′ − m)(p − 1)/2, moreover the semi-stable reduction of Y consists of one irreducible component Y_{1} of genus (m′ − m)(p − 1)/2, which is the projective completion of the above affine curve V_{1}, and which is linked to the point y by a double point.

1.2.6. Now we will study the action of G_{K} on Galois covers of degree p above formal germs of smooth curve at closed points. This action as we will see extends in a functorial way to an action of G_{K} on the corresponding semi-stable models. More precisely, let X := Spf A be the formal germ of an R-curve at a smooth closed point x, and let \overline{X} := Spf \overline{A} where \overline{A} := A⊗_{R} \overline{R}. Let L := Fr \overline{A} be the fractions field of \overline{A}. Then the Galois group G_{K} acts in a natural way on cyclic extentions of degree p of L, and we have a canonical homomorphism:

(1) G_{K} → Aut H_{et}^{1}(Spec L, μ_{p})

The above action (1) is equivalent to the action of G_{K} on normal Galois covers of degree p above \overline{X}. More precisely to each such a cover f : Y → \overline{X} with Y local and normal, which corresponds to the extension of functions fields: Spec L′ → Spec L, and an element σ ∈ G_{K}, one associates the Galois cover f^{σ} : Y^{σ} → \overline{X} of degree p, with Y^{σ} local and normal, which corresponds to the extension of functions fields: Spec L′^{σ} → Spec L. This action of G_{K} on covers extends to an action of G_{K} on corresponding minimal semistable models (because of the minimality condition). Hier by a minimal semi-stable model we mean either a split or a non split one. More precisely, let f̃: Ỹ → Y (resp. f̃′ : Ỹ′ → Y^{σ}) be a minimal semi-stable model of Y (resp. of Y^{σ}), then the element σ ∈ G_{K} maps the semi-stable model f̃ to the semi-stable model f̃′, in the sens that we have a commutative diagram:

Ỹ ⟶^{f̃} Y ⟶^{f} \overline{X}
σ↓  ^{σ}↓  ↓
Ỹ′ ⟶^{f̃^{σ}} Y^{σ} ⟶^{f^{σ}} \overline{X}

where the horizontal arrows are \overline{X} -automorphisms. We first show how the above action of G_{K} induces an action of G_{K} on simple degeneration data via its canonical quotient G_{k}. Our main result then is that this action coincides with the canonical action of G_{k} on degeneration data as defined in 1.2.2, and that the specialisation maps Sp : H_{et}^{1}(Spec L, μ_{p}) → SS-Deg_{p} and Sp : H_{et}^{1}(Spec L, μ_{p}) → SNS-Deg_{p} are G_{K}-equivariant.

First we may assume that the Galois cover f : Y → X′ := X ×_{R} R′, as well as its Galois conjugate via σ, are defined over a finite extension R′ of R. Let π′ be a uniformiser of R′, and let A′ := A ⊗_{R} R′. Let ℘ := (π′) be the ideal of A′ generated by π′, and let Â′_{℘} be the completion of the localisation of A′ at ℘. Let X′_{η} := Spf Â′_{℘} be the boundary of X′, and let X′_{η} → X′ be the canonical morphism. Consider the following cartesian diagram:

Y ⟶^{f} X′
↓  ↓
Y_{η} ⟶^{f_{η}} X′_{η}

Then f_{η} : Y_{η} → X′_{η} is a torsor under a commutative finite and flat R′-group scheme G_{R′} of rank p. Let (G_{k′}, m, h) be the degeneration type of f_{η} where k′ is the residue field of R′. We have also the following cartesian diagram:

Y^{σ} ⟶^{f^{σ}} X′
↓  ↓
Y_{η}^{σ} ⟶^{f_{η}^{σ}} X′_{η}

Moreover the torsor f_{η}^{σ} is the Galois conjugate of the torsor f_{η} by σ as one sees easily. In particular both of these torsors have the same reduction type.

The Galois group G ≃ ℤ/pℤ of the cover f (resp. f^{σ}) acts in a canonical way on Ỹ (resp. Ỹ′ := Ỹ^{σ}). Let X̃ := Ỹ/G which also equals Ỹ′/G. We have a commutative diagram:

Ỹ ⟶ X̃
σ↓  ↓
Ỹ^{σ} ⟶ X̃

Where the horizontal arrows are Galois covers with group G, and σ is an X̃-automorphism. Let Deg(x_{1}) (resp. Deg(x_{2})) be the simple degeneration data associated to the cover f (resp. f^{σ}). Let \overline{σ} be the image of σ in G_{k} via the canonical surjective homomorphism G_{K} → G_{k}. We will prove that Deg(x_{2}) is the transform Deg(x_{1})\overline{σ} of Deg(x, 1) by \overline{σ}, via the canonical action of G_{k} on degeneration data as explained in [Sa], I. The only think we have to check to prove this is that this is true when considering the degenertaion data Deg.4. Let X_{i} be an irreducible component of X̃_{k′} := X̃ ×_{R} k′. Let D_{i} be the set of double points of X̃_{k′} supported by X_{i}, and let S_{i} be the set of smooth points of X_{i} in which specialise some branched points of f. Let U_{i} := X_{i} − {S_{i} ∪ D_{i}}, and let U_{i} be a formal open subset of X̃ with special fibre U_{i}. Let f_{i} : V_{i} → U_{i} (resp. f′_{i} : V′_{i} → U_{i}) be the torsor above U_{i} induced by f (resp. by f^{σ}). Then f′_{i} is the transform f_{i}^{σ} by σ of the torsor f_{i}, via the canonical action of G_{K} on these torsors. Let f_{i,k′} : V_{i,k′} → U_{i,k′} (resp. f′_{i,k′} : V′_{i,k′} → U_{i,k′}) be the special fibre of the torsor f_{i} (resp. f′_{i} := f_{i}^{σ}). Let \overline{σ} be the image of σ in G_{k} via the canonical homomorphism G_{K} → G_{k}. Then f′_{i,k′} is nothing else but the transform f_{i,k′} \overline{σ} of f_{i,k′} via \overline{σ} (cf. [Sa], I). So indeed we have proven the following:

1.2.7. Theorem. Let G_{K} acts on the sets SS-Deg_{p} (resp. SNS-Deg_{p}) of isomorphism classes of simple split (resp. simple non split) degeneration data of rank p, via its canonical quotient G_{k}, and through the natural action of G_{k} on these sets as defined in 1.2.3. Then the surjective specialisation maps Sp : H_{et}^{1}(Spec L, μ_{p}) → SS-Deg_{p} and Sp : H_{et}^{1}(Spec L, μ_{p}) → SNS-Deg_{p} are G_{K}-equivariant.

1.2.8. Remark. It is easy to construct examples of covers f : Y → X as in [Sa-1] 3.1.4, where the special fibre Y_{k} is singular and unibranche at the closed point y of Y, and such that the configuration of the special fibre of a semi-stable model Ỹ of Y is not a tree-like. This indeed answers a question raised by Lorenzini whether such a situation can occur in the inequal characteristic case. More precisely, consider the non split simple degeneration data Deg(x) of type (2, (μ_{p}, −1, 0)) which consists of a graph Γ with two vertices X_{1} and X_{2} linked by a unique edge y, with given \overline{k}-marked points x = x_{1} on X_{1} and \overline{k}-marked point x_{2} on X_{2}, and given étale torsors of rank p : f_{1} : V_{1} → U_{1} := X_{1} − {x_{1}} with conductor m_{1} = 1 at x = x_{1} and f_{2} : V_{2} → U_{2} := X_{2} − {x_{2}} with conductor m_{2} = 1 at x_{2}. Then it follows from 1.2.4 that there exists, after eventually a finite extension of R, a Galois cover f : Y → X of degree p above the formal germ X ≃ Spf R[[T]] at the smooth R-point x, such that the \overline{k}-simple non split degeneration data associated to the above cover f is the above given one. Moreover by construction the singularity of the closed point y of Y is unibranche, and the configuration of the (non-split) semi-stable reduction of Y consists of two projective lines which meet at p-double points (the above cover will be étale in reduction above the double point y), in particular one has p − 1 cycles in this configuration.

1.3. We use the same notations as in 1.1. We consider now the case where x is an ordinary double point. Let f : Y → X be a Galois cover of degree p, where X is the formal germ of an R-curve at an ordinary double point x. In a similar way as in 1.2, and using the results of [Sa] and [Sa-1], one can associate a double degeneration data to f whose definition we summarise in the following:

1.3.1. Definition. Let k′ be an algebraic extension of k. A k′-double non split (resp. split) degeneration data Deg(x) of type (r, (G_{k′,1}, m_{1}, h_{1}), (G_{k′,2}, m_{2}, h_{2})) and rank p consists of the following:

Deg.1. For i ∈ {1, 2}, G_{k′,i} is a finite and flat k′-group scheme of rank p, r ≥ 0 is an integer, m_{i} is an integer prime to p such that r − m_{1} − m_{2} ≥ 0, and h_{i} ∈ 𝔽_{p} equals 0 unless G_{k′,i} = μ_{p} and m_{i} = 0.

Deg.2. Γ := X_{k′} is an oriented tree of projective lines with vertices Vert(Γ) := (X_{i})_{i∈I} (resp. Vert(Γ) := (X_{i})_{i∈I^{sp}}), and Γ̃ = ∪_{i∈Ĩ}X_{i}, where Ĩ ⊂ I, is a geodesic in Γ linking two given vertices X_{i_{1}} and X_{i_{2}}, with a given marked smooth k′-rational point x_{1} ∈ X_{i_{1}} (resp. x_{2} ∈ X_{i_{2}}). We denote by {z_{i,j}}_{j∈D_{i}} the set of double points, or edges, of Γ which are supported by X_{i}.

Deg.3. For each vertex X_{i} of Γ is given a set, may be empty, of smooth marked k′-rational points {x_{i,j}}_{j∈S_{i}}.

Deg.4. For each i ∈ I (resp. i ∈ I^{sp}), is given a torsor f_{i,k′} : V_{i,k′} → U_{i,k′} := X_{i,k′} − {{x_{i,j}}_{j∈S_{i}} ∪ {z_{i,j}}_{j∈D_{i}}} under a commutative finite and flat k′-group scheme G_{k′,i} of rank p, where X_{i,k′} := X_{i} ×_{Spec k} Spec k′, with V_{i,k′} smooth. Let U_{k′} be the open subscheme of X_{k′} obtained by deleting the double and marked points of X_{k′}. Then the above data determines an element of H_{fppf}^{1}(U_{k′})_{p}. Moreover if G_{k′,i} is radicial, and if ω_{i} is the associated differential form, then the set of zeros and poles of w_{i} is necessarily contained in {x_{i,j}}_{j∈S_{i}} ∪{z_{i,j}}_{j∈D_{i}}, as V_{i,k′} is smooth. Also if i ∈ I^{sp}, and S_{i} is non empty, we assume that G_{i,k′} = μ_{p}.

Deg.5. For each i ∈ I (resp. i ∈ I^{sp}), are given for each j ∈ S_{i} integers (m_{i,j}, h_{i,j}), where m_{i,j} is the conductor of the torsor f_{i} at the point x_{i,j}, and h_{i,j} its residue at this point. Let r_{i,j} := m_{i,j} + 1. If i ∈ I^{sp} then r_{i,j} = 1.

Deg.6. For each double point z_{i,j} = z_{i′,j′} ∈ X_{i} ∩ X_{i′} is given the pair (m_{i,j}, h_{i,j}) (resp. (m_{i′,j′}, h_{i′,j′})), where m_{i,j} (resp. m_{i′,j′}) is the conductor of the torsor f_{i,k′} (resp. f_{i′,k′}) at the point z_{i,j} (resp. z_{i′,j′}), and h_{i,j} (resp. h_{i′,j′}) its residue at this point. These data must satisfy m_{i,j} + m_{i′,j′} = 0, and h_{i,j} + h_{i′,j′} = 0. In other words the element of H_{fppf}^{1}(U_{k′})_{p} determined by the data Deg.4 must belong to the subgroup H_{fppf}^{1}(U_{k})_{p}^{kum}. In particular for the double point x_{i_{1},j_{1}} := x_{1} (resp. x_{i_{2},j_{2}} := x_{2}) is given the data (m_{i_{1},j_{1}}, h_{i_{1},j_{1}}) (resp. (m_{i_{2},j_{2}}, h_{i_{2},j_{2}}), such that m_{i_{1},j_{1}} + m_{1} = 0 (resp. m_{i_{2},j_{2}} + m_{2} = 0), and h_{i_{1},j_{1}} + h_{1} = 0 (resp. h_{i_{2},j_{2}} + h_{2} = 0).

Deg.7. For each irreducible component X_{i} of Γ is given an integer δ_{i} ≤ v_{K}(p) which is divisible by p − 1. For each double point z_{i,j} = z_{i′,j′} ∈ X_{i} ∩ X_{i′} of Γ is given an integer e_{i,j} = pt_{i,j} divisible by p, such that with the same notations as above we have δ_{i} − δ_{i′} = m_{i,j}t_{i,j}(p − 1).

Deg.8. Let I_{et} denote the subset of I (resp. of I^{sp}) consisting of those i for which G_{k′,i} is étale, then : (r−m−1)(p−1)/2 = ∑_{i∈I_{et}}(−2+∑_{j∈S_{i}}(m_{i,j} +1)+∑_{j∈D_{i}}(m_{i,j} +1))(p−1)/2. The integer g_{x} := (r − m_{1} − m_{2})(p − 1)/2 is called the genus of the degeneration data Deg(x).

There is a natural notion of isomorphism of double split (resp. non split) degeneration data of a given type over a given algebraic extension of k. We will denote by DS-Deg_{p} (resp. DNS-Deg_{p}) the set of isomorphism classes of \overline{k}-double split (resp. \overline{k}-double non split) degeneration data of rank p. Also as in 1.2.3 one shows that there exists a canonical action of G_{k} on these sets, via the canonical action of G_{k} on the degeneration data Deg.4, and these sets are G_{k}-sets, hence also canonically G_{K}-sets. The following theorem is proven in the same way as in 1.2.4 and 1.2.6.

1.3.2. Theorem. Let X := Spf A be the formal germ of an R-curve at an ordinary double point. Let \overline{A} := A ⊗_{R} \overline{R}, and let L := Fr(\overline{A}). Then the canonical specialisation maps: Sp : H_{et}^{1}(Spec L, μ_{p}) → DS-Deg_{p} and Sp : H_{et}^{1}(Spec L, μ_{p}) → DNS-Deg_{p} are surjective and G_{K}-equivariant.

1.3.3. remark. The surjectivity result in 1.2.4 and 1.2.7 was shown in [He] in the case of degeneration data of genus 0 in the framework of automorphisms of order p of open discs and annuli. Our method based on the techniques developed in [Sa] and [Sa-1] avoid the use of this language and uses only kummer theory.

II. Semi-stable reduction of Galois covers of degree p obove proper curves.

2.0. In this section we will use the same notations as in 1.0. We will consider a proper and semi-stable R-curve X, a Galois cover f : Y → X with group ℤ/pℤ, and with Y normal. We will study the semi-stable reduction of Y as well as the Galois action on these data.

2.1. Let X be a proper and semi-stable R-curve, with smooth and connected generic fibre X_{K} := X ×_{R} K. We assume that the double points of X_{k} are k-rationals. Let f : Y → X be a Galois cover with group G ≃ ℤ/pℤ, and with Y normal. It follows from the theorem of semi-stable reduction for curves (cf. [De-Mu]) that, after eventually a finite extension R′ of R, with fractions field K′ and residue field k′, the curve Y has a semi-stable reduction. More precisely, there exists a birational and proper morphism f̃ : Ỹ → Y′ := Y ×_{R} R′ such that Ỹ_{K′} ≃ Y′_{K′}, and the following conditions hold:

(i) The special fibre Ỹ_{k′} of Ỹ is reduced. (ii) Ỹ_{k′} has only ordinary double points as singularities.

Moreover there exists such a semi-stable model Ỹ which is minimal. In particular the action of G on Y′ extends to an action on Ỹ. Let X̃ be the quotient of Ỹ by G which is a semi-stable model of X′ := X ×_{R} R′. In the following we will choose Ỹ and X̃ as above such that the set of points B_{K} := {x_{i,K}}_{i=1}^{r} consisting of the branch locus of the morphism f_{K} : Y_{K} → X_{K} are rational. Moreover we choose such X̃ and Ỹ which are minimal for these properties. One has the following commutative diagram:

Ỹ ⟶^{g} X̃
f̃↓  g̃↓
Y′ ⟶^{f′} X′

Let (X_{i})_{i∈I} be the irreducible components of the special fibre X′_{k′} of X′. For each i ∈ I, let (x_{i,j})_{j∈S_{i}} be the set (may be empty) of those smooth points of X_{i} in which specialise some points of B_{K}, say in x_{i,j} specialise r_{i,j} points of B_{K}, and let (z_{t})_{t∈J} be the set of double points of X_{k}. For each t ∈ J, let r_{t} be the number of points of B_{K} which specialise in z_{t}, we have r = ∑_{t∈J} r_{t} + ∑_{i∈I} ∑_{j∈S_{j}} r_{i,j}. Let U := X′ − {D ∪ S}, where S := ∪_{i∈I}{x_{i,j}}_{j∈S_{i}} and D = {z_{t}}_{t∈J}. We will denote by {U_{i}}_{i∈I} the set of irreducible components of U.

2.2. Below we will use the notations of 2.0 and 2.1 and will exhibit the degeneration data associated to the semi-stable reduction of Y. Let R′ be a finite extension of R such that the conditions of 2.1 hold. The following data are canonically associated to the Galois cover f : Y → X:

Deg.1. For each irreducible component X_{i} of X′_{k′}, let η_{i} be the generic point of X_{i}, and let X′_{η_{i}} be the spectrum of the completion of the localisation of X′ at η_{i}. We have a canonical morphism : X′_{η_{i}} → X′. Consider the following cartesian diagram:

Y_{η_{i}} ⟶^{f_{η_{i}}} X′_{η_{i}}
↓  ↓
Y′ ⟶^{f′} X′

Then either f_{η_{i}} : Y_{η_{i}} → X′_{η_{i}} is completely split, or is a torsor under a finite and flat commutative R′-group scheme G_{i} of rank p.

Deg.2. For each irreducible component X_{i} of X′_{k}, let U_{i} := X_{i} − {{x_{i,j}}_{j∈S_{i}} ∪ {z_{t}}_{t∈J_{i}}}, where J_{i} ⊂ J denotes the index subset indexing those double points of X′_{k′} which are supported by X_{i}. Then V_{i} := f^{−1}(U_{i}) → U_{i} is an admissible torsor under the finite and flat group scheme G_{k′,i} := G_{i} ×_{R′} k′. Moreover, if G_{k′,i} is étale then V_{i} is smooth, and if G_{k′,i} is radicial then the only singularities of V_{i} lie above the zeros {y_{l}}_{l∈Z_{i}} of the differential form ω_{i} associated to the above torsor (cf. [Sa], I).

Deg.3. For each double point z_{j} ∈ X_{i} ∪ X_{i′}, let X_{j} be the completion of the localisation of X′ at z_{j}, and let X_{j} → X′ be the canonical morphism. Consider the cartesian diagram:

Y_{j} ⟶^{f_{j}} X_{j}
↓  ↓
Y′ ⟶^{f′} X′

Then either the cover f_{j} : Y_{j} → X_{j} is split, or it is a Galois cover of degree p with Y_{j} connected, in which case is associated to f_{j} a double degeneration data Deg(z_{j}). Let m_{j,1} and h_{j,1} (resp. m_{j,2} and h_{j,2}) be the conductor and the residue at the double point z_{j} associated to the torsor V_{i} := f^{−1}(U_{i}) → U_{i} (resp. V_{i′} := f^{−1}(U_{i′}) → U_{i′}). Then the above degeneration data Deg(z_{i}) is of type (r_{j}, (G_{k′,i}, m_{j,1}, h_{j,1}), (G_{k′,i′}, m_{j,2}, h_{j,2})), and genus (r_{j} − m_{j,1} − m_{j,2})(p − 1)/2.

Deg.4. For each irreducible component X_{i} of X′_{k′}, and a smooth point x_{i,j}, j ∈ S_{i}, let X_{i,j} be the completion of the localisation of X′ at x_{i,j}, and let X_{i,j} → X′ be the canonical morphism. Consider the cartesian diagram:

Y_{i,j} ⟶^{f_{i,j}} X_{i,j}
↓  ↓
Y′ ⟶^{f′} X′

The cover f_{i,j} : Y_{i,j} → X_{i,j} is necessarily a Galois cover of degree p with Y_{i,j} connected, and to f_{i,j} is associated a simple degeneration data Deg(x_{i,j}). Let m_{i,j} and h_{i,j} be the conductor and the residue at the smooth point x_{i,j} associated to the torsor V_{i} := f^{−1}(U_{i}) → U_{i}. Then the above degeneration data Deg(x_{i,j}) is of type (r_{i,j}, (G_{k′,i}, m_{i,j}, h_{i,j})), and genus (r_{i,j} − m_{i,j} − 1)(p − 1)/2.

Deg.5. For each i ∈ I such that the group scheme G_{k′,i} is radicial, let {y_{l}}_{l∈Z_{i}} be the zeros of the differential form ω_{i} associated to the above torsor V_{i} := f^{−1}(U_{i}) → U_{i}, and which we may assume to be rational over k′. For each l ∈ Z_{i}, let Z_{l} be the completion of the localisation of X′ at y_{l}, and let Z_{l} → X′ be the canonical morphism. Consider the cartesian diagram:

Z′_{l} ⟶^{f_{l}} Z_{l}
↓  ↓
Y′ ⟶^{f′} X′

The cover Z′_{l} → Z_{l} is a Galois cover of degree p with Z′_{l} connected, and to f_{l} is associated a simple degeneration data Deg(y_{l}). Let m_{l} and h_{l} be the conductor and the residue at the smooth point y_{l} associated to the torsor V_{i} := f^{−1}(U_{i}) → U_{i}. Then the above degeneration data Deg(y_{l}) is of type (0, (G_{k′,i}, m_{l}, h_{l})), and genus (−m_{l} − 1)(p − 1)/2.

The above considerations lead to the following definition of degeneration data associated to a Galois cover of degree p above a proper and semi-stable R-curve.

2.2.1. Definition. Let X_{k} be a proper and semi-stable k-curve, and let {X_{i}}_{i∈I} be the irreducible components of X_{k}. We assume that the double points {z_{t}}_{t∈J} of X_{k} are k-rational. Let k′ be an algebraic extension k. A k′-degeneration data of rank p associated to X_{k}, consists of the following data:

Deg.1. For each irreducible component X_{i} of X_{k} is given a set of smooth k′-rational points {x_{i,j}}_{j∈S_{i}} of X_{k} which are supported by X_{i}.

Deg.2. For each irreducible component X_{i} of X_{k}, let U_{i} := X_{i} − {{x_{i,j}}_{j∈S_{i}} ∪ {z_{t}}_{t∈J_{i}}}, where J_{i} ⊂ J denotes the index subset indexing those double points of X_{k} which are supported by X_{i}. Then we assume given an admissible torsor f_{i} : V_{i} → U′_{i} := U_{i} ×_{k} k′ (cf. [Sa], 4, for the definition of admissibility) under a finite and flat k′-group scheme G_{k′,i}, and we allow the torsor f_{i} to be trivial. If G_{k′,i} is radicial then we assume that the zeros {y_{l}}_{l∈Z_{i}} of the differential form ω_{i} associated to the above torsor f_{i} are k′-rational.

Deg.3. For each double point z_{j} ∈ X_{i} ∩ X_{i′} of X_{k}, let m_{j,1} and h_{j,1} (resp. m_{j,2} and h_{j,2}) be the conductor and the residue at the double point z_{j} associated to the torsor V_{i} := f^{−1}(U_{i}) → U_{i} (resp. V_{i′} := f^{−1}(U_{i′}) → U_{i′}). Let r_{j} be an integer such that r_{j} − m_{j,1} − m_{j,2} ≥ 0. Then we assume given a k′-double degeneration data Deg(z_{j}) of type (r_{j}, (G_{k′,i}, m_{j,1}, h_{j,1}), (G_{k′,i′}, m_{j,2}, h_{j,2})), and genus (r_{j} − m_{j,1} − m_{j,2})(p − 1)/2.

Deg.4. For each irreducible component X_{i} of X_{k′}, and a smooth point x_{i,j}, j ∈ S_{i}, let m_{i,j} and h_{i,j} be the conductor and the residue at the smooth point x_{i,j} associated to the torsor V_{i} := f^{−1}(U_{i}) → U_{i}. Let r_{i,j} be an integer such that r_{i,j} − m_{i,j} − 1 ≥ 0. Then we assume given a k′-simple degeneration data Deg(x_{i,j}) of type (r_{i,j}, (G_{k′,i}, m_{i,j}, h_{i,j})), and genus (r_{i,j} − m_{i,j} − 1)(p − 1)/2.

Deg.5. For each i ∈ I such that the group scheme G_{k′,i} is radicial, let {y_{l}}_{l∈Z_{i}} be the zeros of the differential form ω_{i} associated to the above torsor V_{i} := f^{−1}(U_{i}) → U_{i}, and which we may assume to be rational over k′. For each l ∈ Z_{i}, let m_{l} and h_{l} be the conductor and the residue at the smooth point y_{l} associated to the torsor V_{i} := f^{−1}(U_{i}) → U_{i}. Then we assume given a degeneration data Deg(y_{l}) of type (0, (G_{k′,i}, m_{l}, h_{l})), and genus (−m_{l} − 1)(p − 1)/2.

2.2.2. Remark. The admissibility condition on the torsors {f_{i}}_{i∈I} which is part of the above definition of degeneration data is satisfied when the U_{i} are opens of the projective line, e.g. if X_{k} is a totally degenerate Mumford curve, and in the case where U_{i} is proper.

2.2.3. Example. In this example we consider a semi-stable and proper R-curve X with smooth and geometrically connected generic fibre X_{K}, and whose special fibre X_{k} consists of two irreducible smooth components X_{1} of genus g_{1} ≥ 2 and X_{2} of genus g_{2} ≥ 2 which intersect at the double point x. We suppose moreover that both X_{1} and X_{2} are generic (for this one has to choose R to be rather large).

Consider a Galois cover f : Y → X of degree p with Y normal and such that the Galois cover f_{K} : Y_{K} → X_{K} induced by f above the generic fibre is a μ_{p}-torsor (hence in particular is étale). We assume that the special fibre Y_{k} := Y ×_{R} k of Y is reduced. For i ∈ {1, 2}, let f_{i} : V_{i} → U_{i} := X_{i} − {x} be the torsor above U_{i} induced by the cover f, and let m_{i} be the conductor of f_{i} at the point x. Then only the following two cases can ossur:

Case 1) Either f_{i} is an étale torsor in which case either f_{i} ramify above x or f_{i} extends to an étale torsor above X_{i}. In both cases V_{i} is smooth.

Case 2) Or f_{i} is a torsor under μ_{p} in which case is associated to f_{i} canonically a logarithmic differential form ω_{i}, and the singularities of V_{i} lie above the zeros of ω_{i}. In this case where X_{i} is generic it was shown in [Ra] (proposition 4) that such a differential form ω_{i} has g_{i} − 1 double zeros if p = 2 and 2g_{i} − 2 simple zeros if p ≠ 2. In the later case m_{i} = −2 or m_{i} = −1 depending on whether x is a zero of ω_{i} or not.

In what follows we assume for simplicity that p ≠ 2. Supppose we are in case 2, and let y_{j} be a point of V_{i} above a zero z_{j} of ω_{i} which is different from the point x. Then V_{i} is singular at y_{j}, the singularity at y_{j} is unibranche, and the geometric genus of y_{j} equals (p − 1)/2. Let Deg(z_{j}) be the non split \overline{k}-degeneration data associated to z_{j} via the semistable reduction of Y. Then Deg(z_{j}) is necessarily of type (0, (μ_{p}, −2, 0)), and consists of one projective line P_{i} with a marked \overline{k}-point x_{i} = x and an étale torsor T_{i} → P_{i} − \{x_{i}\} with conductor 2 at the point x_{i}.

Above the double point x three situations can occur:

Case a) Both f_{1} and f_{2} are étale in which case f is necessarily étale above x.

Apart from case a) Y must be local above, we will denote by y the unique point of Y above x.

Case b) Say f_{1} is étale and f_{2} is radicial, and assume for simplicity that Y has two branches at the point y. In this case the geometric genus of y equals (m_{1} + 2)(p − 1)/2 if ω_{2} has a zero at x, or (m_{1} + 1)(p − 1)/2 otherwise.

Case c) Both f_{1} and f_{2} are radicial in which case the geometric genus of y equals 2(p − 1), 3(p − 1)/2 or p − 1 depending on whether both f_{i} have a zero at the point x, or just one of them do, or none of them.

Let Deg(x) be the double non split \overline{k}-degeneration data associated to x via the semistable reduction of Y. assume for example that we are in case c) and that the genus of y equals p − 1. Then the following two cases can occur:

First case: either Deg(x) consists of one projective line P with two marked \overline{k}-points x_{1} = x and x_{2} = x, and an étale torsor T → P − {x_{1}, x_{2}} with conductor 2 at each of x_{1} and x_{2}. In this case the special fibre of the semi-stable reduction of Y is a tree like which consist of the components V_{1} and V_{2} which are linked by the projective completion of V (which has genus p − 1) at the points above x, and at each of the points of V_{i} above a zero of ω_{i} is linked a curve of genus (p − 1)/2.

Second case: Deg(x) consists of two projective line P_{1} and P_{2} with a marked \overline{k}-points x_{1} = x and x_{2} = x on each one, and an étale torsor T_{i} → P_{i} − {x_{i}} with conductor 2 at x_{i}. In this case the special fibre of the semi-stable reduction of Y contains (p − 1) cycles and it consists of the components V_{1} and V_{2} which are linked by the projective completion of T_{1} and T_{2} (each has genus (p − 1)/2) at the points above x, moreover both T_{1} and T_{2} meet at p double points, and at each of the points of V_{i} above a zero of ω_{i} is linked a curve of genus (p − 1)/2 as in the preceeding case.

2.3. There is a natural notion of isomorphism of degeneration data of rank p associated to a proper and semi-stable k-curve X_{k}, over a given algebraic extension of k. We will denote by Deg_{p}(X_{k}) the set of isomorphism classes of \overline{k}-degeneration data of rank p associated to X_{k}. The set Deg_{p}(X_{k}) is equipped in a natural way with a G_{k}-action, via the natural action of G_{k} on the above data which was explained in [Sa] and I, and hence is a G_{k}-set. The following result is analogous to the result of realisation for degeneration data obtained in paragraph I in the local situation.

2.3.1. Theorem. Let X_{k} be a proper and semi-stable k-curve. Let R be a complete discrete valuation ring of inequal characteristics with residue field k, and fractions field K. Suppose given a \overline{k}-degeneration data deg(X_{k}) of rank p associated to X_{k} (in other words suppose given an element of Deg_{p}(X_{k})). Then there exists, after eventually a finite extension of R, a proper and semi-stable R-curve X̃ with smooth generic fibre and a special fibre isomorphic to X_{k}, and a Galois cover f : Ỹ → X̃, such that the degeneration data associated to f as in 2.2 is isomorphic to the given data deg(X_{k}). The above cover f is not unique in general, moreover it can be constructed such that the cover f_{K} : Ỹ_{K} → X̃_{K} induced by f on the generic fibres is ramified above r = ∑_{t∈J} r_{t} + ∑_{i∈I} ∑_{j∈S_{j}} r_{i,j} geometric points of X_{K} (hier r_{t} and r_{i,j} are the one given in the definition of the degeneration data in 2.2.1).

The next result expreses the compatibility of degeneration with the Galois action.

2.3.2. Theorem. Let R be a complete discrete valuation ring of inequal characteristics with residue field k, and fractions field K, and let X be a proper and semi-stable R-curve with smooth and geometrically connected generic fibre X_{K} and with special fibre X_{k}. Let \overline{K} be an algebraic closure of K, and let L be the function field of the geometric generic fibre X\overline{K} := X ×_{R} \overline{K} of X. Then the canonical specialisation map Sp : H_{et}^{1}(Spec L, μ_{p}) → Deg_{p}(X_{k}) is G_{K}-equivariant, where hier we consider the natural action of G_{K} on Deg_{p}(X_{k}) via its canonical quotient G_{k}.

Note that the above specialisation map can not be surjective in this case since the set Deg_{p}(X_{k}) with fixed ramification data on the “generic fibre” is infinite in general. The proof of the above two results is very similar to the one obtained in paragraph I in the local case and are left to the reader. They are based on formal patching techniques and the results in [Sa] and [Sa-1]. However in order to prove 2.3.1 one will also need the following result on lifting of admissible torsors with given degeneration data at the “critical points”. More precisely let R be as in 1.0. Let X be a formal smooth R-curve, whose special fibre X_{k} is irreducible. Let f_{k} : Y_{k} → X_{k} be an admissible torsor, under a finite and flat kgroup scheme G_{k} of rank p, which is radicial. Let ω be the associated differential form, and let Z := {x_{j}}_{j∈J} be the set of zeros of ω which are contained in X_{k}, which we call the critical points of f_{k}, in particular the singularities of Y_{k} lie above the points of Z. Assume that f : Y → X is a torsor under a finite and flat R-group scheme G which lifts f_{k}. To each point x_{j} ∈ Z is associated via f, a simple degeneration data Deg(x_{j}) of type (0, (G_{k}, m_{j}, h_{j})), where m_{j} is the conductor of f_{k} at x_{j}, and h_{j} its residue at this point. The following result shows that we can choose such a lifting f of f_{k} which gives rise to a given set of degeneration data at the critical points.

2.3.3. Proposition / Definition. Let f_{k} : Y_{k} → X_{k} be an admissible torsor under a finite and flat k-group scheme G_{k} of rank p, which is radicial, and let Z := {x_{j}}_{j∈J} be the set of critical point of f_{k} which we assume to be k-rational. For each point x_{j} ∈ Z consider the pair (m_{j}, h_{j}), where m_{j} is the conductor of f_{k} at x_{j}, and h_{j} its residu at this point. Suppose given for each critical point x_{j}, a simple degeneration data Deg(x_{j}) of type (0, (G_{k}, m_{j}, h_{j})). Then there exists, after eventually a finite extension of R, a smooth formal R-curve X̃ with a special fibre X̃_{k} := X̃ ×_{R} k which is isomorphic to X_{k}, and a torsor f̃ : Ỹ → X̃ under a finite and flat R-group scheme G of rank p, such that the restriction of f̃ to U_{k} := X_{k} − Z is isomorphic to the restriction of f_{k} to U_{k}, and such that the simple degeneration data assosiated to each point x_{j} ∈ Z, via f̃, is isomorphic to Deg(x_{j}). We call such a torsor f̃ a lifting of f_{k}, with the given degeneration data {Deg(x_{j})}_{j∈Z} at the critical points {x_{j}}_{j∈Z}.

Proof. The torsor f_{k} : Y_{k} → X_{k} is admissible hence by definition can be lifted, after eventually a ramified extension of R, to a torsor f : Y′ → X under a finite and flat R-group scheme of rank p, which is either μ_{p} or H_{n} for 0 < n < v_{K}(λ) where v_{K} is the normalised valuation of the fractions field of R (such a lifting is not unique). Also for each point x_{j} ∈ Z one can find a Galois cover f_{j} : Y_{j} → X_{j} of degree p, where X_{j} is the formal germ of X at x_{j}, and such that the degeneration data associated to x_{j} via f_{j} is isomorphic to Deg(x_{j}) (cf. 1.2.3). Let U_{k} := X_{k} − Z, and let g_{k} be the restriction of f_{k} to U_{k}. Then f induces a lifting g : V → U of g_{k}, which is a torsor under a finite and flat R-group scheme of rank p, and where U is obtained from X by deleting the formal fibres at the points {x_{j}}_{j∈Z}. Now a patching of the covers g and f_{j} along the formal fibres at the points {x_{j}}_{j} will produce the desired torsor f̃: Ỹ → X̃.

2.3.4. Example. Hier we use the same notations as in the example 2.2.3. First we will explain the Galois action on a μ_{p}-torsor f\overline{K} : Y\overline{K} → X\overline{K} above the geometric generic fibre X\overline{K} of X. We assume that the above torsor induces in reduction the situation in case c) where g_{y} = p − 1 and where the semi-stable reduction of Y\overline{K} is a tree like. Let σ ∈ G_{K}, and let \overline{σ} be the image of σ in G_{k}. Consider the μ_{p}-torsor f\overline{K}^{σ} : Y\overline{K}^{σ} → X\overline{K} which is the conjugate of f\overline{K} via σ. Then using the result in 2.3.2 one can describe the semi-stable reduction of Y\overline{K}^{σ}. More precisely the graph of the semi-stable reduction of Y\overline{K}^{σ} is isomorphic to the one of Y\overline{K}, in particular it is also a tree like, and it consists of the components V \overline{σ} ; i, for i = 1, 2, where V_{i}\overline{σ} is the conjugate of V_{i} via \overline{σ}, which are linked by an irreducible curve T \overline{σ} of genus p − 1 which is the conjugate of T by \overline{σ}. Moreover at each point of V_{i}\overline{σ} above a zero of ω_{i} is linked a curve of genus (p − 1)/2 which is the conjugate by \overline{σ} of the curve that is linked to the corresponding point (via the action of \overline{σ}) of V_{i} in the graph of semi-stable reduction of Y\overline{K}.

Also one can establish the converse of the situation in 2.2.3, i.e. reconstruct a μ_{p}- torsor f_{K} as above from the degeneration data, using 2.3.1. More precisely, let X_{k} be the semi-stable curve considered in 2.2.3 and consider the non split degeneration data Deg(X_{k}) of rank p which consists in the following:

For i = 1, 2 is given a μ_{p}-torsor Y_{i} → X_{i} associated to the logarithmic differential form ω_{i}, and assume that the zeros {z_{j,i}}_{j} of ω_{i} lie outside the point x. For each i, j suppose given a non split simple degeneration data Deg(z_{j,i}) of type (0, (μ_{p}, −2, 0)) and which consists of one projective line P_{j,i} with one \overline{k}-marked point z_{j,i} and an étale torsor above P_{j,i} − {z_{j,i}} with conductor 2 at the point z_{j,i}. At the double point x is given a non split double degeneration data of type (0, (μ_{p}, −2, 0), (μ_{p}, −2, 0)) and which consists of one projective line P and two \overline{k}-marked points x_{1} = x and x_{2} = x and an étale torsor above P − {x_{1}, x_{2}} with conductor 2 at both points x_{1} and x_{2}. Then by 2.3.1 one can find a proper and semi-stable curve X̃ with smooth generic fibre and a special fibre X̃_{k} := X̃ ×R_{k} isomorphic to X_{k}, and (eventually after a finite extension of R) a Galois cover f̃: Ỹ → X̃ of degree p, such that the degeneration data of rank p associated to X_{k} via the semi-stable reduction of Ỹ is isomorphic to the above given data Deg(X_{k}).

References

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[Gr] A. Grothendieck, Séminaire de géométrie algébrique SGA-1, Lecture Notes 224, Springer Verlag, (1971).

[He] Y. Henrio, Arbres de Hurwitz et automorphismes d’ordre p des disques et couronnes p-adiques formels. Thèse de doctorat, Bordeaux France (1999).

[Ra] M. Raynaud, p-Groupes et reduction semi-stable des courbes, The Grothendieck Festschrift, vol. 3, 179-197, Birkhäuser, (1990).

[Sa] M. Saïdi, Torosrs under finite and flat group schemes of rank p with Galois action, math.AG/0106246.

[Sa-1] M. Saïdi, Wild ramification and vanishing cycles formula, math.AG/0106249.

Mohamed Saïdi

Departement of Mathematics University of Durham Science Laboratories South Road Durham, DH1 3LE, UK saidi@durham.ac.uk