Wild ramification and a vanishing cycles formula

Mohamed Saïdi

Abstract

In this paper we prove an explicit formula which compares the dimensions of the spaces of vanishing cycles in a Galois cover of degree p between formal germs of curves over a complete discrete valuation ring of inequal characteristics (0, p). This formula can be easily generalised to the case of a Galois cover with group which is nilpotent or which has a normal p-sylow subgroup. The results of this paper play a key role in [Sa-1] where is studied the semi-stable reduction of Galois covers of degree p above semi-stable curves over a complete discrete valuation ring of inequal characteristics (0, p), as well as the Galois action on these covers.

0. Introduction. Let R be a complete discrete valuation ring of inequal characteristics, with uniformiser π, fraction field K, and algebraically closed residue field k of characteristic p. In this paper we investigate Galois covers of degree p between formal germs of R-curves at closed points. Our main result is the following formula which compares the dimensions of the spaces of vanishing cycles at the corresponding closed points. More precisely we have the following:

Theorem (2.4). Let X := Spf Ô_{x} be the formal germ of an R-curve at a closed point x, with X_{k} reduced. Let f : Y → X be a Galois cover of group ℤ/pℤ with Y normal and local. Assume that the special fibre Y_{k} := Y ×_{R} k of Y is reduced. Let {℘_{i}}_{i∈I} be the minimal prime ideals of Ô_{x} which contain π and which correspond to the branches (η_{i})_{i∈I} of the special fibre X_{k} := X ×_{R} k of X at x, and let X_{i} := Spf Ô_{℘_{i}} be the formal completion of the localisation of X at ℘_{i}. For each i ∈ I, the above cover f induces a torsor f_{i} : Y_{i} → X_{i} under a finite and flat R-group scheme G_{i} of rank p, above the boundary X_{i}. For each i ∈ I, let (G_{k,i}, m_{i}, h_{i}) be the reduction type of f_{i} as defined in [Sa] 3.2. Let y be the closed point of Y. Then one has the following “local Riemman-Hurwitz formula”:

2g_{y} − 2 = p(2g_{x} − 2) + d_{η} − d_{s}

Where g_{y} (resp. g_{x}) denotes the genus of the singularity at y (resp. x), d_{η} is the degree of the divisor of ramification in the morphism f_{η} : Y_{η} → X_{η} induced by f on the generic fibre, and d_{s} := ∑_{i∈I^{rad}}(m_{i} − 1)(p − 1) + ∑_{i∈I^{et}}(m_{i} − 1)(p − 1), where I^{rad} is the subset of I consisting of those i for which G_{k,i} is radicial, and I^{et} is the subset of I consisting of those i for which G_{k,i} is étale and m_{i} ≠ 0. Hier G_{i,k} denotes the special fibre of the group scheme G_{i}.

In particular the genus g_{y} of y depends only on the genus g_{x} of x, the ramification datas on the generic fibre in the above morphism f : Y → X, and its degeneration type on the boundaries of the formal fibre X. The above formula can be easily extended to the case of a Galois cover with group G which is nilpotent, or a group which has a normal p-Sylow subgroup. Our method to prove such a formula is to construct, using formal patching techniques à la Harbater, a compactification f̃: Y → X of the above cover f : Y → X (cf. 2.3.2). The formula follows then by comparing the genus of the special and the generic fibres of Y in this compactification.

As an application, and using the above formula one can obtain interesting results in the case where X is the formal germ of a semi-stable R-curve (cf. 3.1 and 3.2), in particular one can predict in this case if Y is semi-stable or not. We give several examples which illustrate this situation namely the case of Galois covers of degree p between formal germs of semi-stable R-curves (cf. 3.1.3, 3.1.4 and 3.2.4). In particular one can classify étale Galois covers of degree p between annuli (cf. 3.2.5). The above results play a key role in [Sa-1] in order to exhibit and realise the “degeneration datas” associated to Galois covers of degree p above a proper semi-stable R-curve.

I. Formal and rigid patching.

In what follows we explain the procedure which allows to construct (Galois)-covers of curves in the setting of formal or rigid geometry by glueing together covers of formal affine or affinoid rigid curves with covers of formal fibres at closed points of the special fibre. We refer the reader to the exposition in [Pr] for a discussion of patching results and for detailled references on the subject.

Let R be a complete discrete valuation ring with fractions field K, residue field k, and uniformiser π. Let X be an admissible formal R-scheme which is an R-curve, by which we mean that the special fibre X_{k} := X×_{R}k is a reduced one dimensional k-scheme of finite type. Let Z be a finite set of closed points of X_{k}. For a point x ∈ Z, let X_{x} := Spf Ô_{X,x} be the formal completion of X at x, which is the formal fibre at the point x. Also let X′ be a formal open subset of X whose special fibre is X_{k} − Z. For each point x ∈ Z, let {℘_{i}}_{i=1}^{n} be the set of minimal prime ideals of Ô_{X,x} which contain π, they correspond to the branches (η_{i})_{i=1}^{n} of the completion of X_{k} at x, and let X_{x,i} := Spf Ô_{x,℘_{i}} be the formal completion of the localisation of X_{x} at ℘_{i}. The ring Ô_{x,℘_{i}} is a complete discrete valuation ring. The set {X_{x,i}}_{i=1}^{n} is the set of boundaries of the formal fibre X_{x}. For each i ∈ {1, n} we have a canonical morphism X_{x,i} → X_{x}.

1.1. Definition. With the same notations as above a (G)-cover patching data for the pair (X, Z) consists of the following: a ) A finite (Galois) cover Y′ → X′ (with group G). b ) For each point x ∈ Z, a finite (Galois) cover Y_{x} → X_{x} (with group G).

The above datas a) and b) must satisfy to the following condition: c ) If {X_{x,i}}_{i=1}^{n} are the boundaries of the formal fibre at the point x, then for each i ∈ {1, n} is given a (G-equivariant) isomorphism σ_{i} : Y_{x} ×_{X_{x}} X_{x,i} ≃ Y′ ×_{X′} X_{x,i}.

1.2. Proposition / Formal patching. Given a (G)-cover patching data as in 1.1 there exists a unique, up to isomorphism, (Galois) cover Y → X (with group G) which induces the above (G)-covers in a) (resp. in b) when restricted to X′ (resp. when pulled back to X_{x} for each point x ∈ Z).

The proof of 1.2 is an easy consequence of theorem 3.4. in [Pr], and which is due to Ferrand and Raynaud.

1.3. Remark. With the same notations as above let x ∈ Z and let X̃_{k} be the normalisation of X_{k}. There is a one to one correspondance between the set of points of X̃_{k} above x and the set of boundaries of the formal fibre at the point x. Let x_{i} be the point of X̃_{k} above x which corresponds to the boundary X_{x,i}, for i ∈ {1, n}. Assume that the point x ∈ X_{k}(k) is rational. Then the completion of X̃_{k} at x_{i} is isomorphic to the spectrum of a ring of formal power series k[[t_{i}]] in one variable over k, where t_{i} is a local parameter at x_{i}. The complete local ring Ô_{x,℘_{i}} is a two dimensional discrete valuation ring whose residue field is isomorphic to k((t_{i})). Let T_{i} be an element of Ô_{x,℘_{i}} which lifts t_{i}, such an element is called a parameter of Ô_{x,℘_{i}}. Then it follows from [Bo] that there exists an isomorphism Ô_{x,℘_{i}} ≃ R[[T_{i}]]{T_{i}^{−1}}, where R[[T_{i}]]{T_{i}^{−1}} is the ring of formal power series ∑_{i∈ℤ} a_{i}T^{i} with lim_{i→−∞} |a_{i}| = 0, and where | | is an absolute value of K associated to its valuation.

1.4. Rigid patching. The analog of the above result is well known in rigid geometry, which is not surprising because of the link between rigid and formal geometry (cf. [Ra], [Bo-Lu]). We explain briefly the patching procedure in this context locally. Let X := Spm A be an affinoid reduced curve, and let X be a formal model of X with special fibre X_{k}. Let x be a closed point of X_{k}, and let X_{x} be the formal fibre of X at x, which is a non quasi-compact rigid space and which consists of the set of points of X which reduce to the point x. The structure of the boundary of X_{x} is well known and depends functorially on the normalisation of the complete local ring Ô_{X_{k},x} (cf. [Bo-Lu-1]). Namely this boundary decomposes into a disjoint union of semi-open annuli X_{i} := X_{x,i} one corresponding to each minimal prime ideal η_{i} of Ô_{X_{k},x}. Let X′ := X − X_{x} which is a quasi-compact rigid space. Let f′ : Y′ → X′ and f_{x} : Y_{x} → X_{x} be (Galois)-covers (with group G). The (G)-cover f′ : Y′ → X′ extends to (G)-covers f_{i} : Y_{i} → X_{i} of each of the components of the boundary of X_{x}, and the germ of such an extension is unique (cf. [Ra]). A (G)-patching data in this context are (G-equivariant) isomorphisms between the germs of f_{i} : Y_{i} → X_{i} and the restriction of the initial (G)-cover f_{x} : Y_{x} → X_{x} to X_{i}. The rigid patching result is that given a (G)-patching data as above then there exists a unique, up to isomorphism, (G)-cover f : Y → X which induces the above covers above X′ and X_{x} when restricted to this analytic subspaces.

1.5. Local-global principle. As a direct consequence of the above patching results one obtains a local-global principle, which is certainly well known to the experts, for lifting of covers of curves. More precisely we have the following:

1.6. Proposition. let X be a proper and flat algebraic (or formal) R-curve and let Z := {x_{i}}_{i=1}^{n} be a finite set of closed points of X. Let f_{k} : Y_{k} → X_{k} be a finite generically separable (Galois)-cover (of group G) whose branch locus is contained in Z. Assume that for each i ∈ {1, n} there exists a (Galois)-cover f_{i} : Y_{i} → Spf Ô_{X,x_{i}} (of group G) which lifts the cover Ŷ_{k,i} → Spec Ô_{X_{k},x_{i}} induced by f_{k}, where Ŷ_{k,i} denotes the completion of Y_{k} above x_{i}. Then there exists a unique, up to isomorphism, (Galois)-cover f : Y → X (of group G) which lifts the cover f_{k}, and which is isomorphic to the cover f_{i} when pulled back to Spf Ô_{X,x_{i}}, for each i ∈ {1, n}.

Proof. After passing to the formal completion of X along its special fibre we may reduced to the case where X is a formal R-curve. We treat the case where Z = {x} consists of one point (the general case is similar). Let U_{k} := X_{k} − {x}, and let U be a formal open in X whose special fibre equals U_{k}. The étale cover f′_{k} : V_{k} → U_{k} induced by f_{k} above U_{k} can be lifted, uniquely, by the theorems of lifting of étale covers (cf. [Gr]) to an étale formal cover f′ : V → U. Let {℘_{i}}_{i=1}^{n} be the minimal prime ideals of Ô_{X,x} which contain π, and let X_{i} := X_{x,i} := Spf Ô_{x,℘_{i}} be the formal completion of the localisation of Ô_{X,x} at ℘_{i}. We have canonical morphisms X_{i} → X, and X_{i} → Spf Ô_{X,x}. The cover f′ (resp the given cover f_{i} : Y_{i} → Spec Ô_{X,x_{i}}) induces (by pull back) a cover f′_{i} : Y_{i} → X_{i} (resp. a cover f_{i} : Y_{i} → X_{i}). For each i ∈ {1, n}, the cover f_{i} and f′_{i} by construction are isomorphic when restricted to the special fibre Spec k((t_{i})) of X_{i}. Since both f_{i} and f′_{i} are étale and X_{i} is local and complete we deduce that they are isomorphic. Hence we obtain a patching data which allows us to patch the covers f′ : V → U and f_{i} : Y_{i} → Spf Ô_{X,x_{i}} in order to obtain a cover f : Y → X with the required properties. Now thanks to the formal GAGA theorem this cover is algebraic f : Y → X and has the desired properties. Moreover if the starting datas are Galois then the constructed cover is also Galois with the same Galois group.

1.7. Remark. Although the formal patching result 1.2 and the rigid patching result 1.4 are equivalent, we opted in this paper to use the formal patching result and the framework of formal geometry since this seems to be more convenient for most readers. However it should be clear that one could also use the framework of rigid geometry and adapt the content of this paper to this setting.

II. Computation of vanishing cycles.

The main result of this section is 2.4 which gives a formula which compares the dimensions of the space of vanishing cycles in a Galois cover f̃ : Y → X of group ℤ/pℤ between formal germs of R-curves, where R is a complete discrete valuation ring of inequal characteristic which contains a primitive p-th root of unity, where p is the residue characteristic, in terms of the degeneration type of f̃ above the boundaries of X.

2.1. In this section we consider a complete discrete valuation ring R of inequal characteristic, 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 of R, k the residue field of R, and λ := ζ − 1. We also denote by v_{K} the valuation of K which is normalised by v_{K}(π) = 1. We assume that the residue field k is algebraically closed. By a (formal) R-curve we mean a (formal) R-scheme of finite type which is normal, flat, and whose fibres have dimension 1. For an R-scheme X, we denote by X_{K} := X ×_{Spec R} Spec K the generic fibre of X, and X_{k} := X ×_{Spec R} Spec k its special fibre. In what follows by a (formal) germ X of an R-curve we mean that X := Spec O_{X,x} is the (resp. X := Spf Ô_{X,x} is the formal completion of the) spectrum of the local ring of an R-curve X at a closed point x. Let O_{x} be the local ring of X_{k} at x. Let δ_{x} := dim_{k} Õ_{x}/O_{x} where Õ_{x} is the normalisation of O_{x} in its total ring of fractions, and let r_{x} be the number of maximal ideals in Õ_{x}. The contribution to the arithmetic genus of the point x is by definition g_{x} := δ_{x} − r_{x} + 1. We will call the integer g_{x} the genus of the point x. The following lemma is easy to prove (cf. for example [Bo-Lu-1]).

2.1.1. Lemma. Let X_{k} be a proper reduced algebraic curve over k. Let X̃_{k} → X_{k} be the normalisation of X_{k}, and let {X_{i}}_{i∈I} be the irreducible components of X̃_{k}. Let {x_{j}}_{j∈J} be the singular point of X_{k}, which we assume to be rational. Let g(X_{k}) (resp. g(X_{i})) be the arithmetic genus of X_{k} (resp. the arithmetic genus of X_{i}). Then g(X_{k}) = ∑_{i∈I} g(X_{i}) + ∑_{j∈J} g_{x_{j}}.

2.2. Let f : Y → X be a finite cover between R-curves. Assume that the special fibres X_{k} and Y_{k} are reduced. Let y be a closed point of Y and let x be its image in X, which we assume to be a rational point. Let (x_{j})_{j∈J} be the points of the normalisation X̃_{k} of X_{k} above x, and for a fixed j let (y_{i,j})_{i∈I_{j}} be the points of the normalisation Ỹ_{k} of Y_{k} which are above x_{j}. Assume that the morphism f_{k} : Y_{k} → X_{k} is generically étale. Under this assumption we have the following local Riemann-Hurwitz formula which is due to Kato (cf. [Ka], and [Ma-Yo]):

(1) (g_{y} + δ_{y} − 1) = n(g_{x} + δ_{x} − 1) + d_{K} − d_{k}^{w}

Where n is the local degree at y, which is the degree of the morphism Spec Ô_{Y,y} → Spec Ô_{X,x} between the completion of the local rings of Y (resp X) at the point y (resp. x), d_{K} is the degree of the divisor of ramification in the morphism Spec(Ô_{Y,y} ⊗_{R} K) → Spec(Ô_{X,x} ⊗_{R} K). Let d_{i,j}^{w} := v_{x_{j}}(δ_{y_{i,j},x_{j}}) − e_{i,j} + 1, where δ_{y_{i,j},x_{j}} is the discriminant ideal of the extension Ô_{X̃_{k},x_{j}} → Ô_{Ỹ_{k},y_{i,j}} of complete discrete valuation ring, and e_{i,j} its ramification index. The integer d_{k}^{w} is equal to the sum ∑_{i,j} d_{i,j}^{w}. In 2.4 we will obtain a formula similair to (1) in the case where f is Galois of group ℤ/pℤ and which includes the case where f_{k} is generically radicial.

2.3. Compactification process. Let X := Spf Ô_{X,x} be the formal germ of an

R-curve at a closed point x. Let f̃: Y → X be a Galois cover of group ℤ/pℤ with Y local. We assume that the special fibre of Y_{k} is reduced (this can alaways be achieved after a finite extension of R). We will construct a compactification of the above cover f̃ which will allows us to compute the arithmetic genus of the closed point of Y. More precisely we will construct a Galois cover f : Y → X of degree p, between proper algebraic R-curves, a closed point y ∈ Y and its image x = f(y), such that the formal germ of X (resp. Y) at x (resp. at y) equals X (resp. Y), and such that the Galois cover f_{x} : Spf Ô_{Y,y} → Spf Ô_{X,x} induced by f between the formal germs at y and x is isomorphic to the above given cover f̃: Y → X. The construction of such a compactification is well known in the case where f̃ has étale reduction type on the boundaries (cf [Ma-Yo] and [Ra-2]). In the case of radicial reduction type of degree p on the boundaries one is able to carry out such a construction using the formal patching result in 1.2 and the result 3.1 in [Sa]. In fact it suffices to be able to treat the case of one boundary, which is easily done using the next proposition and the formal patching result.

2.3.1. Proposition. Let D := Spf R < 1/T > be the formal closed disc centered at ∞. Let D := Spf R[[T]]{T^{−1}}, and let D → D be the canonical morphism. Let f̃: Y → D be a non trivial torsor under a finite and flat R-group scheme of rank p, such that the special fibre of Y is reduced. Then there exists a Galois cover f : Y → D with group ℤ/pℤ whose pull back to D is isomorphic to the above given torsor f̃. More precisely, with the same notations as in [Sa] 3.1 we have the following depending on the several cases that occur there:

case a ) consider the cover f : Y → D given generically by an equation Z^{p} = λ^{p}T^{m}+1. This cover is an étale torsor above D under the group scheme H_{v_{K}(λ)} and induces an étale torsor f_{k} : Y_{k} → D_{k} in reduction. Moreover the genus of the smooth compactification of Y_{k} equals (m − 1)(p − 1)/2.

case b-1 ) Consider the cover f : Y → D given generically by an equation Z^{p} = T^{−r} where r := p − h. This cover is ramified at the generic fibre only above the point at infinity, the finite morphism f_{k} : Y_{k} → X_{k} is a μ_{p}-torsor outside infinity, and Y_{k} is smooth at infinity. Moreover the genus of the smooth compactification of Y_{k} equals 0.

case b-2 ) Consider the cover f : Y → D given generically by an equation Z^{p} = T^{−α}(T^{−m} + 1), where α is an integer such that m + α ≡ 0 mod (p). This cover is ramified at the generic fibre above infinity and above the distincts m-th root of unity. The finite morphism f_{k} : Y_{k} → D_{k} is a μ_{p}-torsor outside infinity and the distincts m-th root of unity, and Y_{k} is smooth. Moreover the genus of the smooth compactification of Y_{k} equals 0.

case c ) First if m ≤ 0 consider the cover f : Y → D given generically by an equation Z^{p} = 1 + π^{np}T^{m}. This cover is a torsor under the group scheme H_{n}, and its special fibre f_{k} : Y_{k} → X_{k} is a torsor under α_{p}. The affine curve Y_{k} has a unique singular point y which is the point above infinity and g_{y} = (−m − 1)(p − 1)/2. Secondly if m ≥ 0 consider the cover f : Y → D given generically by an equation Z^{p} = T^{−α}(T^{−m} + π^{pn}) where α is as in b-2. This cover is ramified above infinity and the m distinct roots of π^{−np}. The finite morphism f_{k} : Y_{k} → X_{k} is an α_{p}-torsor outside infinity, and the special fibre Y_{k} of Y is smooth. Moreover in both cases the smooth compactification of the normalisation of Y_{k} has genus 0.

Proof. Case a is straitforward. In case b-1 one has only to justify that Y_{k} is smooth at infinity. the cover f is given by the equation Z^{p} = T^{−r}, and after using the Bezout identity one reduces to an equation Z′^{p} = T^{−1} from which it follows directly that the complete local ring at the point above infinity is B = R[[Z′]] hence this point is smooth. In case b-2 one deduces in a similar way as above that Y_{k} is smooth. Case c: if m is negatif the α_{p}-torsor f_{k} : Y_{k} → X_{k} is given locally for the étale topology above the point at infinity by an equation z^{p} = t^{−m} where t := T mod (π). The computation of the arithmetic genus of the singularity above infinity follows then by a direct calculation (cf. example [Ra-1], or [Sai], 2.9). If m is positive and in order to see that Y_{k} is smooth one considers the Galois cover f′ : Y′ → P above the formal R-projective line P, obtained by gluing D with the formal unit closed disc D′ := Spf R < T > centered at 0, and given by the (same) equation Z^{p} = 1 + π^{np}T^{m}. The genus of the generic fibre of Y′ is (m − 1)(p − 1)/2. The finite morphism f′_{k} : Y′_{k} → P_{k} is an α_{p}-torsor outside infinity, and outside 0 coincides with the morphism f_{k}. Above 0 this torsor is given locally for the étale topology by an equation z^{p} = t^{m} where t := T mod (π), hence the arithmetic genus above 0 equals (m − 1)(p − 1)/2 from which one deduces that Y_{k} is smooth.

2.3.2. Proposition. Let X := Spf Ô_{x} be the formal germ of an R-curve at a closed point x, and let {X_{i}}_{i}^{n} be the boundaries of X. Let f̃: Y → X be a Galois cover of group ℤ/pℤ with Y local. Assume that Y_{k} and X_{k} are reduced. Then there exists a Galois cover f : Y → X of degree p, between proper algebraic R-curves, a closed point y ∈ Y and its image x = f(y), such that the formal germ of X (resp. Y) at x (resp. at y) equals X (resp. Y), and such that the Galois cover Spf Ô_{Y,y} → Spf Ô_{X,x} induced by f between the formal germs at y and x is isomorphic to the above given cover f̃: Y → X. Moreover the formal completion of X along its special fibre has a covering which consists of n closed formal discs D_{i} which are patched with X along the boundaries D_{i}, and the special fibre X_{k} of X consists of n smooth projective lines which intersect at the point x.

Proof. Let {℘_{i}}_{i=1}^{n} be the minimal prime ideals of Ô_{x} which contain π and which correspond to the branches (η_{i})_{i=1}^{n} of X_{k} at x, and let D_{i} := Spf Ô_{℘_{i}} be the formal completion of the localisation of X at ℘_{i}. If T_{i} is a lifting of a uniformising parameter of the branche (η_{i}) of X_{k} at x then Ô_{℘_{i}} is isomorphic to R[[T_{i}]]{T_{i}^{−1}}. For each i ∈ {1, n} consider a formal closed disc D_{i} := Spf R < 1/T_{i} > centered at infinity, and the canonical morphism D_{i} := Spf R[[T_{i}]]{T_{i}^{−1}} → D_{i}. As a consequence of the formal patching result, which is valid for coherent sheaves (cf. [Pr] theorem 3.4), one can patch X := Spf Ô_{x} with the D_{i}, via the choice for each i of an automorphism of D_{i}, in order to construct a proper formal R-curve X, and a closed point x ∈ X, such that the formal completion of X at x equals X. The special fibre X_{k} of X is a union of n smooth k-projective lines which intersect at the point x. Now the given cover f̃ induces a torsor f_{i} : Y_{i} → D_{i}, under a finite and flat R-group scheme of rank p for each i, and by the above lemma 3.3.1 one can find Galois covers Y_{i} → D_{i} of degree p which after pull back to D_{i} coincide with f_{i}, for each i ∈ {1, n}. The formal patching result again allows us then to patch these covers in order to construct a Galois cover f : Y → X of degree p with the desired properties. The formal R-curve is proper. By the formal GAGA theorems X is algebraic and the Galois cover f : Y → X is also algebraic.

The next result is the main one of this paper, it gives a formula which compares the dimensions of the space of vanishing cycles in a Galois cover of degree p between formal fibres.

2.4. Theorem. Let X := Spf Ô_{x} be the formal germ of an R-curve at a closed point x, with X_{k} reduced. Let f̃ : Y → X be a Galois cover of group ℤ/pℤ with Y local, and Y_{k} reduced. Let {℘_{i}}_{i∈I} be the minimal prime ideals of Ô_{x} which contain π and which correspond to the branches (η_{i})_{i∈I} of X_{k} at x, and let X_{i} := Spf Ô_{℘_{i}} be the formal completion of the localisation of X at ℘_{i}. For each i ∈ I, the above cover f̃ induces a torsor f̃_{i} : Y_{i} → X_{i} under a finite and flat R-group scheme of rank p, above the boundary X_{i}. For each i ∈ I, let (G_{k,i}, m_{i}, h_{i}) be the reduction type of f̃_{i} (cf. [Sa] 3.2). Let y be the closed point of Y. Then one has the following “local Riemman-Hurwitz formula”:

(2) 2g_{y} − 2 = p(2g_{x} − 2) + d_{η} − d_{s}

Where d_{η} is the degree of the divisor of ramification in the morphism f̃_{K} : Y_{K} → X_{K} induced by f̃, where X_{K} := Spec(Ô_{x} ⊗_{R} K) and Y_{K} := Spec(Ô_{Y,y} ⊗_{R} K), and d_{s} := ∑_{i∈I^{rad}}(m_{i} − 1)(p − 1) + ∑_{i∈I^{et}}(m_{i} − 1)(p − 1), where I^{rad} is the subset of I consisting of those i for which G_{k,i} is radicial, and I^{et} is the subset of I consisting of those i for which G_{k,i} is étale and m_{i} ≠ 0.

Proof. By 2.3.2 one can compactify the given morphism f̃. More precisely we constructed a Galois cover f : Y → X of degree p between proper algebraic R-curves, a closed point y ∈ Y and its image x = f(y), such that the formal germ of X (resp. Y) at x (resp. at y) equals X (resp. Y), and such that the Galois cover Spf Ô_{Y,y} → Spf Ô_{X,x} induced by f between the formal germs at y and x is isomorphic to the given cover f̃ : Y → X. The special fibre of X consists (by construction) of n-ditincts smooth projective lines which meet at the closed point x. The formal completion of X along its special fibre has a covering which consists of |I| formal closed unit discs which are patched with the formal fibre X along the boundaries X_{i}. The above formula (2) follows then by comparing the arithmetic genus of the generic fibre Y_{K} of Y and the one of the special fibre Y_{k}. By the precise informations given in 3.3.1 one can easily deduce that g(Y_{K}) = pg_{x} + (1 − p) + d_{η}/2 + ∑_{i∈I_{b}}(−m_{i} + 1)(p − 1)/2 + ∑_{i∈I_{c,>}}(−m_{i} + 1)(p − 1) where I_{b} is the subset of I consisting of those i for which the degeneration data correspond to the one in case b of [Sa] 3.1, and I_{c,>} is the subset of I consisting of those i for which the degeneration data correspond to the one in case c in [Sa] 3.1, and with m positif. On the other hand one has g(Y_{k}) = g_{y} + ∑_{i∈I_{a,>}}(−m_{i} − 1)(p − 1)/2 + ∑_{i∈I_{c,<}}(−m_{i} + 1)(p − 1), where I_{a,>} is the subset of I consisting of those i for which the degeneration data correspond to the one in case a of 2.4.1 with m_{i} ≠ 0, and I_{c,<} is the subset of I consisting of those i for which the degeneration data correspond to the one in case c in [Sa] 3.1, and with m_{i} < 0. Now since Y is flat g(Y_{K}) = g(Y_{k}) from which directly follows the formula (2).

III. Galois covers of degree p above germs of semi-stable curves. In

what follows we use the same notations as in 2.1. As a consequence of the above result 2.4 we will deduce some interesting results in this section in the case of a Galois cover Y → X, where X is the formal germ of a semi-stable R-curve at a closed point. These results will play an important role in the paper [Sa-1] in order to exhibit, and realise, the degeneration datas which describe the semi-stable reduction of Galois covres of degree p.

3.1. We start with the case of a Galois cover of degree p above a germ of a smooth point.

3.1.1. Proposition. Let X := Spf R[[T]] be the germ of a formal R-curve at a smooth point x. Let η be the generic point of the special fibre X_{k} of X. The completion of the localisation of X at η is X_{η} := Spf R[[T]]{T^{−1}}, which is the boundary of X. Let f : Y → X be a Galois cover of degree p, with Y local. Assume that the special fibre of Y is reduced. Let y be the unique closed point of Y_{k}. Let δ_{K} := r(p − 1) be the degree of the divisor of ramification in the morphism f : Y_{K} → X_{K}. We distinguish two cases: case 1 ) Y_{k} is unibranche at y. Let (G_{k}, m, h) be the degeneration type of f above the boundary X_{η}. Then necessarily r − m − 1 ≥ 0, and g_{y} = (r − m − 1)(p − 1)/2. case 2 ) Y_{k} has p-branches at y. Then the cover f has an étale completely split reduction of type (ℤ/pℤ, 0, 0) on the boundary, i.e. the induced torsor above Spf R[[T]]{T^{−1}} is trivial, in which case g_{y} = (r − 2)(p − 1)/2.

With the same notations as in 3.1.1, and as an immediate consequence, one can immediately see whether the point y is smooth or not. More precisely we have the following:

3.1.2. Corollary. We use the same notations as in 3.1.1 Then y is a smooth point, which is equivalent to g_{y} = 0, if and only if r = m+1. This in the case of radicial reduction type on the boundary is equivalent to r = − ord(ω), where ω is the associated differential form. In particular if the reduction is of multiplicative type on the boundary, i.e. G_{k} = μ_{p}, then g_{y} = 0 only if r = 1 or r = 0, since ord(ω) ≥ −1 in this case. Also if r = 1 and g_{y} = 0 then necessarily G_{k} = μ_{p}.

Next we give some examples of Galois covers of degree p above the formal germ of a smooth point which cover all the possibilities for the genus and the degeneration type on the boundary. Both in 3.1.3 and 3.1.4 we use the same notations as in 3.1.1. We first begin with examples with genus 0.

3.1.3. Examples. The following are examples given by explicit equations of the different cases, depending on the possible degeneration type, of Galois covers f : Y → X of degree p above X = Spf R[[T]], and where g_{y} = 0.

1 ) For m > 0 an integer prime to p, consider the cover given generically by the equation X^{p} = 1 + λ^{p}T^{−m}. Hier r = m + 1, and this cover has a reduction of type (ℤ/pℤ, m, 0) on the boundary.

2 ) For h ∈ 𝔽_{p}^{∗}, consider the cover given generically by the equation X^{p} = T^{h}. Hier r = 1, and this cover has a reduction of type (μ_{p}, 0, h) on the boundary.

3 ) Consider the cover given generically by the equation X^{p} = 1 + T. Hier r = 0, and this cover has a reduction of type (μ_{p}, −1, 0) on the boudary.

4 ) For n < v_{K}(λ), and m < 0, consider the cover given generically by the equation X^{p} = 1 + π^{np}T^{m}. Hier r = −m + 1, and this cover has a reduction of type (α_{p}, −m, 0) on the boudary.

5) For n < v_{K}(λ), consider the cover given generically by the equation X^{p} = 1+π^{np}T. Hier r = 0, and this cover has a reduction of type (α_{p}, −1, 0) on the boudary.

Next we give examples of Galois covers of degree p above formal germs of smooth points which lead to a singularity with positive genus.

3.1.4. Examples. The following are examples given by explicit equations of the different cases, depending on the possible reduction type, of Galois covers f : Y → X of degree p above X = Spf R[[T]], and where g_{y} > 0.

1 ) For m > 0 an integer prime to p, and m′ > m, consider the cover given generically by the equation X^{p} = 1 + λ^{p}(T^{−m} + πT^{−m′}). Hier r = m′ + 1, and this cover has a reduction of type (ℤ/pℤ, m, 0) on the boundary. Moreover the genus g_{y} of the closed point y of Y equals (m′ − m)(p − 1)/2.

2 ) For h ∈ 𝔽_{p}^{∗}, and m > 0 an integer prime to p, consider the cover given generically by the equation X^{p} = T^{h′}(T^{m} +a), where h′ is a positif integer such that m+h′ ≡ h mod (p), and a ∈ πR. Hier r = m + 1, and this cover has a reduction of type (μ_{p}, 0, h) on the boundary. Moreover the genus g_{y} of the closed point y of Y equals m(p − 1)/2.

3 ) For a positif integer m′, an integer h such that m′ + h ≡ 0 mod (p), and a ∈ πR, consider the cover given generically by the equation X^{p} = T^{h}(T^{m′} + a)(1 + T^{m}). Hier r = m′ + 1, and this cover has a reduction of type (μ_{p}, −m, 0) on the boudary. Moreover the genus g_{y} of the closed point y of Y equals (m′ + m)(p − 1)/2.

4 ) For n < v_{K}(λ), and integers m > 0 prime to p and m < m′, consider the cover given generically by the equation X^{p} = 1 + π^{np}(T^{−m} + πT^{−m′}). Hier r = m′ + 1, and this cover has a reduction of type (α_{p}, −m, 0) on the boudary. Moreover the genus g_{y} of the closed point y of Y equals (m′ + m)(p − 1)/2.

Note that in cases 5, 6 of 3.1.3, and 4 of 3.1.4, and in order to realise these covers above X, for a given n, one needs in general to perform a ramified extension of R.

3.2. Next we examine the case of Galois covers of degree p above formal germs at double points.

3.2.1. Proposition. Let X := Spf R[[S, T]]/(ST − π^{e}) be the formal germ of an Rcurve at an ordinary double point x of thikeness e. The special fibre of X consists of two irreducible components X_{1} and X_{2} with generic points η_{1} and η_{2}, corresponding to the prime ideal (π, T) (resp. (π, S)) of R[[S, T]]/(ST − π^{e}). The completion of the localisation of X at η_{1} (resp. η_{2}) is isomorphic to X_{1} := Spf R[[S]]{S^{−1}} (resp. X_{2} := Spf R[[T]]{T^{−1}}). These are the two boundaries of X. Let f : Y → X be a Galois cover of group ℤ/pℤ with Y local. Assume that the special fibre of Y is reduced. We assume that Y_{k} has two branches at the point y. Let δ_{K} := r(p − 1) be the degree of the divisor of ramification in the morphism f : Y_{K} → X_{K}. Let (G_{k,i}, m_{i}, h_{i}) be the type of reduction on the two boundaries of X, for i = 1, 2. Then necessarily r − m_{1} − m_{2} ≥ 0, and g_{y} = (r − m_{1} − m_{2})(p − 1)/2.

3.2.2. Proposition. We use the same notations as in 3.2.1. We consider the remaining cases: case 1) Y_{k} has p + 1 branches at y. We can assume that Y is completely split above η_{1}. Let (G_{k,2}, m_{2}, h_{2}) be the reduction type on the second boundary of X. Then necessarily r − m_{2} − 1 ≥ 0 and g_{y} = (r − m_{2} − 1)(p − 1)/2. case 2) Y_{k} has 2p branches at y. In this case Y is completely split above η_{1} and η_{2} and g_{y} = (r − 2)(p − 2)/2.

With the same notations as in 3.2.1, and as an immediate consequence, one can recognise whether the point y is a double point or not. More precisely we have the following.

3.2.3. Corollary. We use the same notations as in 3.2.1. Then y is an ordinary double point, which is equivalent to g_{y} = 0, if and only if x is an ordinary double point of thikeness divisible by p, and r = m_{1} + m_{2}. Moreover if g_{y} = 0, if r = 0, and if (G_{k,i}, m_{i}, h_{i}) is the reduction type on the boundary for i = 1, 2, then necessarily h_{1} + h_{2} = 0.

Proof. We need only to justify the last assertion. If both h_{1} and h_{2} equals 0 there is nothing to prove. Otherwise assume h_{1} ≠ 0. Then m_{1} = m_{2} = 0 (because g_{y} = 0 = m_{1} + m_{2}), and one sees easily that necessarily G_{k,1} = G_{k,2} = μ_{p}. So the cover f : Y → X is in this case a μ_{p}-torsor and its reduction f_{k} : Y_{k} → X_{k} is also μ_{p}-torsor given by an equation t^{p} = u, and ω := du/u is the associated differential form. The restriction ω_{i} of ω to the i-th branch , for i = 1, 2, is the differential form associated to the μ_{p}-torsor f_{i} : Y_{i} → X_{i} induced by f above the boundary X_{i} of X. The equality h_{1} + h_{2} = 0 follows then from the fact that the sum of the residues of ω on each branch equals 0, which is a property of the regular differential form at a double point.

Next we give some examples of Galois covers of degree p, above the formal germ of a double point, which leads to singularities with genus 0, i.e. double points, and such that r = 0. These examples will be used in the paper [Sa-1] in order to realise the “degeneration datas”.

3.2.4. Examples. The following are examples, given by explicit equations, of the different cases, depending on the possible type of reduction on the boundaries, of Galois covers f : Y → X of degree p above X = Spf R[[S, T]]/(ST − π^{e}), with r = 0, and where g_{y} = 0, for a suitable choice of e and R. Note that e = pt must be divisible by p. In all the following examples we have r = 0.

1 ) p-Purity: if f as above has an étale reduction type on the boundaries, and r = 0, then f is necessarily étale, and hence is completely split since X is strictly henselian.

2 ) Consider the cover given generically by an equation X^{p} = T^{h}, which leads to a reduction on the boundaries of type (μ_{p}, 0, h) and (μ_{p}, 0, −h).

3 ) For a fixed integer m > 0 prime to p, and after eventually a ramified extension of R choose t such that tm < v_{K}(λ), and consider the cover given generically by an equation X^{p} = 1 + S^{m}, which leads to a reduction on the boundaries of type (μ_{p}, −m, 0) and (α_{p}, m, 0).

4 ) For a fixed integer m > 0 prime to p, and after eventually a ramified extension of R choose t such that t = v(λ)/m, and consider the cover given generically by an equation X^{p} = λ^{p}/T^{m} + 1, which leads to a reduction on the boundaries of type (ℤ/pℤ, m, 0) and (μ_{p}, −m, 0).

5 ) For a fixed integer m > 0 prime to p, and after eventually a ramified extension of R, choose t such that tm < v_{K}(λ), and consider the cover given generically by an equation X^{p} = 1 + λ^{p}S^{−m}, which leads to a reduction on the boundaries of type (ℤ/pℤ, m, 0) and (α_{p}, −m, 0).

6 ) For a fixed integer m > 0 prime to p, and after eventually a ramified extension of R choose t and n such that tm + n < v_{K}(λ), and consider the cover given generically by an equation X^{p} = 1 + π^{np}S^{m}, which leads to a reduction on the boundaries of type (α_{p}, −m, 0) and (α_{p}, m, 0).

In fact one can describe Galois covers of degree p above formal germs at double points, which are étale above the generic fibre, and with genus 0. Namely they are all of the form given in the examples 3.2.4. In particular these covers are uniquely determined, up to isomorphism, by their degeneration type on the boundaries. More precisely we have the following:

3.2.5. Proposition. Let X be the formal germ of a semi-stable R-curve at an ordinary double point x. Let f : Y → X be a Galois cover of degree p, with Y_{k} reduced and local, and with f_{K} : Y_{K} → X_{K} étale. Let X_{i}, for i = 1, 2, be the boundaries of X. Let f_{i} : Y_{i} → X_{i} be the torsors induced by f above X_{i}, and let δ_{i} be the corresponding degree of the different (cf. 2.3.5). Let y be the closed point of Y. Assume that g_{y} = 0. Then there exists an isomorphism X ≃ Spf R[[S, T]]/(ST − π^{tp}) such that, if say X_{2} is the boundary corresponding to the prime ideal (π, S), the following holds:

a ) The cover f is generically given by an equation X^{p} = T^{h}, with h ∈ 𝔽_{p}^{∗}, which leads to a reduction on the boundaries of X of type (μ_{p}, m = 0, h) and (μ_{p}, m = 0, −h). Hier t > 0 can be any integer. In this case δ_{1} = δ_{2} = v_{K}(p).

b ) The cover f is generically given by an equation X^{p} = 1+T^{m}, for m > 0 an integer prime to p, such that tm < v_{K}(λ). In particular in this case necessarily t < v_{K}(λ). This cover leads to a reduction on the boundaries of X of type (α_{p}, m, 0) and (μ_{p}, −m, 0). In this case δ_{2} = v_{K}(p) = δ_{1} + (p − 1)tm

c ) The cover f is generically given by an equation X^{p} = 1 + T^{m}, with m > 0 an integer prime to p, such that tm = v_{K}(λ). In particular in this case t divides v_{K}(λ). This cover leads to a reduction on the boundaries of X of type (ℤ/pℤ, m, 0) and (μ_{p}, −m, 0). In this case δ_{2} = v_{K}(p) = δ_{1} + (p − 1)tm, and δ_{1} = 0.

d ) The cover f is generically given by an equation X^{p} = 1 + λ^{p}T^{−m}, with m > 0 an integer prime to p such that tm < v_{K}(λ), which leads to a reduction on the boundaries of X of type (ℤ/pℤ, m, 0) and (α_{p}, −m, 0). In particular we have necessarily t < v_{K}(λ). In this case δ_{1} = δ_{2} + (p − 1)tm, and δ_{2} = 0.

e ) The cover f is generically given by an equation X^{p} = 1 + π^{np}T^{m}, for a positif integer m prime to p, with n < v_{K}(λ), and n + tm < v_{K}(λ), which leads to a reduction on the boundaries of type (α_{p}, −m, 0) and (α_{p}, m, 0). In particular we have necessarily t < v_{K}(λ). In this case δ_{2} = δ_{1} + (p − 1)tm, and δ_{1} = (p − 1)(v_{K}(λ) − (n + tm)).

In all cases we have δ_{2} − δ_{1} = mt(p − 1).

Proof. We explain briefly the proof. Since g_{y} = 0, the thikness e = pt at the double point x of X is divisible by p. Say X ≃ Spf A′, where A′ = R[[T′, S′]]/(S′T′ − π^{tp}). The μ_{p}-torsor f_{K} : Y_{K} → X_{K} is given by an equation X^{p} = u_{K} where, u_{K} is a unit on X_{K}. Such a unit can be uniquely written as u_{K} := π^{n}T′^{m}u, where n and m are integers, and u ∈ R[[T′, S′]]/(S′T′ − π^{tp}) is a unit. Note first that necessarily n ≡ 0 mod(p), since Y_{k} is reduced. Assume first that gcd(m, p) = 1. Let t′ := T′ mod(π), let s′ := S′ mod(π), and let ū := u mod(π) which is a unit of k[[s′, t′]]/(s′t′). Let T := T′u^{1/m}. Then X ≃ Spf A, where A = R[[S, T]]/(ST − π^{tp}) for a suitable S ∈ A. Let B := A[X, Y]/(X^{p} − T^{m}, Y^{p} − S^{m}, XY − π^{t}). Then B is a finite flat algebra over A which is integrally closed (because B/πB is reduced), and Y = Spf B. The cover f : Y → X is thus generically given by an equation X^{p} = T^{m}, and we are in case a. Assume now that m ≡ 0 mod(p). Hier we have two cases:

case 1) ū is not a p-power. Then it is easy to see, that after changing the Kummer generator of the torsor f_{K}, one can assume that u is such that ū = 1 + t′^{m}v̄, where m is a positif integer prime to p and v̄ is a unit of k[[s′, t′]]/(s′t′). In particular ū = 1 + (t′v̄^{1/m})^{m} = 1+t^{m}, where t := t′v̄^{1/m}. Let T := T′v^{1/m}, where v := (u−1)/T′^{m}. We have X ≃ Spf R[[S, T]]/(ST −π^{tp}) for a suitable S ∈ A. The cover f : Y → X is thus generically given by an equation X^{p} = 1 + T^{m}, and we are in case b. Let X_{1} := Spf R[[S]]{S^{−1}} be the boundary of X corresponding to the ideal (π, T). The torsor f_{1} : Y_{1} → X_{1} induced by f above X_{1} is generically given by an equation X^{p} = 1 + π^{pt}S^{−m}, which imply that t ≤ v_{K}(λ), since f_{1} is not completely split. Moreover we are in case b if t < v_{K}(λ), and in case c if t = v_{K}(λ).

case 2) ū is a p-power. In this case, and after changing the Kummer generator of the torsor f_{K}, we can assume that this torsor is given by an equation X^{p} = 1 + π^{n′}v, where v ∈ A′ does not belong to the ideal πA′, and v is not a p-power mod π. Also one can check, after localisation and completion at the ideal (π, S′) as above (namely by looking what happend above the boundary X_{1}), that necessarily n′ ≤ pv_{K}(λ), and n′ = np is divisible by p (cf. for example [Gr-Ma] proof of 1.1, and [He] proof of 1.6, chap 5). Moreover after changing the Kummer generator of the torsor f_{K}, one can assume that v̄ = t′^{m}v̄′, where m is an integer prime to p, and v′ is a unit. Let T := T′(v/T′^{m})^{1/m}. We have X ≃ Spf R[[S, T]]/(ST − π^{tp}) for a suitable S ∈ A′. The cover f : Y → X is thus generically given by an equation X^{p} = 1 + π^{np}T^{m}, and we are in case d or e. Let X_{1} be as above the boundary of X corresponding to the ideal (π, T). The torsor f_{1} : Y_{1} → X_{1} induced by f above X_{1} is given by an equation X^{p} = 1 + π^{p(n+t)}S^{−m}, which imply that t + n ≤ v_{K}(λ) since f_{1} is not completely split. Moreover we are in case d if t + n = v_{K}(λ) and in case e if n + t < v_{K}(λ).

3.3. Variation of the different. (Compare with [He], 5.2) The following lemma, which is a direct consequence of 3.2.5, describes how does the degree of the different vary from one boundary to another in a cover f : Y → X as in 3.2.5.

3.3.1. Proposition. Let X be the formal germ of a semi-stable R-curve at an ordinary double point x. Let f : Y → X be a Galois cover of degree p, with Y_{k} reduced and local, and with f_{K} : Y_{K} → X_{K} étale. Let y be the closed point of Y. Assume that g_{y} = 0, which imply necessarily that the thikness e = pt of the double point x is divisible by p. For each integer 0 < t′ < t, let X_{t′} → X be the blow-up of X at the ideal (π^{pt′}, T). The special fibre of X_{t′} consists of a projective line P_{t′} which meets two germs of double points x and x′. Let η be the generic point of P_{t′}, and let v_{η} be the corresponding discrete valuation of the function field of X. Let f_{t′} : Y_{t′} → X_{t′} be the pull back of f, which is a Galois cover of degree p, and let δ(t′) be the degree of the different induced by this cover above v_{η} (cf. [Sa] 3.1). Also let X_{i}, for i = 1, 2, be the boundaries of X. Let f_{i} : Y_{i} → X_{i} be the torsors induced by f above X_{i}, let (G_{k,i}, m_{i}, h_{i}) be their degeneration type, and let δ_{i} be the corresponding degree of the different (cf. [Sa] 3.1). Say δ_{1} = δ(0), δ_{2} = δ(t), and δ(0) ≤ δ(t). We have m := −m_{1} = m_{2} say is positif. Then the following holds:

1) If δ(t′) < v_{K}(p), for every 0 ≤ t′ ≤ t. Then for 0 ≤ t_{1} ≤ t_{2} ≤ t we have δ(t_{2}) = δ(t_{1}) + m(p − 1)(t_{2} − t_{1}), and δ(t′) is an increasing function of t′.

2) If there exists 0 ≤ t′ ≤ t such that δ(t′) = v_{K}(p). Then there exists 0 ≤ t_{1} ≤ t_{2} ≤ t such that δ(t′) = v_{K}(p) is constant for t_{1} ≤ t′ ≤ t_{2}, δ(t′) is increasing as t′ increases from 0 to t_{1}, and δ(t′) decreases as t′ increases from t_{2} to t.

3.3.2. Remark. In [Gr-Ma] and [He] where studied order p-automorphisms of open p-adic discs and p-adic annuli. In their aproach one writes such an automorphism as a formal series and one deduce some results, e.g. the only if part of 3.1.2 and 3.2.3, using the Weirstrass preparation theorem. The aproach adobted hier, and which consists on directly computing the vanishing cycles first, I believe provides another way to study such automorphisms. Namely these are those covers above formal fibres of semi-stable R-curves with genus 0, and one can easily writes down Kummer equations which lead to such covers as in 3.1.3, 3.2.4, and 3.2.5.

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