Cyclic p-groups and semi-stable reduction of curves in equal characteristic p > 0. I

Mohamed Saïdi

Abstract

In this paper we study the semi-stable reduction of p and p^{2}-cyclic covers of curves in equal characteristic p > 0. The main tool we use is the classical Artin-Schreier-Witt theory for p^{n}-cyclic covers in characteristic p. Although the results of this paper concern only the cases of degree p and p^{2}-cyclic cover we develop the techniques and the framework in which the general cyclic case can be studied.

0. Introduction. This paper is a further attempt to understand the degeneration of wildly ramified Galois covers of algebraic varieties. Let R be a complete discrete valuation ring with fraction field K and residue field k of characteristic p > 0. Let X be a smooth (or more generally semi-stable) R-scheme and let Y → X be a finite Galois cover with group G. The basic problem we are interested in is to understand the reduction of Y and in particular its semi-stable reduction if it exists (e.g. case of curves). The case where the cardinality of G is prime to p is by now classical and well understood (cf. [SGA-1], and [S-2] for the case of semi-stable curves). Although the case of higher dimensional semi-stable schemes is not available in the literature it can be developed using the ideas in [S-2] and higher dimensional formal patching techniques. The case where the cardinality of G is divisible by p is much less understood and presents more technical difficulties du to a phenomena of wild ramification that appears. Raynaud first attempted to understand this situation in [R] where a rather general “qualitative” result is proven in the case where G is a p-group and X is a proper and smooth curve. Beside this the only case which is “explicitly” understood is the basic case where G is cyclic of order p and the ring R has inequal characteristics (cf. [G-Ma], [H], [L], [Ma], [S], [S-1], [S-3]...). Though one also has some results in inequal characteristics in the special case where p strictly divides the cardinality of G (cf. [R-1] and [We]): this case is essencially a by-product of the above discussed cases. One of the main objectives of this work is to go beyond these basic cases. In this paper we study the case where G is a cyclic p-group and R has equal characteristics. The above situation in equal characteristics p has been inexplored so far despite the important potential applications of such a theory (cf. explanations below). Though this case is in principle easier than the case of inequal characteritics, but still the two situations are well related, as it is known to experts (this is part of the philosophy that the Artin-Schreier-Witt theory can be viewed as a degeneration of the Kummer theory). Also the results in equal characteristics, beside their own interest, can shed some lights on the situation in inequal characteristics. Although the results stated in this paper concern only the cases of degree p and p^{2}-cyclic cover we develop, pursueing the ideas and the set-up developed in [S], [S-1], [S-3], the techniques and the framework in which the general p^{n}-cyclic case can be studied. In what follows we review the content and results of this paper.

In section I we recall the classical Artin-Schreier-Witt theory, which is the main tool we use in this paper, and which provides (generically) explicit equations for p^{n}-cyclic covers in characteristic p. In section II we consider the following situation: let X be a formal R-scheme of finite type which is normal connected and flat over R, with geometrically integral fibres, and let f : Y → X be a p^{n}-cyclic cover, with Y normal, which is an étale torsor on the generic fibre of X. We assume that the special fibre Y_{k} of Y is reduced. We are interested in describing the map f and its special fibre f_{k} : Y_{k} → X_{k}. One of our main results is the following:

Theorem 2.2.1. Assume deg(f) = p. Then the cover f : Y → X has the structure of a torsor under a finite and flat R-group scheme of rank p over X.

Moreover we give an explicit description of the group schemes which appear as the group of the torsor in this situation (cf. 2.1) and in particular we provide “integral” equations for the torsor f which also provide (by reduction) equations for its special fibre. Next we study the case of p^{2}-cyclic covers. Our main result is theorem 2.4.3. One of the main consequences of 2.4.3 is that the analog of the above theorm doesn’t hold: namely one can find (generic) examples of f as above of degree p^{2} which doesn’t have the structure of a torsor under a finite and flat R-group scheme of rank p^{2} over X (cf. 2.4.5). Such examples were not known in the literature before. We are however able in 2.4.3 to find “integral” equations for f which provide by reduction equations for its special fibre f_{k} : Y_{k} → X_{k}. In other terms we explain how the Artin-Schreier-Witt equations of degree p^{2} degenerate. The proof of 2.4.3 is rather involved and uses the technical lemma 2.4.2. Moreover we exhibit the cases as above where f has the structure of a torsor under a finite and flat R-group scheme of rank p^{2} over X (cf. 2.4.3 and 2.4.4), in which case we explicit the group schemes which appear as groups of the torsor. These group schemes are basically obtained by “twisting” the Artin-Schreier-Witt theory (cf. 2.3 for more details). We are also able to associate some degeneration data to the cover f which determine explicitly the cover f_{k} (cf. 2.4.8). These degeneration data will play an important role in VI were we are able to reconstruct p^{2}-cyclic covers of formal germs of R-curves from the “ degeneration data”. Finally we classify in II p and p^{2}-cyclic covers above boundaries of formal germs of R-curves depending on their reduction type (cf. 2.5.1 and 2.6.1).

In section III we develop in equal characteristic p the technique of computation of vanishing cycles as initiated in [S-1] (cf. also [Ma-Y]) and which is based on the technique of compactification of covers between formal germs of R-curves (cf. 3.2). Our main result is theorem 3.2.3 which gives an explicit formula which compares the dimensions of the spaces of vanishing cycles in a p-cyclic cover between formal germs of R-curves and which depends on the degeneration type of the cover on the boundaries. This formula can easily be generalised to the case of p^{2}-cyclic covers (cf. proposition 4.1.1). We also study examples of p and p^{2}-cyclic covers above formal germs of semi-stable curves (i.e. above a smooth or a double point). Theses examples play an important role in the next sections V and VI. We are for example able to classify p and p^{2}-cyclic covers f : Y → X between formal germs of double points wich are étale above the generic fibre X_{K} of X (cf. 3.3.9 and 4.2.11).

In section V and VI we consider the following local situation: let f : Y → X be a cyclic cover of degree p or p^{2} above the formal germ X ≃ Spf R[[T]] of an R-curve at a smooth point. We study the semi-stable reduction of f. We are able to exhibit in 5.2.2 and 6.2.1 “degeneration” data which completly determine the geometry of a semi-stable model of Y. These data are of geometric and combinatorial nature and leave over the residue field k of R (cf. 5.2.2 and 6.2.1 for more details). One of the main results of this paper is that we are able to reconstruct the cover f from these data. More precisely let Deg be the set of isomorphism classes of the degeneration datas or rank p (resp. rank p^{2}) as defined in 6.2.1 (resp. 5.2.2) and let L be the function field of the geometric fibre of X. Then we prove using the results in III and IV the following:

Theorem. 5.2.4, 6.2.3 Let G be a cyclic p-group of order p (resp. of order p^{2}). Then there exists a canonical specialisation map Sp : H_{et}^{1}(Spec L, G) → Deg which is surjective.

Although the above result is local the techniques and results we use in the proof of 5.2.4 and 6.2.3 can be used to prove a global result of lifting for finite covers of degree p or p^{2} between semi-stable curves (such a result is proven in [M] for p-covers).

Finally let’s mention some potential applications of this work. The results of this paper can be used in order to determine the semi-stable reduction of the Drinfeld modular curves and that of Galois covers of the projective line in characteristic p with Galois groups having a cyclic p-Sylow subgroup of order p or p^{2}. These results also can be used in the study of automorphism groups of curves in positive characteritics. One of the motivations of this work was also to explore the possibility of defining reasonable Hurwitz spaces for wildly ramified Galois covers of curves. It seems that the only case where such a definition is possible is the case of (Galois) covers of degree p: the right moduli problem to classify being torsors of rank p. Unfortunately our results show that when we degenerate Galois covers of degree higher than p then we loose in general the torsor structure. this makes it really difficult to imagine a reasonable definition of what the Hurwitz spaces should be in general. May be the right point of view to adopt here is to restrict simply to covers where one keeps the torsor structure.

Although the results of this paper concern only the cyclic groups of order p and p^{2} they can be generalised, using the set-up and techniques we develop, to the general p^{n}- cyclic case. I hope to come back to this question in the future. This work was done during my visit to the Max-Planck-Institut Für Mathematik in Bonn. I would like very much to thank the directors of the Institut for their invitation and for the wonderful working atmosphere.

I. Artin-Schreier-Witt theory of p^{n}-cyclic covers in characteristic p.

1.1. In this section we review the Artin-Schreier-Witt theory (first developed in [W]) which provides explicit equations describing cyclic covers of degeree p^{n} in characteristic p. We refer the reader to the modern treatment of the theory as in [D-G].

Let X be a scheme of characteristic p and denote by X_{et} the étale site on X. Let n > 0 be an integer. We denote by W_{n,X} (or simply W_{n} if there is no confusion) the sheaf of Witt vectors of length n on X_{et} (cf. [D-G], chapitre 5, 1). In the sequel any addition or substraction of Witt vectors will mean the addition and substarction in the sense of Witt theory. We denote by F the Frobenius endomorphism of W_{n,X} which is locally defined by F.(x_{1}, x_{2}, ..., x_{n}) = (x_{1}^{p}, x_{2}^{p}, ..., x_{n}^{p}), for a Witt vector (x_{1}, x_{2}, ..., x_{n}) of length n, and by Id the identity automorphism of W_{n,X}. The following sequence is exact on X_{et}:

(1) 0 → (ℤ/p^{n}ℤ)_{X} ⟶^{i_{n}} W_{n,X} ⟶^{F − Id} W_{n,X} → 0

where (ℤ/p^{n}ℤ)_{X} denotes the constant sheaf (ℤ/p^{n}ℤ) on X_{et} and i_{n} is the natural monomorphism which applies 1 ∈ ℤ/p^{n}ℤ to 1 ∈ W_{n} (cf [G-D], chapitre 5, 5.4). From the long cohomology exact sequence associated to (1) one deduces the following exact sequence:

(2) W_{n,X}(X) ⟶^{F − Id} W_{n,X}(X) → H_{et}^{1}(X, ℤ/p^{n}ℤ) → H_{et}^{1}(X, W_{n}) ⟶^{F − Id} H_{et}^{1}(X, W_{n})

Assume that X = Spec A is affine in which case we have H_{et}^{1}(Spec A, W_{n}) = 0 and hence an isomorphism: H_{et}^{1}(Spec A, ℤ/p^{n}ℤ) ≃ W_{n,A}(A)/(F − Id)(W_{n,A}(A)). This isomorphism has the following interpretation: to an étale ℤ/p^{n}ℤ-torsor f : Y → X = Spec A above X corresponds a Witt vector (a_{1}, a_{2}, ..., a_{n}) ∈ W_{n,A}(A) of length n which is uniquely determined modulo addition of elements of the form F.(b_{1}, b_{2}, ..., b_{n})−(b_{1}, b_{2}, ..., b_{n}). Moreover the equations F.(x_{1}, x_{2}, ..., x_{n})−(x_{1}, x_{2}, ..., x_{n}) = (a_{1}, a_{2}, ..., a_{n}) where the x_{i} are indeterminates are equations for the torsor f. More precisely there is a canonical factorisation

of f as Y = Y_{n} ⟶^{f_{n}} Y_{n−1} ⟶^{f_{n−1}} ... ⟶^{f_{2}} Y_{1} ⟶^{f_{1}} Y_{0} := X where each Y_{i} = Spec B_{i} is affine

and f_{i} : Y_{i} := Spec B_{i} → f_{i−1} := Spec B_{i−1} is the étale ℤ/pℤ-torsor corresponding to the algebra extension B_{i+1} := B_{i}[x_{i}]. In the general case (where H_{et}^{1}(X, W_{n}) ≠ 0) the above equations provide local equations for an étale ℤ/p^{n}ℤ-torsor in characteristic p.

1.2. Examples. In what follows X is a scheme of characteristic p.

1.2.1. ℤ/pℤ-Torsors. Let f : Y → X be an étale ℤ/pℤ-torsor. Then locally f is given by an equation x^{p} − x = a where a is a regular function on X which is uniquely defined up to addition of elements of the form b^{p} − b for some regular function b.

1.2.2. ℤ/p^{2}ℤ-Torsors. Let f : Y → X be an étale ℤ/p^{2}ℤ-torsor. Then we have a canonical factorisation of f as: Y =: Y_{2} ⟶^{f_{2}} Y_{1} ⟶^{f_{1}} X where f_{2} and f_{1} are étale ℤ/pℤtorsors. The torsor f is locally given by equations of the form:

F.(x_{1}, x_{2}) − (x_{1}, x_{2}) := (x_{1}^{p} − x_{1}, x_{2}^{p} − x_{2} − p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}x_{1}^{pk}(−x_{1})^{p−k}) = (a_{1}, a_{2})

for some regular functions a_{1} and a_{2} on X and the Witt vector (a_{1}, a_{2}) is uniquely determined up to addition (in the Witt theory) of vectors of the form:

(b_{1}^{p}, b_{2}^{p}) − (b_{1}, b_{2}) := (b_{1}^{p} − b_{1}, b_{2}^{p} − b_{2} − p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}b_{1}^{pk}(−b_{1})^{p−k})

Thus locally the torsor f_{1} is defined by the equation:

x_{1}^{p} − x_{1} = a_{1}

and f_{2} by the equation:

x_{2}^{p} − x_{2} = a_{2} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}x_{1}^{pk}(−x_{1})^{p−k}

Moreover if we replace the vector (a_{1}, a_{2}) by the vector (a_{1}, a_{2}) + (b_{1}^{p}, b_{2}^{p}) − (b_{1}, b_{2}) the above equations are replaced by:

x_{1}^{p} − x_{1} = a_{1} + b_{1}^{p} − b_{1}

and:

x_{2}^{p} − x_{2} = a_{2} + b_{2}^{p} − b_{2} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}x_{1}^{pk}(−x_{1})^{p−k} − p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}b_{1}^{pk}(−b_{1})^{p−k}−

p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}(b_{1}^{p} − b_{1})^{k}(a_{1})^{p−k}

respectively.

II. Degeneration of p and p^{2}-cyclic covers in equal characteristic

p > 0.

In all this paragraph we use the following notations: R is a complete discrete valuation ring of equal characteristic p > 0 with residue field k and fraction field K := Fr R. We denote by π a uniformising parameter of R.

2.1. The group schemes M_{n} (cf. also [M], 3.2). Let n ≥ 0 be an integer and let 𝔾_{a,R} = Spec R[X] be the additive group scheme over R. The map:

ϕ_{n} : 𝔾_{a,R} → 𝔾_{a,R}

given by:

X → X^{p} − π^{(p−1)n}X

is an isogeny of group schemes. The kernel of ϕ_{n} is denoted by M_{n,R} := M_{n}. We have M_{n} := Spec R[X]/(X^{p} − π^{(p−1)n}X) and M_{n} is a finite and flat R-group scheme of rank p. Further the following sequence is exact in the fppf topology:

(3) 0 → M_{n} → 𝔾_{a,R} ⟶^{ϕ_{n}} 𝔾_{a,R} → 0

If n = 0 then the sequence (3) is the Artin-Schreir sequence which is exact in the étale topology and M_{0} is the étale constant group scheme (ℤ/pℤ)_{R}. If n > 0 the sequence (3) has a generic fibre which is isomorphic to the étale Artin-Schreier sequence and a special fibre isomorphic to the radicial exact sequence:

(4) 0 → α_{p,k} → 𝔾_{a,k} ⟶^{x^{p}} 𝔾_{a,k} → 0

Thus if n > 0 the group scheme M_{n} has a generic fibre which is étale isomorphic to (ℤ/pℤ)_{K} and its special fibre is isomorphic to the infinitesimal group scheme α_{p,k}. Let X be an R-scheme. The sequence (3) induces a long cohomology exact sequence:

(5) 𝔾_{a,R}(X) ⟶^{ϕ_{n}} 𝔾_{a,R}(X) → H_{fppf}^{1}(X, M_{n}) → H_{fppf}^{1}(X, 𝔾_{a,R}) ⟶^{ϕ_{n}} H_{fppf}^{1}(X, 𝔾_{a,R})

The cohomology group H_{fppf}^{1}(X, M_{n}) classifies the isomorphism classes of fppf-torsors with group M_{n} above X. The above sequence allows the following description of M_{n}- torsors: locally a torsor f : Y → X under the group scheme M_{n} is given by an equation T^{p} − π^{(p−1)n}T = a where T is an indeterminate and a is a regular function on X, which is uniquely defined up to addition of elements of the form b^{p} − π^{(p−1)n}b, for some regular function b. In particular if H_{fppf}^{1}(X, 𝔾_{a,R}) = 0 (e.g. if X is affine) then an M_{n}-torsor above X is globally defined by an equation as above.

2.2. Degeneration of étale ℤ/pℤ-torsors. In what follows let X be a formal

R-scheme of finite type which is normal connected and flat over R. Let X_{K} := X ×_{R} K (resp. X_{k} := X ×_{R} k) be the generic (resp. special) fibre of X. By generic fibre of X we mean the associated K-rigid space (cf. [B-L]). We assume further that the special fibre X_{k} is integral. Let η be the generic point of the special fibre X_{k} and let O_{η} be the local ring at η which is a discrete valuation ring with fraction field K(X): the function field of X. Let f_{K} : Y_{K} → X_{K} be a non trivial étale ℤ/pℤ-torsor with Y_{K} geometrically connected and let K(X) → L be the corresponding extension of function fields. The main result of this section is the following:

2.2.1. Theorem. Assume that the ramification index above O_{η} in the extension K(X) → L equals 1. Then the torsor f_{K} : Y_{K} → X_{K} extends to a torsor f : Y → X under a finite and flat R-group scheme of rank p with Y normal. Let δ be the degree of the different above η in the extension K(X) → L. Then the following cases occur:

a) δ = 0 in which case f is an étale torsor under the group scheme M_{0} and f_{k} : Y_{k} → X_{k} is then an étale ℤ/pℤ-torsor.

b) δ > 0 in which case δ = n(p − 1) for a certain integer n ≥ 1 and f is a torsor under the group scheme M_{n}. Further f_{k} : Y_{k} → X_{k} is a radicial torsor under the k-group scheme α_{p}.

Note that starting from a torsor f_{K} : Y_{K} → X_{K} as in 2.2.1 the condition that the ramification index above O_{η} equals 1 is always satisfied after eventually a finite extension of R (cf. e.g. [E]).

Proof. We denote by v the discrete valuation of K(X) corresponding to the valuation ring O_{η} which is normalised by v(π) = 1 (note that π is a uniformiser of O_{η}). We first start with the special case where H_{et}^{1}(X_{K}, 𝔾_{a,K}) = 0. In this case the torsor f_{K} is given by an Artin-Schreier equation of the form T^{p} − T = a_{K} where a_{K} is a regular function on X_{K} and we have a_{K} = π^{m}a where m ∈ ℤ is an integer and a is a regular function on X (i.e. a function with v(a) = 0). First note that necessarily m ≤ 0 for if m > 0 then a_{K} = b^{p} −b where b = aπ^{m} +(aπ^{m})^{p} +(aπ^{m})^{p^{2}} +...+(aπ^{m})^{p^{i}} +.. (since X_{K} is complete for the π-adic topology the above expression converges) and this contradicts the fact that f_{K} is a non trivial torsor. If m = 0 then the equation T^{p} − T = a defines an étale ℤ/pℤ-torsor f : Y → X above X which coincides with f_{K} on the generic fibre and we are in the case a). In this case the étale torsor f_{k} : Y_{k} → X_{k} is given by the Artin-Schreier equation T^{p} − T = ā where ā is the image of a modulo π. Next we treat the case where m < 0. In this case −m = np is necessarily divisible by p for otherwise the extension K(X) → L is totally ramified above O_{η}. Assume now first that the image ā of a modulo π via the canonical map Γ(X, O(X)) → Γ(X, O(X))/πΓ(X, O(X)) is not a p-power. Consider the cover f : Y → X given by the equation T̃^{p} − π^{n(p−1)}T̃ = a. Then f is an fppf-torsor under the group scheme M_{n} which coicides with f_{K} on the generic fibre (consider the change of variables T := T̃/π^{n}) and its special fibre f_{k} : Y_{k} → X_{k} is the α_{p}-torsor given by the equation t^{p} = ā.

In the case where ā is a p-power then two cases can occur: either ā is a p^{s}-power for any positive integer s in which case necessarily a ∈ R and the torsor f_{K} is just the extension of scalars of X_{K} from K to the extension K′ of K defined by the equation T^{p} − T = π^{m}a and this contradicts the assumption that Y_{K} is geometrically connected. Hence this case doesn’t occur. So necessarily there exists a positive integer r such that ā is a p^{r} power but not a p^{r+1} power. We assume for simplicity that r = 1 (the general case r > 1 is treated in a similar way). Let ā = b̄^{p} so that a = b^{p} + πb̃ where b and b̃ are functions on X_{K} and b is a function which reduces to b̄ modulo π. Our equation is then of the form T^{p} −T = (b/π^{n})^{p} +b̃/π^{(pn−1)} and after adding (b/π^{n})−(b/π^{n})^{p} to the right hand side which doesn’t change the torsor f_{K} we get the equation T^{p} − T = (b/π^{n}) + b̃/π^{(pn−1)} which can also be written in the form T^{p} − T = (b/π^{n}) + b′/π^{n′} where b′ is a function with v(b′) = 0 and n′ ≤ pn − 1. If n > n′ then n = ps is necessarily divisible by p and the equation T̃^{p} − π^{s(p−1)}T̃ = b + π^{n−n′}b′ defines a torsor f : Y → X under the group scheme M_{s} which coincides with f_{K} on the generic fibre and its special fibre f_{k} : Y_{k} → X_{k} is the α_{p}-torsor given by the equation t̃^{p} = b̄. In the case where n′ ≥ n then n′ = s′p is necessarily divisible by p. In this case if b̄′ (resp. b̄′ + b̄ in case n′ = n) is not a p power (where b̄ and b̄′ denote the reduction of b resp. b′ modulo π) then the equation T̃^{p} − π^{s′(p−1)}T̃ = π^{n′−n}b + b′ defines a torsor f : Y → X under the group scheme M_{s′} which coincides with f_{K} on the generic fibre and its special fibre f_{k} : Y_{k} → X_{k} is the α_{p}-torsor given by the equation t̃^{p} = b̄′ (resp. t̃^{p} = b̄′ + b̄ in case n = n′). Otherwise if b̄′ (or b̄′ + b̄ in case n = n′) is a p-power then we repeat the same procedure as above and since n and n′ decrease at each step this process must stop at some step and we end up with an equation of the form T̃^{p} − π^{r(p−1)}T̃ = b̃ where b̃ is a function whose reduction modulo π is not a p-power, for some positive integer r. Hence the required result.

The argument in the general case is similar to the one used in [S] proof of 2.4. More precisely in general there exists an open covering (U_{i})_{i} of X and regular functions ã_{i} ∈ Γ(U_{i,K}, O_{X}) (where U_{i,K} := U_{i} ×_{R} K) which are defined up to addition of functions of the form b_{i}^{p} −b_{i} and such that the torsor f_{K} is defined by the equation T_{i}^{p} −T_{i} = ã_{i} above U_{i,K}. Now the above discussion shows that after some modifications (of the type used above) the torsor f_{K} can be defined above each open U_{i,K} by an equation T̃_{i} − π^{n_{i}(p−1)}T̃ = a_{i} for some uniquely detemined integer n_{i} ≥ 0 such that in case n_{i} > 0 the image ā_{i} of a_{i} modulo π is not a p-power and such that the degree of the different δ_{i} above the generic point η of U_{i,k} := U_{i} ×_{R} k equals n_{i}(p − 1). From this we deduce that all n_{i} =: n are equal. Then the M_{n}-torsor f : Y → X given locally by the equation T̃_{i} − π^{n(p−1)}T̃_{i} = a_{i} above the open U_{i} coincides on the generic fibre with the torsor f_{K}.

2.2.2. It follows from 2.2.1 that an étale ℤ/pℤ-torsor above the generic fibre X_{K} of X induces canonically a degeneration data which is a torsor above the special fibre X_{k} of X under a finite and flat k-group scheme which is either étale or of type α_{p}. Reciprocally we have the following result of lifting of such a degeneration data.

2.2.3. Proposition. Assume that X is affine. Let f_{k} : Y_{k} → X_{k} be a torsor under a finite and flat k-group scheme which is étale (resp. of type α_{p}). Then f_{k} can be lifted to a torsor f : Y → X under a finite and flat R-group scheme of rank p which is étale (resp. isomorphic to M_{n} where n > 0).

Proof. Since X is affine the torsor f_{k} is given by an equation x^{p} − x = ā where ā is a regular function on X_{k} (resp. an equation x^{p} = ā where ā is a regular function on X_{k}). Let a be a regular function on X which reduces to ā modulo π. The equation X^{p} − X = a (resp. X^{p} − π^{n(p−1)}X = a where n > 0 is an integer) defines a cover f : Y → X above X which has the structure of a torsor under the étale group scheme (ℤ/pℤ)_{R} (resp. under the group scheme M_{n}) and which clearly induces the torsor f_{k} on the special fibre X_{k}.

2.2.4. Remark. If X is no more affine one can find examples, where X is actually a proper and smooth R-curve, of an α_{p}-torsor above the special fibre X_{k} of X which can not be lifted to a torsor above X under a finite and flat R-group scheme of rank p which is étale above the generic fibre of X. If X_{k} is a proper and smooth curve over k such a lifting is however possible after eventually replacing X by another R-curve X′ with special fibre X_{k}.

2.3. The group schemes W_{m_{1},m_{2}}. Let m_{1} and m_{2} be two positive integers such that m_{2} − pm_{1} ≥ 0. We define the twisted R-Witt group scheme W_{m_{1},m_{2}} of length two as follows: scheme theoretically W_{m_{1},m_{2}} ≃ 𝔾_{a,R}^{2} and the group law is defined by:

(x_{1}, x_{2}) + (y_{1}, y_{2}) := (x_{1} + y_{1}, x_{2} + y_{2} − p^{−1}π^{m_{2}−pm_{1}} ∑_{k=1}^{p−1} \binom{p}{k}x_{1}^{k}y_{1}^{p−k})

The generic fibre (W_{m_{1},m_{2}})_{K} of W_{m_{1},m_{2}} is isomorphic to the Witt group scheme W_{2,K} via the map:

(W_{m_{1},m_{2}})_{K} → W_{2,K}
(x_{1}, x_{2}) ⟶ (x_{1}/π^{m_{1}}, x_{2}/π^{m_{2}})

and its special fibre (W_{m_{1},m_{2}})_{k} is isomorphic either to the Witt group scheme W_{2,k} if m_{2} − pm_{1} = 0 or to the group scheme 𝔾_{a,k}^{2} otherwise. Note finally that we have an exact sequence:

0 → 𝔾_{a} ⟶^{V} W_{m_{1},m_{2}} ⟶^{R} 𝔾_{a} → 0

where V : 𝔾_{a} → W_{m_{1},m_{2}} is the vershiebung homomorphism defined by V (x) = (0, x) and R : W_{m_{1},m_{2}} → 𝔾_{a} is the projection R(x_{1}, x_{2}) = x_{1}.

2.3.1. The group schemes H_{m_{1},m_{2}}. We use the same notations as in 2.3. The following maps I_{m_{1},m_{2}} and F are group scheme homomorphisms:

I_{m_{1},m_{2}} : W_{m_{1},m_{2}} → W_{pm_{1},pm_{2}}
(x_{1}, x_{2}) → (π^{m_{1}(p−1)}x_{1}, π^{m_{2}(p−1)}x_{2})

and:

F : W_{m_{1},m_{2}} → W_{pm_{1},pm_{2}}
(x_{1}, x_{2}) → (x_{1}^{p}, x_{2}^{p})

Consider the following isogeny:

φ_{m_{1},m_{2}} := F − I_{m_{1},m_{2}} : W_{m_{1},m_{2}} → W_{pm_{1},pm_{2}}

which is given by:

(x_{1}, x_{2}) → (x_{1}^{p} −π^{m_{1}(p−1)}x_{1}, x_{2}^{p} −π^{m_{2}(p−1)}x_{2} −p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}π^{m_{2}p−m_{1}(pk+p−k)}x_{1}^{pk}(−x_{1})^{p−k})

We define the group scheme H_{m_{1},m_{2}} to be the kernel of the above isogeny. Thus we have an exact sequence:

(6) 0 → H_{m_{1},m_{2}} → W_{m_{1},m_{2}} ⟶^{F −I_{m_{1},m_{2}}} W_{pm_{1},pm_{2}} → 0

and H_{m_{1},m_{2}} is a finite and flat commutative R-group scheme of rank p^{2}. Further we have the following commutative diagram:

0  0  0
↓  ↓  ↓
0 ⟶ M_{m_{2}} ⟶ 𝔾_{a}  ⟶^{ϕ_{m_{2}}}  𝔾_{a}  ⟶ 0
↓  ↓  ↓
0 ⟶ H_{m_{1},m_{2}} ⟶ W_{m_{1},m_{2}} ⟶^{φ_{m_{1},m_{2}}} W_{pm_{1},pm_{2}} ⟶ 0
↓  ↓^{R}  ↓^{R}
0 ⟶ M_{m_{1}} ⟶ 𝔾_{a}  ⟶^{ϕ_{m_{1}}}  𝔾_{a}  ⟶ 0
↓  ↓  ↓
0  0  0

The group scheme H_{m_{1},m_{2}} is then an extension of the group scheme M_{m_{1}} by M_{m_{2}}. Its generic fibre (H_{m_{1},m_{2}})_{K} is isomorphic to the étale constant group ℤ/p^{2}ℤ and its special fibre (H_{m_{1},m_{2}})_{k} is the group scheme H_{k} which is an extension of ℤ/pℤ by α_{p} if m_{1} = 0 and m_{2} > 0 (such an extension is necessarily split), and the group scheme G_{k} which is an extension of α_{p} by α_{p} if m_{1} > 0. We have the following exact sequences:

(7) 0 → H_{k} → 𝔾_{a,k}^{2} ⟶^{(x_{1}^{p}−x_{1},x_{2}^{p})} 𝔾_{a,k}^{2} → 0

and:

(8) 0 → G_{k} → 𝔾_{a,k}^{2} ⟶^{(x_{1}^{p},x_{2}^{p})} 𝔾_{a,k}^{2} → 0

Let X be an R-scheme. The sequence (6) induces a long cohomology exact sequence

(9): W_{m_{1},m_{2}}(X) ⟶^{φ_{m_{1},m_{2}}} W_{pm_{1},pm_{2}}(X) → H_{fppf}^{1}(X, H_{m_{1},m_{2}}) → H_{fppf}^{1}(X, W_{m_{1},m_{2}}) ⟶^{ϕ_{n}}

H_{fppf}^{1}(X, W_{pm_{1},pm_{2}}).

The cohomology group H_{fppf}^{1}(X, H_{m_{1},m_{2}}) classifies the isomorphism classes of fppftorsors with group H_{m_{1},m_{2}} above X. The above exact sequence allows the following description of H_{m_{1},m_{2}}-torsors: locally a torsor f : Y → X under the group scheme H_{m_{1},m_{2}} is given by the equations:

T_{1}^{p} − π^{m_{1}(p−1)}T_{1} = a_{1}

and:

T_{2}^{p} − π^{m_{2}(p−1)}T_{2} = a_{2} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}π^{pm_{2}−m_{1}(pk+p−k)}T_{1}^{pk}(−T_{1})^{p−k}

where T_{1} and T_{2} are indeterminate and a_{1}, a_{2} are regular functions on X. Its special fibre is the H_{k}-torsor given by the equations:

t_{1}^{p} − t_{1} = ā_{1}

and:

t_{2}^{p} = ā_{2}

if m_{1} = 0, and the G_{k}-torsor given by the equations:

t_{1}^{p} = ā_{1}

and:

t_{2}^{p} = ā_{2}

otherwise. In particular if H_{fppf}^{1}(X, 𝔾_{a,R}) = 0 (e.g. if X is affine) then an H_{m_{1},m_{2}}-torsor above X is globally defined by an equation as above.

2.4. Degeneration of étale ℤ/p^{2}ℤ-torsors. Our aim in this section is to describe explicitely the degeneration of étale ℤ/p^{2}ℤ-torsors. In what follows we use the same notations as in 2.2. Let f_{K} : Y_{K} → X_{K} be a non trivial ℤ/p^{2}ℤ-torsor. Let K(X) → L be the cyclic p^{2}-extension of function fields corresponding to the torsor f_{K} which canonically factorises as K(X) → L_{1} → L_{2} := L where K(X) → L_{1} is a cyclic p-extension. We assume that the ramification index above the generic point η of X_{k} in the extension K(X) → L equals 1. There exists a canonical factorisation Y_{K} = Y_{2,K} ⟶^{f_{2,K}} Y_{1,K} ⟶^{f_{1,K}} X_{K} of f_{K} where f_{i,K} is a ℤ/pℤ-torsor for i ∈ {1, 2}. Moreover by 2.2.1 the torsor f_{2,K} (resp. f_{1,K}) extends to a torsor f_{2} : Y_{2} → Y_{1} (resp. f_{1} : Y_{1} → X) under a finite and flat R-group scheme of rank p and the composite f := f_{1} ∘ f_{2} is a finite and flat cover which coincides on the generic fibre with f_{K}. We assume that the special fibre Y_{2,k} of Y_{2} is irreducible. In particular above the generic point η there exists a unique generic point η_{1} in Y_{1} which lies above η. We denote by δ (resp. δ_{1} and δ_{2}) the degree of different in the extension L above the point η (resp. the degree of different in the extension L_{1} above the point η and that of the different in the extension L_{2} above the point η_{1}). Note that δ = δ_{1} + δ_{2}.

2.4.1. We start with the following technical lemma 2.4.2 which will be used later on. In what follows we assume that X = Spec A is affine and that f_{1} : Y_{1} := Spec B → X is a torsor under the group scheme M_{n} for some integer n > 0. Thus f_{1} is given by an equation T^{p} − π^{n(p−1)}T = v where v ∈ A is such that its image v̄ ∈ \overline{A} := A/πA is not a p-power. In particular the special fibre f̄_{1} : Y_{1,k} = Spec \overline{B} → X_{k} = Spec \overline{A} of the torsor f (here \overline{B} := B/πB and \overline{A} := A/πA) is the α_{p}-torsor given by the equation t^{p} = v̄ and \overline{B} is a free \overline{A}-algebra with basis \{1, t, t^{2}, ..., t^{p−1}\}. We need to characterise elements of A which become p-powers modulo π in \overline{B} but are not necessarily p-powers modulo π in \overline{A}.

2.4.2. Lemma. Let u ∈ A. Assume that the image ū of u is a p-power in \overline{B}. Then u = f(v) + πu′ where u′ ∈ A and f(v) belongs to the additive subgroup A_{v} := A^{p} ⊕ A^{p}.v ⊕ .... ⊕ A^{p}.v^{p−1} of A. Moreover let f(v) := a_{0}^{p} + a_{1}^{p}v + ... + a_{p−1}^{p}v^{p−1} ∈ A_{v} and let m be an integer. Consider the element g := f(v)π^{−pm} ∈ B_{K}. Then g = (a_{o}^{p} + a_{1}^{p}(T^{p} − π^{n(p−1)}T) + ... + a_{p−1}^{p}(T^{p} − π^{n(p−1)}T)^{p−1})π^{−pm} and after addition of elements of B_{K} of the form b^{p} − b one can transform g in g̃ := π^{−m}(a_{0} + a_{1}T + ... + a_{p−1}T^{p−1}) + π^{−(pm−n(p−1))}(− ∑_{j=1}^{p−1} ja_{j}^{p}T^{p(j−1)}T) + π^{−(pm−2n(p−1))}h(T) where h(T) ∈ B. Moreover the image of − ∑_{j=1}^{p−1} ja_{j}^{p}T^{p(j−1)}T = −a_{1}^{p}T − 2a_{2}^{p}T^{p+1} − ... − (p − 1)a_{p−1}^{p}T^{p(p−2)+1} in \overline{B} is not a p-power.

Proof. We have \overline{B} = \overline{A} ⊕ \overline{A}.t ⊕ ... ⊕ \overline{A}.t^{p−1}, hence \overline{B}^{p} = \overline{A}^{p} ⊕ \overline{A}^{p}.v̄ ⊕ ... ⊕ \overline{A}.v̄^{p−1} and the first assertion of the lemma follows. Now let g := (a_{0}^{p} + a_{1}^{p}v + ... + a_{p−1}^{p}v^{p−1})π^{−pm} ∈ B_{K} then since T^{p} − π^{n(p−1)}T = v in B_{K} we can write g = π^{−pm}(a_{o}^{p} + a_{1}^{p}(T^{p} − π^{n(p−1)}T) + ... + a_{p−1}^{p}(T^{p} − π^{n(p−1)}T)^{p−1}) in B_{K}. After developing the terms (T^{p} − π^{n(p−1)}T)^{j} for j ∈ {1, p−1} according to the binomial expansion and rearanging the terms with the same power of π we get that g = π^{−pm}(a_{0}^{p} + a_{1}^{p}T^{p} + ... + a_{p−1}^{p}T^{p(p−1)}) + π^{−(pm−n(p−1))}(−a_{1}^{p}T − 2a_{2}^{p}T^{p+1} − ... − (p − 1)a_{p−1}^{p}T^{p(p−2)+1}) + π^{−(pm−2n(p−1))}h(T) where h(T) ∈ B. Finally after adding (a_{0} + a_{1}T + ... + a_{p−1}T^{p−1})/π^{m} − (a_{0}^{p} + a_{1}^{p}T^{p} + ... + a_{p−1}^{p}T^{p(p−1)})/π^{pm} we get the desired expression for g̃.

The next theorem is the main result of this section. It describes locally and explicitely the degeneration of étale ℤ/p^{2}ℤ-torsors. Although we loose the structure of torsors in this case of cyclic covers of degree p^{2} (cf. Proposition 2.4.5) we are able to find “canonical integral equations” which describe the reduction of p^{2}-cyclic covers in equal characteristic p.

2.4.3. Theorem. We use the same notations as in 2.4. Assume that X = Spec A is affine. Then the torsor f_{K} can be described by an equation of the form:

(T_{1}^{p}, T_{2}^{p}) − (T_{1}, T_{2}) = (π^{m_{1}}a_{1}, π^{m_{2}}a_{2})

where a_{1} and a_{2} are regular functions on X_{K} with v(a_{1}) = v(a_{2}) = 0, m_{1} ≤ 0, m_{2} is an integer, with the following cases which occur:

a) m_{1} = 0 and m_{2} ≥ 0. In this case f is an étale ℤ/p^{2}ℤ-torsor above X given by the equation:

(T_{1}^{p}, T_{2}^{p}) − (T_{1}, T_{2}) = (a_{1}, π^{m_{2}}a_{2})

Its special fibre f_{k} : Y_{k} → X_{k} is the étale ℤ/p^{2}ℤ-torsor given by the equation:

(t_{1}^{p}, t_{2}^{p}) − (t_{1}, t_{2}) = (ā_{1}, \overline{π^{m_{2}}a_{2})}

and δ = δ_{1} = δ_{2} = 0.

b) m_{1} = 0, m_{2} = −pm′_{2} < 0, and a_{2} is not a p-power modulo π. In this case f is a torsor under the R-group scheme W_{0,m′_{2}} given by the equations:

T_{1}^{p} − T_{1} = a_{1}

and

T̃_{2}^{p} − π^{m′_{2}(p−1)}T̃_{2} = a_{2} + p^{−1}π^{−m_{2}} ∑_{k=1}^{p−1} \binom{p}{k}T_{1}^{pk}(−T_{1})^{p−k}

Its special fibre is the torsor under the k-group scheme (W_{0,m′_{2}})_{k} ≃ H_{k} given by the equations: t_{1}^{p} − t_{1} = ā_{1}, and t̃_{2}^{p} = ā_{2}, where ā_{1} (resp. ā_{2}) is the image of a_{1} (resp. of a_{2}) modulo π. In this case δ_{1} = 0 and δ = δ_{2} = m′_{2}(p − 1).

c) m_{1} = −pm′_{1} < 0 and the image ā_{1} of a_{1} modulo π is not a p-power. In this case f_{1} is an M_{m′_{1}}-torsor given by the equation:

T̃_{1}^{p} − π^{m′_{1}(p−1)}T̃_{1} = a_{1}

and its special fibre f_{1,k} : Y_{1,k} → X_{k} is the α_{p}-torsor given by the equation t̃_{1}^{p} = ā_{1}. We have δ_{1} = m′_{1}(p − 1). For f_{2} one has the following cases:

c-1) m′_{1}(p(p − 1) + 1) > −m_{2} (resp. m′_{1}(p(p − 1) + 1) = −m_{2}) In this case m′_{1} = pm′′_{1} is necessarily divisible by p and if m̃_{1} := m′′_{1}(p(p − 1) + 1) then f_{2} is a torsor under M_{m̃_{1},R} given by the equation:

T̃_{2}^{p} − π^{m̃_{1}(p−1)}T̃_{2} = π^{m̃_{1}p+m_{2}}a_{2} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}T̃_{1}^{pk}(−T̃_{1})^{p−k}π^{m′_{1}((p(p−1)+1)−pk−p+k)}

Its special fibre is the α_{p}-torsor given by the equation: t̃_{2}^{p} = −t̃_{1}^{p(p−1)+1} (resp. the equation: t̃_{2}^{p} = −t̃_{1}^{p(p−1)+1} + ā_{2}).

Otherwise −m_{2} > m′_{1}(p(p − 1) + 1) in which case −m_{2} = pm′_{2} is necessarily divisible by p and we have the following description for π^{m_{2}}a_{2}:

π^{m_{2}}a_{2} = f_{1}(a_{1})/π^{pm′_{2}} + f_{2}(a_{1})/π^{pm′_{2}−t_{1}} + ... + f_{r}(a_{1})/π^{pm′_{2}−t_{1}−...−t_{r−1}} + g/π^{pm′_{2}−t_{1}−...−t_{r}}

where f_{i}(a_{1}) belongs to the subgroup A_{a_{1}} of A (cf. 2.4.2), g ∈ A, and the t_{i} are positive integers (note that g and the f_{i} can be 0). Moreover the torsor f_{2,K} is given by the equation: T_{2}^{p} − T_{2} = f_{1}(a_{1})/π^{pm′_{2}} + f_{2}(a_{1})/π^{pm′_{2}−t_{1}} + ... + f_{r}(a_{1})/π^{pm′_{2}−t_{1}−...−t_{r−1}} +g/π^{pm′_{2}−t_{1}−...−t_{r}} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}T̃_{1}^{pk}(−T̃_{1})^{p−k}π^{−m′_{1}(pk+p−k)}.

And the following distinct cases occur:

c-2) pm′_{2} − (p − 1)m′_{1} > sup(m′_{1}(p(p − 1) + 1), pm′_{2} − t_{1} − ... − t_{r}) (resp. pm′_{2} − (p − 1)m′_{1} = m′_{1}(p(p − 1) + 1) > pm′_{2} − t_{1} − ... − t_{r}). In this case pm′_{2} − m′_{1}(p − 1) = m′′_{2}p is divisible by p and δ_{2} = m′′_{2}(p − 1). Let f_{1}(a_{1}) := c_{0}^{p} + c_{1}^{p}a_{1} + ... + c_{p−1}^{p}a_{1}^{p−1}. Then f_{2} is a torsor under M_{m′′_{2}} and its special fibre is the α_{p}-torsor given by the equation: t̃_{2}^{p} = −c̄_{1}^{p}t̃_{1} − 2c̄_{2}^{p}t̃_{1}^{p+1} − ... − (p − 1)c̄_{p−1}^{p}t̃_{1}^{p(p−2)+1} (resp. the equation: t̃_{2}^{p} = −c̄_{1}^{p}t̃_{1} − 2c̄_{2}^{p}t̃_{1}^{p+1} − ... − (p − 1)c̄_{p−1}^{p}t̃_{1}^{p(p−2)+1} − t̃_{1}^{p(p−1)+1}) where c̄_{i} is the image of c_{i} modulo π.

c-3) pm′_{2}−t_{1}−...−t_{r} > sup(pm′_{2}−(p−1)m′_{1}, m′_{1}(p(p−1)+1)) (resp. pm′_{2}−(p−1)m′_{1} = pm′_{2} − t_{1} − ... − t_{r} > m′_{1}(p(p − 1) + 1)) and the image of g modulo π is not a p-power in O(Y_{1,k}). In this case pm′_{2} − t_{1} − ... − t_{r} = pm′′_{2} is divisible by p, δ_{2} = m′′_{2}(p − 1), and f_{2} is an M_{m′′_{2}}-torsor. Its special fibre is the α_{p}-torsor given by the equation: t̃_{2}^{p} = ḡ (resp. the equation: t̃_{2}^{p} = −c̄_{1}^{p}t̃_{1} − 2c̄_{2}^{p}t̃_{1}^{p+1} − ... − (p − 1)c̄_{p−1}^{p}t̃_{1}^{p(p−2)+1} + ḡ).

c-4) m′_{1}(p(p−1)+1) > sup(pm′_{2}−t_{1}−...−t_{r}, pm′_{2}−(p−1)m′_{1}) (resp. pm′_{2}−t_{1}−...−t_{r} = m′_{1}(p(p − 1) + 1) > pm′_{2} − (p − 1)m′_{1}) In this case m′_{1} = pm′′_{1} and if m̃_{1} := m′′_{1}(p(p − 1) + 1) then f_{2} is a torsor under M_{m̃_{1},R}. Its special fibre is the α_{p}-torsor given by the equation: t̃_{2}^{p} = −t̃_{1}^{p(p−1)+1} (resp. the equation: t̃_{2}^{p} = −t̃_{1}^{p(p−1)+1} + ḡ).

c-5) pm′_{2}−t_{1}−...−t_{r} = m′_{1}(p(p−1)+1) = pm′_{2}−(p−1)m′_{1}. In this case m′_{1} = pm′′_{1} and if m̃_{1} := m′′_{1}(p(p−1)+1) then f_{2} is a torsor under M_{m̃_{1},R}. Its special fibre is the α_{p}-torsor given by the equation: t̃_{2}^{p} = −c̄_{1}^{p}t̃_{1} − 2c̄_{2}^{p}t̃_{1}^{p+1} − ... − (p − 1)c̄_{p−1}^{p}t̃_{1}^{p(p−2)+1} − t̃_{1}^{p(p−1)+1} + ḡ.

In all the above cases if f_{1} (resp. f_{2}) is a torsor under the group scheme Mm̃_{1} (resp. Mm̃_{2}) then necessarily m̃_{2} ≥ m̃_{1}(p(p − 1) + 1)/p. Moreover in all the cases c-2, c-3, c-4, and c-5 above the functions c̄_{1}, c̄_{2},..., c̄_{p−1} (resp. ḡ) are uniquely determined (resp. is uniquely determined up to addition of h̄^{p} where h̄ is a regular function on X_{k}). In the case c-1 the function ā_{2} is uniquely determined up to addition of b̄^{p} where b̄ is a regular function on X_{k}.

Proof. The torsor f_{K} is given by the Artin-Schreier-Witt theory by an equation of the form (T_{1}^{p}, T_{2}^{p}) − (T_{1}, T_{2}) = (ã_{1}, ã_{2}) where ã_{1} and ã_{2} are regular functions on X_{K}. We can write ã_{1} = π^{m_{1}}a_{1} (resp. ã_{2} = π^{m_{2}}a_{2}) with v(a_{1}) = v(a_{2}) = 0 and it follows from 2.2.1 that we have necessarily m_{1} ≤ 0. If m_{1} = 0 and m_{2} ≥ 0 we are in case a) and the assertion there is clear. Assume that m_{1} = 0 and m_{2} < 0. Then it follows from 2.2.1 that m_{2} = −pm′_{2} is necessarily divisible by p and after eventually some modifications (as in the proof of 2.2.1) we may assume that a_{2} is not a p-power modulo π (here one uses the fact that a regular function u on X which is not a p-power modulo π in X_{k} can not become a p-power in Y_{1,k} since f_{1,k} is an étale torsor hence not radicial). Then f is defined by the equations:

T_{1}^{p} − T_{1} = a_{1}

and

T̃_{2}^{p} − π^{m′_{2}(p−1)}T̃_{2} = a_{2} + p^{−1}π^{−m_{2}} ∑_{k=1}^{p−1} \binom{p}{k}T_{1}^{pk}(−T_{1})^{p−k}

where T̃_{2} := π^{m′_{2}}T_{2}. The rest of the assertion in case b) follows then easily. Assume now that m_{1} < 0. Then the assertion concerning f_{1} follows from 2.2.1. Assume first that m_{2} ≥ 0. The assertion concerning f_{2} follows then easily after adapting the equation defining the torsor f_{2,K} to the change of variables T_{1} = T̃_{1}/π^{m′_{1}} and we are in the case c-1). In this case m′_{1} = pm′′_{1} is divisible by p and the cover f is given by the equations:

T̃_{1}^{p} − π^{m′_{1}(p−1)}T̃_{1} = a_{1}

and

T̃_{2}^{p} − π^{m̃_{1}(p−1)}T̃_{2} = π^{m̃_{1}p+m_{2}}a_{2} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}T̃_{1}^{pk}(−T̃_{1})^{p−k}π^{m̃_{1}p−m′_{1}(pk+p−k)}

where m̃_{1} := m′′_{1}(p(p − 1) + 1). From which we deduce that its special fibre is given by the equations: t̃_{1}^{p} = ā_{1} and t̃_{2}^{p} = −t̃_{1}^{p(p−1)+1} where t̃_{2} (resp. t̃_{1}) is the image of T̃_{2} (resp. image of T̃_{1}) modulo π.

So lastly we assume that m_{2} < 0. Then f_{2,K} is given by the equation:

T_{2}^{p} − T_{2} = π^{m_{2}}a_{2} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}T̃_{1}^{pk}(−T̃_{1})^{p−k}π^{−m′_{1}(pk+p−k)}

The highest power of π in the denominators of the summands in:

p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}T̃_{1}^{pk}(−T̃_{1})^{p−k}π^{−m′_{1}(pk+p−k)}

is m′_{1}(p(p − 1) + 1) and in order to understand the reduction of the torsor f_{2} we have to compare this to m_{2}. Assume first that m′_{1}(p(p − 1) + 1) > −m_{2}. Then it follows from 2.2.1 that m′_{1} = m′′_{1}p must be divisible by p. Let m̃_{1} := m′′_{1}(p(p − 1) + 1). Then we are in case c-1) and f_{2} is a torsor under the group scheme M_{m̃_{1},R} defined by the equation:

T̃_{2}^{p} − π^{m̃_{1}(p−1)}T̃_{2} = π^{m̃_{1}p+m_{2}}a_{2} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}π^{m̃_{1}p−m′_{1}(pk+p−k)}T̃_{1}^{pk}(−T̃_{1})^{p−k}

Its special fibre is the α_{p}-torsor given by the equation: t̃_{2}^{p} = −t̃_{1}^{p(p−1)+1}. Assume next that m′_{1}(p(p − 1) + 1) < −m_{2} (the case where m′_{1}(p(p − 1) + 1) = −m_{2} is easily treated and is left to the reader). Then it follows from 2.2.1 that m_{2} = −pm′_{2} is divisible by p and two cases occur depending on whether or not the image ā_{2} of a_{2} modulo π is or is not a p power in O(Y_{1,k}). If ā_{2} is not a p power in O(Y_{1,k}) then f_{2} is a torsor under M_{m′_{2},R} given by the equation:

T̃_{2}^{p} − π^{m′_{2}(p−1)}T̃_{2} = a_{2} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}π^{pm′_{2}−m_{1}(pk+p−k)}T̃_{1}^{pk}(−T̃_{1})^{p−k}

Its special fibre is the α_{p}-torsor given by the equation t̃_{2}^{p} = ā_{2} and we are in the case c-3). Assume that ā_{2} is a p power in O(Y_{1,k}). Then either a_{2} is already a p-power in O(X_{k}) in which case we can transform, using the kind of transformations used in the proof of 2.2.1, the term π^{m_{2}}a_{2} into πm̃_{2} ã_{2} where 0 > m̃_{2} > m_{2} and ã_{2} ∈ A. Or a_{2} is not a p-power in O(X_{1}) but becomes a p-power in O(Y_{1,k}). In the later case it follows from 2.4.2 that a_{2} = f_{1}(a_{1}) + π^{t_{1}}g_{1} where f_{1}(a_{1}) := c_{0}^{p} + c_{1}^{p}a_{1} + ... + c_{p−1}^{p}a_{1}^{p−1} belongs to the subgroup A_{a_{1}} of A, t_{1} > 0, and g_{1} ∈ A. Moreover the term π^{m_{2}}a_{2} = f_{1}(a_{1})/π^{pm′_{2}} +g_{1}/π^{pm′_{2}−t_{1}} can be transformed to f̃_{1}(T_{1})/π^{pm′_{2}−m′_{1}(p−1)} +g_{1}/π^{pm′_{2}−t_{1}} where the image \overline{f̃_{1}(T_{1})} := −c̄_{1}^{p}t̃_{1} −2c̄_{2}^{p}t̃_{1}^{p+1} −...−(p−1)c̄_{p−1}^{p}t̃_{1}^{p(p−2)+1} of f̃_{1}(T_{1}) modulo π is not a p-power (cf. loc. cit.). At this point we can repeat the same argument as above. Namely if in the first case the image ¯ ã_{2} modulo π is not a p-power in O(Y_{1,k}) then we conclude as above that we are either in case c-3) if m̃_{2} ≥ m′_{1}(p(p − 1) + 1) in which case m̃_{2} = pm′′_{2} and f_{2} is a torsor under M_{m′′_{2},R} whose special fibre is the α_{p}-torosr given by the equation t̃_{2}^{p} = ¯ ã_{2}. Otherwise we repeat the same process as above. And in the second case if pm′_{2} − (p − 1)m′_{1} > sup(pm′_{2} − t_{1}, m′_{1}(p(p − 1) + 1)) then pm′_{2} − (p − 1)m′_{1} =: pm′′_{2} is divisible by p and f_{2} is a torsor under the group scheme M_{m′′_{2},R} defined by the equation:

T̃_{2}^{p} − π^{m′′_{2}(p−1)}T̃_{2} = f̃_{1}(T_{1}) + π^{pm′′_{2}−pm′_{2}+t_{1}}g_{1} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}π^{pm′′_{2}−(pk+p−k)m_{1}} T̃_{1}^{pk}(−T̃_{1})^{p−k}

Its special fibre is the α_{p}-torsor given by the equation: t̃_{2}^{p} = −c̄_{1}^{p}t̃_{1} − 2c̄_{2}^{p}t̃_{1}^{p+1} − ... − (p − 1)c̄_{p−1}^{p}t̃_{1}^{p(p−2)+1} and we are in the case c-2). If m′_{1}(p(p − 1) + 1) > sup(pm′_{2} − t_{1}, pm′_{2} − (p − 1)m′_{1}) then m′_{1}(p(p − 1) + 1) is divisible by p, f_{2} is a torsor under the group scheme M_{(m′_{1}(p(p−1)+1))/p,R}, its special fibre is the α_{p}-torsor given by the equation: t̃_{2}^{p} = −t̃_{1}^{p(p−1)+1} and we are in the case c-4). If pm′_{2} − t_{1} > sup(m′_{1}(p(p − 1) + 1), pm′_{2} − (p − 1)m′_{1}) and the image ḡ_{1} of g_{1} in O(Y_{1,k}) is not a p-power then pm′_{2} − t_{1} =: pm′′_{2} is divisible by p, f_{2} is a torsor under the group scheme M_{m′′_{2},R} defined by the equation:

T̃_{2}^{p} − π^{m′′_{2}(p−1)}T̃_{2} = πm̃_{2} f̃_{1}(T_{1}) + g_{1} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}π^{pm′′_{2}−(pk+p−k)m_{1}} T̃_{1}^{pk}(−T̃_{1})^{p−k}

where m̃_{2} := pm′′_{2} − pm′_{2} − (p − 1)m′_{1}, and its special fibre is the α_{p}-torsor given by the equation: t̃_{2}^{p} = ḡ_{1} and we are in the case c-3). Now in the general case we repeat the same argument as above if in the first case the image ¯ ã_{2} of ã_{2} modulo π is a p-power or if in the second case pm′_{2} − t_{1} > sup(m′_{1}(p(p − 1) + 1), pm′_{2} − (p − 1)m′_{1}) and the image ḡ_{1} of g_{1} in O(Y_{1,k}) is a p-power. As the exponent of π in the denominators in the equation defining f_{2,K} decreases at each step we conclude that this process must stop after finitely many steps and we end up with an equation as claimed in the statement c) and the rest of the conclusion follows then easily.

2.4.4. Remark. Assume that we are in the case c-3) of 2.4.3 and that t_{1} = ... = t_{r} = 0 and f_{1} = ... = f_{r} = 0. Then f is a torsor under the R-group scheme H_{m′_{1},m′_{2}} given by the equations:

T̃_{1}^{p} − π^{m′_{1}(p−1)}T̃_{1} = a_{1}

and:

T̃_{2}^{p} − π^{m′_{2}(p−1)}T̃_{2} = g + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}T̃_{1}^{pk}(−T̃_{1})^{p−k}π^{pm′_{2}−pk−p+k}

Its special fibre is the (H_{m′_{1},m′_{2}})_{k} ≃ H_{k}-torsor given by the equations: t̃_{1}^{p} = ā_{1} and t̃_{2}^{p} = ḡ.

The next proposition states that in general the cover f_{2} in 2.4.3 doesn’t have the structure of a torsor under a finite and flat R-group scheme of rank p^{2}.

2.4.5. Proposition. We use the same notations as in 2.4.3. Assume that we are in the case c-1) and that p > 2. Then the finite cover f doesn’t have the structure of a torsor under a finite and flat R-group scheme of rank p^{2}.

Proof. Recall that in the case c-1 of 2.4.3 the étale torsor f_{K} is given by the equations (1): (T_{1}^{p}, T_{2}^{p}) − (T_{1}, T_{2}) = (a_{1}/π^{pm′_{1}}, a′_{2} := π^{m_{2}}a_{2}). Let b_{1} ∈ A. Then the equations: (S_{1}^{p}, S_{2}^{p})−(S_{1}, S_{2}) = (π^{−pm′_{1}}a_{1}, a′_{2})+(π^{−pm′_{1}}b_{1}^{p}, 0)−(π^{−m′_{1}}b_{1}, 0) are also defining equations for the torsor f_{K}. The cover f is then given by the equations (1):

T̃_{1}^{p} − π^{m′_{1}(p−1)}T̃_{1} = a_{1}

and

T̃_{2}^{p} − π^{m̃_{1}(p−1)}T̃_{2} = π^{m̃_{1}p+m_{2}}a_{2} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}T̃_{1}^{pk}(−T̃_{1})^{p−k}π^{m̃_{1}p−m′_{1}(pk+p−k)}

And f is also given by the equations (2):

S̃_{1}^{p} − π^{m′(p−1)}S̃_{1} = a_{1} + b_{1}^{p} − π^{m(p−1)}b_{1}

and: S_{2}^{p} − S_{2} = a′_{2} + p^{−1}(∑_{k=1}^{p−1} \binom{p}{k}S̃_{1}^{pk}(−S̃_{1})^{p−k}π^{−m′_{1}(pk+p−k)} − ∑_{k=1}^{p−1} \binom{p}{k}b_{1}^{pk}(−b_{1})^{p−k}π^{−m′_{1}(pk+p−k)} − ∑_{k=1}^{p−1} \binom{p}{k}(b_{1}^{p} − π^{m′_{1}(p−1)}b_{1})^{k}(a_{1})^{p−k}π^{−m′_{1}p^{2}}). By reducing modulo π both equations (1) and (2) we obtain equations (1)’ and (2)’ for the cover f_{k} on the level of special fibres. Suppose that f has the structure of a torsor under a finite and flat R-group scheme G_{R} of rank p^{2}. Then the special fibre f_{k} of f is a torsor under the special fibre G_{k} of G which is a group scheme of rank p^{2} and an extension of α_{p} by α_{p}. In particular both equations (1)’ and (2)’ that we obtain for f_{k} must define the same α_{p}-torsor f_{1,k} and f_{2,k}. This is the case for f_{1,k} since both equations are t̃_{1}^{p} = ā_{1} and s̃_{1}^{p} = ā_{1} + b̄_{1}^{p} but we will see that this is not the case for f_{2,k} if p > 2. Indeed one can see after some (easy) computations that the equations we obtain for f_{2,k} are: t̃_{2}^{p} = −t̃_{1}^{p(p−1)+1} and s̃_{2}^{p} = −s̃_{1}^{p(p−1)+1}+b̄_{1}^{p(p−1)+1}+∑_{k=1}^{p−1}(b̄_{1}^{p(k−1)+1}(s̃_{1}−b̄_{1})^{p(p−k)}+b̄_{1}^{pk}(s̃_{1}− b̄_{1})^{p(p−k−1)+1})), where s̃_{2} is the image of πm̃_{1}S_{2} modulo π, which gives s̃_{2} = −s̃_{1}^{p(p−1)+1} + b̄_{1}^{p(p−1)+1} + ((s̃_{1} − b̄_{1})^{p−1} + b̄_{1}^{p−1}) ∑_{k=1}^{p−1} b̄_{1}^{p(k−1)+1}(s̃_{1} − b̄_{1})^{p(p−k−1)+1}. An easy verification shows that these two equations define non isomorphic α_{p}-torsors (although they define isomorphic covers). If for example p = 3 then the two equations are: t̃_{2}^{3} = −t̃_{1}^{7} and s̃_{2}^{3} = −s̃_{1}^{7} + b̄_{1}^{7} + ((s̃_{1} − b̄_{1})^{2} − b̄_{1}^{2})(b̄_{1}(s̃_{1} − b̄_{1})^{4} + b̄_{1}^{4}(s̃_{1} − b̄_{1})). Using that s̃_{1} = t̃_{1} + b̄_{1} we get s̃_{2}^{3} = −(t̃_{1}+b̄_{1})^{7}+b̄_{1}^{7}+(t̃_{1}^{2}−b̄_{1}^{2})(b̄_{1}t̃_{1}^{4}+b̄_{1}^{4}t̃_{1}) thus s̃_{2}^{3} = −t̃_{1}^{7}−34t̃_{1}^{3}b̄_{1}^{4}−8t̃_{1}b̄_{1}^{6} = −t̃_{1}^{7}+2t̃_{1}^{3}b̄_{1}^{4}+t̃_{1}b̄_{1}^{6} and since 2t̃_{1}^{3}b̄_{1}^{4} + t̃_{1}b̄_{1}^{6} = 2ā_{1}b̄_{1}^{4} + t̃_{1}b̄_{1}^{6} is not a cube we deduce that the two equations do not define the same α_{3}-torsor.

2.4.6. Corollary. Assume p > 2. Let X be a formal affine R-scheme of finite type with X_{k} irreducible. Then there exists a p^{2}-cyclic cover f : Y → X which is étale on the generic fibre X_{K} of X and such that f doesn’t have the structure of a torsor under a finite and flat R-group scheme of rank p^{2}.

Proof. One can easily find examples of cyclic p^{2}-covers f : Y → X which have the structure of an étale torsor above X_{K} and such that we are in the case c-1) of 2.4.3. The result follows then by 2.4.5.

2.4.7. Question. With the same notations as in 2.4.3 is it possible to find necessary and sufficient conditions, for example on δ_{1} and δ_{2}, such that f has the structure of a torsor under a finite and flat R-group scheme of rank p^{2}?

Next we define the “degeneration data” arising from the reduction of an étale ℤ/p^{2}ℤtorsor.

2.4.8. Definition. Let f_{K} : Y_{K} → X_{K} be an étale ℤ/p^{2}ℤ-torsor with X = Spec A affine as in 2.4.3. Then we define the degeneration type of the torsor f_{K} as follows: f_{K} has a degeneration of type A, or of type (étale, étale), if we are in the case a) of 2.4.3, a degeneration of type B, or of type (étale, radicial), if we are in the case b) of 2.4.3 and a degeneration of type C, or of type (radicial, radicial), if we are in the case c) of 2.4.3.

Further we define the degeneration data associated to a degeneration type as follows:

a) A degeneration data of type A consists of an element of H_{et}^{1}(X_{k}, ℤ/p^{2}ℤ).

b) A degeneration data of type B consists of an element of H_{fppf}^{1}(X_{k}, G_{k}) where G_{k} is the finite commutative group scheme extension of ℤ/pℤ by α_{p} as defined in 2.3.1.

c) A degeneration data of type C consists of an element of H_{fppf}^{1}(X_{k}, H_{k})⊕Γ(X_{k}, O_{X_{k}})^{p−1}. where H_{k} is the finite commutative group scheme extension of α_{p} by α_{p} as defined in 2.3.1

2.4.9. Proposition. Assume that X is affine as in 2.4.3 and let f_{K} : Y_{K} → X_{K} be an étale ℤ/p^{2}ℤ-torsor which has a degeneration of type A (resp. B or C). Then f_{K} induces canonically a degeneration data of type A (resp. of type B or C).

Proof. This is a direct consequence of 2.4.3. If f_{K} has a degeneration of type A then the special fibre f_{k} of f is an étale ℤ/p^{2}ℤ-torsor and the assertion follows in this case. Assume that f_{K} has a degeneration of type B. Then the special fibre f_{k} of f is canonically a G_{k}- torsor and the assertion follows in this case too. Finally assume that the torsor f_{K} has a degeneration of type C. Then it follows from 2.4.3 that the special fibre f_{k} of the cover f is defined by the equations: t̃_{1}^{p} = ā_{1} and t̃_{2}^{p} = −c̄_{1}^{p}t̃_{1} − 2c̄_{2}^{p}t̃_{1}^{p+1} − ... − (p − 1)c̄_{p−1}^{p}t̃_{1}^{p(p−2)+1} − t̃_{1}^{p(p−1)+1} + ḡ (resp. t̃_{1}^{p} = ā_{1} and t̃_{2}^{p} = −c̄_{1}^{p}t̃_{1} − 2c̄_{2}^{p}t̃_{1}^{p+1} − ... − (p − 1)c̄_{p−1}^{p}t̃_{1}^{p(p−2)+1} + ḡ, or t̃_{1}^{p} = ā_{1} and t̃_{2}^{p} = ḡ) where c̄_{1},...,c̄_{p−1} (resp. ḡ) are functions on X_{k} (eventually equal to 0) which are uniquely determined (resp. determined up to addition of element of the form h̄^{p} where h̄ is a function on X_{k}). The pair (ā_{1}, ḡ) defines then canonically an element of H_{fppf}^{1}(X_{k}, H_{k}) and the tuple (c̄_{1}, ..., c̄_{p−1}) an element of Γ(X_{k}, O_{X_{k}})^{p−1}. Thus we get canonically an element of H_{fppf}^{1}(X_{k}, H_{k}) ⊕ Γ(X_{k}, O_{X_{k}})^{p−1} associated to f_{K} in this case.

2.4.10. It follows from 2.4.8 that an étale ℤ/p^{2}ℤ-torsor above the generic fibre X_{K} of X induces canonically a degeneration data of type either A, B or C. Reciprocally we have the following result of lifting of such a degeneration data.

2.4.11. Proposition. Assume given a degeneration data, say D, of type either A, B or C as in 2.4.6. Then there exists a ℤ/p^{2}ℤ-torsor f_{K} : Y_{K} → X_{K} such that the degeneration data associated to f_{K} via 2.4.8 equals D.

Proof. The proof in the case where the degeneration data is of type A or B is similar to the proof in 2.2.3. Assume that the degeneration data is of type C and consists of the pair (ā_{1}, ā_{2}), where ā_{1} and ā_{2} are functions on X_{k} which are not p-powers, and the tuple of functions (c̄_{1}, ..., c̄_{p−1}). Let a_{1} and a_{2} (resp. c_{1},...,c_{p−1}) be regular functions on X which lift ā_{1} and ā_{2} (resp. which lifts c̄_{1},...,c̄_{p−1}). Let n = pn′ = p^{2}n′′ > 0 be an integer. Consider the ℤ/p^{2}ℤ-torsor given by the equations: (T_{1}^{p}, T_{2}^{p}) − (T_{1}, T_{2}) = (a_{1}π^{−n′p}, f(a_{1})π^{−pm} + a_{2}π^{−pm+n′(p−1)}) where f(a_{1}) = c_{1}a_{1} + ... + c_{p−1}a_{1}^{p−1} and m = n′p.

Then it follows easily from the proof of 2.4.3 that the degeneration data associated to f_{K} via 2.4.7 equals D. In this lifting we have δ_{1} = n′(p−1) and δ_{2} = n′′(p(p−1)+1)(p−1). We have also the other following possibility for the lifting. Namely consider the ℤ/p^{2}ℤ-torsor given by the equations: (T_{1}^{p}, T_{2}^{p})−(T_{1}, T_{2}) = (a_{1}π^{−n′p}, f(a_{1})π^{−pm} +gπ^{−pm+n′(p−1)}) where m is a positive integer such that mp−n′(p−1) > n′(p(p−1)+1) and mp−n′(p−1) = pm′. In this later case we have δ_{1} = n′(p − 1) and δ_{2} = m′(p − 1).

2.5. Degeneration of ℤ/pℤ-torsors on the boundaries of formal fibres.

In this section we assume that the residue field k of R is algebraically closed. In what follows we explain the degeneration of ℤ/pℤ-torsors on the boundary X ≃ Spf R[[T]]{T^{−1}} of formal fibres of germs of formal R-curves. Here R[[T]]{T^{−1}} denotes the ring of formal power series ∑_{i∈ℤ} a_{i}T^{i} with lim_{i→−∞} |a_{i}| = 0 where | | is an absolute value of K associated to its valuation. Note that R[[T]]{T^{−1}} is a complete discrete valuation ring with uniformising parameter π and residue field k((t)) where t ≡ T mod π. The function T is called a parameter of the formal fibre X. The following result, which describes the degeneration of ℤ/pℤ-torsors above the formal fibre Spf R[[T]]{T^{−1}}, will be used in the next paragraph III in order to prove a formula comparing the dimensions of the spaces of vanishing cycles in a Galois cover of degree p between formal germs of R-curves.

2.5.1. Proposition. Let A := R[[T]]{T^{−1}} and let f : Spf B → Spf A be a non trivial Galois cover of degree p. Assume that the ramification index of the corresponding extension of discrete valuation rings equals 1. Then f is a torsor under a finite and flat R-group scheme G_{R} of rank p. Let δ be the degree of the different in the above extension. Then the following cases occur:

a ) δ = 0. In this case f is a torsor under the étale group (ℤ/pℤ)_{R} and for a suitable choice of the parameter T of A the torsor f is given by an equation X^{p} − X = T^{m} for some negative integer m which is prime to p. In this case X^{1/m} is a parameter for B.

b ) 0 < δ = n(p − 1) for some positive integer n. In this case f is a torsor under the group scheme M_{n,R}. Moreover for a suitable choice of the parameter T the torsor f is given by an equation X^{p} − π^{n(p−1)}X = T^{m} with m ∈ ℤ is prime to p. In this case X^{1/m} is a parameter for B.

Proof. First it follows from 2.2.1 that we are either in case a) or in case b). We start first with the case a). In this case f is an étale torsor given by an equation X^{p} − X = u = ∑_{i∈ℤ} a_{i}T^{i} ∈ A. On the level of special fibres the torsor f_{k} := Spec B/πB → Spec A/πA is the étale torsor given by the equation x^{p} − x = ∑_{i≥m} ā_{i}t^{i} ∈ A where ā_{i} is the image of a_{i} modulo π and m ∈ ℤ is some integer. Assume for example that the integer m = pm′ is divisible by p. Then after adding ā_{m}^{1/p}t^{m′} − ā_{m}t^{m} into the defining equation for f_{k} we can replace ā_{m}t^{m} by ā_{m}^{1/p}t^{m′}. Repeating this process eventually we can finally assume that the integer m is prime to p in which case ∑_{i≥m} ā_{i}t^{i} = t^{m}v̄ and u = T^{m}v where v ∈ A is a unit whose image modulo π equals v̄. Further the integer m is then necessarily negative since the residue field extension f_{k} := Spec B/πB → Spec A/πA must ramify. Finally after extracting an m-th root of v and replacing T by T v^{1/m} we arrive to an equation of the form X^{p} − X = T^{m}. Secondly assume that we are in the case b). Then f is a torsor under the finite and flat group scheme M_{n,R}, for some positive integer n, given by an equation X^{p} − π^{n(p−1)}X = u = ∑_{i∈ℤ} a_{i}T^{i} ∈ A and u is not a p-power modulo π. In this case on the level of special fibres the torsor f_{k} := Spec B/πB → Spec A/πA is the α_{p}-torsor given by the equation x^{p} = ∑_{i≥m} ā_{i}t^{i} ∈ A/πA where ā_{i} is the image of a_{i} modulo π and m ∈ ℤ is some integer. Assume for example that the integer m = pm′ is divisible by p. Then after adding ā_{m}^{1/p}t^{m′} − ā_{m}t^{m} into the defining equation for f_{k} we can eliminate the term ā_{m}t^{m} from the defining equation for f_{k} and since ∑_{i≥m} ā_{i}t^{i} ∈ A/πA is not a p-power we can repeat this process eventually and assume after finitely many steps that m is prime to p. In this case ∑_{i≥m} ā_{i}t^{i} = t^{m}v̄ and u = T^{m}v where v ∈ A is a unit whose image modulo π equals v̄. Finally after extracting an m-th root of v and replacing T by T v^{1/m} we arrive to an equation of the form X^{p} − π^{n(p−1)}X = T^{m}.

2.5.2. Definition. With the same notations as in 2.5.1 we define the conductor of the torsor f to be the integer −m. further we define the degeneration type of the torsor f to be (0, m) in the case a) and (n, m) in the case b).

2.5.3. Remark. The above proposition implies in particular that Galois covers f : Spf B → Spf A as in 2.5.1 are classified by their degeneration type as defined in 2.5.2. More precisely given two such Galois covers which have the same degeneration type then there exists a (non canonical) Galois equivariant isomorphism between both covers.

2.6. Degeneration of ℤ/p^{2}ℤ-torsors on the boundaries of formal fibres.

We use the same notations and hypothesis as in 2.5. Next we explain the degeneration of étale ℤ/p^{2}ℤ-torsors on the boundaries of formal fibres.

2.6.1. Proposition. Let A := R[[T]]{T^{−1}} and let f̃ : Y := Spf B → X := Spf A be a non trivial cyclic Galois cover of degree p^{2} with Y_{k} irreducible. Assume that the ramification index of the corresponding extension of discrete valuation rings equals 1. Then f̃ factorises canonically as f̃ = f_{1} ∘ f_{2} with f_{2} : Y → Y_{1} := Spf B_{1} and f_{1} : Y_{1} → X where f_{i} is a Galois cover of degree p. Let δ (resp. δ_{1} and δ_{2}) be the degree of the different in the above extension (resp. in f_{1} and f_{2}). Then the following cases occur:

a ) δ = 0. In this case f̃ is a torsor under the étale R-group G = (ℤ/p^{2}ℤ)_{R} and for a suitable choice of the parameter T of A the torsor f̃ is given by an equation:

(X_{1}^{p}, X_{2}^{p}) − (X_{1}, X_{2}) = (1/T^{m_{1}}, f(T))

where m_{1} is a positive integer prime to p and f(T) ∈ A is such that its image f̄(t) modulo π equals ∑_{i≥−m_{2}} c_{i}t^{i} where m_{2} ≥ 0 is prime to p. In this case the torsor f_{1} (resp. f_{2}) has a reduction of type (0, −m_{1}) (resp. (0, m̃_{2} := −pm_{2} + m_{1}(p − 1)) if m_{2} ≥ pm_{1} and (0, m̃_{2} := −m_{1}(p(p − 1) + 1)) otherwise).

b) δ_{1} = 0 and δ_{2} > 0. In this case f̃ is a torsor under the group scheme W_{(0,n)} for some positive integer n and for a suitable choice of the parameter T of A the cover f̃ is generically given by the equations:

X_{1}^{p} − X_{1} = 1/T^{m_{1}}

and:

X_{2}^{p} − X_{2} = f(T)/π^{pn} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}X_{1}^{pk}(−X_{1})^{p−k}

with f(T) ∈ A is such that its image f̄(t) = ∑_{i∈I} c_{i}t^{i} modulo π is not a p-power. Let m_{2} := inf{i ∈ I, gcd(i, p) = 1}. We call the integer m_{2} the conductor of f(T). Then the torsor f_{1} (resp. f_{2}) has a reduction of type (0, −m_{1}) (resp. (n, m̃_{2}) where m̃_{2} := pm_{2} + m_{1}(p − 1)).

c) δ_{1} < 0. In this case the cover f̃ is generically given for a suitable choice of the parameter T by the equations:

X_{1}^{p} − X_{1} = T^{m_{1}}/π^{n_{1}p}

for some integer m_{1} prime to p and:

X_{2}^{p} − X_{2} = f(T)/π^{pn_{2}} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}X_{1}^{pk}(−X_{1})^{p−k}

where f(T) = ∑_{i∈I} a_{i}T^{i} ∈ A is not a p-power modulo π. Let m_{2} := inf{i ∈ I, gcd(i, p) = 1, and a_{i} is a unit} be the conductor of f. Then the following distinct cases occur:

c-1) pn_{2} − (p − 1)n_{1} > n_{1}(p(p − 1) + 1). Then pn_{2} − (p − 1)n_{1} = pn′_{2} is divisible by p and the torsor f_{1} (resp. f_{2}) has a reduction of type (n_{1}, m_{1}) (resp. (n′_{2}, m_{2}p − m_{1}(p − 1))).

c-2) pn_{2} − (p − 1)n_{1} < n_{1}(p(p − 1) + 1). Then n_{1}(p(p − 1) + 1) = pn′′_{2} is divisible by p and the torsor f_{1} (resp. f_{2}) has a reduction of type (n_{1}, m_{1}) (resp. (n′′_{2}, m_{1}(p(p − 1) + 1))).

c-3) pn_{2} − (p − 1)n_{1} = n_{1}(p(p − 1) + 1). Then n_{1}(p(p − 1) + 1) = pn′′_{2} is divisible by p and the torsor f_{1} (resp. f_{2}) has a reduction of type (n_{1}, m_{1}) (resp. (n′′_{2}, m̃_{2}) where m̃_{2} = inf(m_{1}(p(p − 1) + 1), m_{2}p − m_{1}(p − 1)))).

Proof. First it follows from 2.4.3 that the above cases are all the possible cases of degeneration of the ℤ/p^{2}ℤ-torsor f_{K}. We start with the case a) and assume that δ = 0. In this case f̃ is an étale torsor given by an equation (X_{1}^{p}, X_{2}^{p}) − (X_{1}, X_{2}) = (g(T), f(T)) where g(T) (resp. f(T)) are elements of A. Moreover by 2.5.1 we can assume that for a suitable choice of the parameter T we have g(T) = 1/T^{m_{1}} for some positive integer m_{1} prime to p. One has to compute the conductors of the torsors f_{1} and f_{2}. As for f_{1} we have T = X_{1}^{−p/m_{1}}(1 − X_{1}^{(1−p)})^{−1/m_{1}} from which it follows that Z := X_{1}^{−1/m_{1}} is a parameter for Y_{1} and we have T = Z^{p}(1 − Z^{m_{1}(p−1)})^{−1/m_{1}}. The torsor f_{2} is given by the equation X_{2}^{p} −X_{2} = f(T)+p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}X_{1}^{pk}(−X_{1})^{p−k} and after adding elements of the form b^{p} −b where b ∈ A we can assume that the image f̄(t) of f(T) modulo π equals ∑_{i≥−m_{2}} c_{i}t^{i} where m_{2} ≥ 0 is prime to p. Now we have 1/t^{m_{2}} = 1/z^{pm_{2}}(1 − z^{m_{1}(p−1)})^{m_{2}/m_{1}} = 1/z^{pm_{2}} − (m_{2}/m_{1})1/z^{pm_{2}−m_{1}(p−1)} + ..., where z is the image of Z modulo π, which can be replaced after some transformation by 1/z^{m_{2}} −(m_{2}/m_{1})1/z^{pm_{2}−m_{1}(p−1)} +.... On the other hand the summand in p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}X_{1}^{pk}(−X_{1})^{p−k} which gives the highest contribution to the different is −X_{1}^{p(p−1)+1} = 1/z^{m_{1}(p(p−1)+1)}. from this we deduce that the conductor for the torsor f_{2} equals −pm_{2} + m_{1}(p − 1) if m_{2} ≥ pm_{1} and −m_{1}(p(p − 1) + 1) otherwise.

Assume next that we are in the case b). In this case f̃ is generically given by an equation (X_{1}^{p}, X_{2}^{p}) − (X_{1}, X_{2}) = (g(T), f(T)/π^{pn}) where g(T) (resp. f(T)) are elements of A, n is a positive integer, and the image of f(T) modulo π is not a p-power. Moreover by 2.5.1 we can assume that for a suitable choice of the parameter T we have g(T) = 1/T^{m_{1}} for some positive integer m_{1} prime to p. Consider the image f̄(t) of f(T) modulo π which equals ∑_{i≥m_{2}} c_{i}t^{i}. Then we can assume after some transformations which eliminate the p-powers that the integer m_{2} is prime to p. The special fibre of the torsor f_{2} is given by the equation: x̃_{2}^{p} = f̄(t) = ∑_{i≥m_{2}} c_{i}t^{i}. Now t^{m_{2}} = z^{pm_{2}}(1 − z^{m_{1}(p−1)})^{−m_{2}/m_{1}} = z^{pm_{2}} + (m_{2}/m_{1})z^{pm_{2}+m_{1}(p−1)} + ... which can be transformed, after eliminating the term z^{pm_{2}} which is a p-power, to (m_{2}/m_{1})z^{pm_{2}+m_{1}(p−1)} + .... From this we deduce that the conductor of f_{2} equals pm_{2} + m_{1}(p − 1).

Finally assume that we are in the case c). In this case the cover f̃ is given by 2.4.3 and 2.5.1, and for a suitable choice of the parameter T, by the equations:

X_{1}^{p} − π^{n_{1}(p−1)}X_{1} = T^{m_{1}}

and

X_{2}^{p} − X_{2} = f(T)/π^{pn_{2}} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}X_{1}^{pk}(−X_{1})^{p−k}π^{−n_{1}(p(p−k)+k)}

where f(T) = ∑_{i∈I} a_{i}T^{i} ∈ A. Note that the image f̄(T) = ∑_{i∈I} ā_{i}t^{i} modulo π is necessarily a p-power in B_{1}/πB_{1} since the image t of T is. We write f(T) = f((X_{1}^{p} − π^{n_{1}(p−1)}X_{1})^{1/m_{1}}) = ∑_{i∈I} a_{i}(X_{1}^{p} − π^{n_{1}(p−1)}X_{1})^{i/m_{1}}.

The expression f((X_{1}^{p} − π^{n_{1}(p−1)}X_{1})^{1/m_{1}}) can be transformed after addition of elements of the form b^{p} − b (where b ∈ B_{1}) to:

∑_{i} a_{i}^{1/p}X_{1}^{i/m_{1}}π^{−n_{2}} − ∑_{i}(i/m_{1})a_{i}X_{1}^{ip/m_{1}}X_{1}^{1−p}π^{−pn_{2}+n_{1}(p−1)} + g(X_{1})π^{−n′}

where g(X_{1}) ∈ B_{1} and n′ < pn_{2} − n_{1}(p − 1) is a positive integer. Assume first that n_{2}p − n_{1}(p − 1) > n_{1}(p(p − 1) + 1) then n_{2}p − n_{1}(p − 1) = n′_{2}p is necessarily divisible by p and f_{2} is a torsor under the group scheme M_{n′_{2},R}. Its special fibre f_{2,k} is the α_{p}-torsor given by the equation x̃_{2}^{p} = − ∑_{i≥m_{2}}(i/m_{1})ā_{i}x_{1}^{ip/m_{1}}x_{1}^{1−p} where ā_{i} (resp. x_{1}) is the image of a_{i} (resp. X_{1}) modulo π and m_{2} := inf{i ∈ I, gcd(i, p) = 1, and a_{i} is a unit} is the conductor of f. Since x_{1}^{1/m_{1}} is a parameter for B_{1}/πB_{1} we deduce immediately that the conductor for the torsor f_{2} equals m_{2}p − m_{1}(p − 1). Next assume that n_{2}p − n_{1}(p − 1) < n_{1}(p(p − 1) + 1) then n_{1}(p(p − 1) + 1) = n′′_{2}p is necessarily divisible by p and f_{2} is a torsor under the group scheme M_{n′′_{2},R}. Its special fibre f_{2,k} is the α_{p}-torsor given by the equation x̃_{2}^{p} = −x_{1}^{p(p−1)+1} from which we deduce immediately that the conductor of f_{2} equals m_{1}(p(p − 1) + 1). Finally consider the last case where n_{2}p − n_{1}(p − 1) = n_{1}(p(p − 1) + 1) in which case n_{1}(p(p − 1) + 1) = n′′_{2}p is necessarily divisible by p and f_{2} is a torsor under the group scheme M_{n′′_{2},R}. Its special fibre f_{2,k} is the α_{p}-torsor given by the equation x̃_{2}^{p} = −x_{1}^{p(p−1)+1} − ∑_{i≥m_{2}}(i/m_{1})ā_{i}x_{1}^{ip/m_{1}}x_{1}^{1−p} from which we deduce that the conductor of f_{2} equals inf(m_{1}(p(p − 1) + 1), m_{2}p − m_{1}(p − 1)) which equals m_{1}(p(p − 1) + 1) if m_{2} ≤ pm_{1} and m_{2}p − m_{1}(p − 1) otherwise.

2.6.2. Definition. With the same notations as in 2.6.1 let (n_{1}, m_{1}) (resp. (n_{2}, m_{2})) be the degeneration type of the torsor f_{1} (resp. f_{2}). Then we define the degeneration type of the cover f̃ to be {(n_{1}, m_{1}), (n_{2}, m_{2})}. Note that the inequality n_{2} ≥ n_{1}(p(p − 1) + 1)/p holds by 2.4.3.

2.6.3. Remark. Contrary to what happend for p-cyclic covers (cf. Remark 2.5.3) it is no more true that p^{2}-cyclic covers above formal boundaries are determined by their degeneration type as defined in 2.6.2.

2.6.4. Definition. Let {(n_{1}, m_{1}), (ñ_{2}, m̃_{2})} be a pair of pair of integers. We say that this pair is admissible if the following holds: either n_{1} = ñ_{2} = 0 in which case m_{1} and m̃_{2} are negative and m̃_{2} = inf(m_{1}(p(p − 1) + 1), m_{2}p − m_{1}(p − 1)) for some negative integer m_{2} prime to p or n_{1} = 0 and ñ_{2} ≠ 0 in which case m_{1} ≤ −1 and m̃_{2} = pm_{2} + m_{1}(p − 1) for some integer m_{2} prime to p or finally n_{1} ≠ 0 in which case either ñ_{2} > n_{1}(p(p − 1) + 1)/p and m̃_{2} = pm_{2} − m_{1}(p − 1) for some integer m_{2} prime to p or ñ_{2} = n_{1}(p(p − 1) + 1)/p in which case m̃_{2} = m_{1}(p(p − 1) + 1) or m̃_{2} = m_{2}p − m_{1}(p − 1)) for some integer m_{2} prime to p with m_{2} < pm_{1}. It follows from 2.6.1 that the degeneration type {(n_{1}, m_{1}), (ñ_{2}, m̃_{2})} of the cover f̃ is an admissible pair.

III. Computation of vanishing cycles and examples for cyclic pcovers.

The main result of this section is Theorem 3.2.3 which gives a formula which compares the dimensions of the spaces of vanishing cycles in a Galois cover f̃: Y → X with group ℤ/pℤ between formal germs of R-curves where R is a complete discrete valuation ring of equal characteristic p > 0 in terms of the degeneration type of f̃ above the boundaries of X as defined in 2.5.2. This formula enables one in principle to compare the dimensions of the spaces of vanishing cycles in a Galois cover f̃ : Y → X with group a p-group as we will illustrate for the case of cyclic covers of degree p^{2} in IV. In all this section we use the following notations: R is a complete discrete valuation ring of equal characteristic p > 0. We denote by K the fraction field of R by π a uniformising parameter and k the residue field. We also denote by v_{K} the valuation of K which is normalised by v_{K}(π) = 1. We assume also that the residue field k is algebraically closed.

3.1. 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. We refer to [S-1] 3.1 for the definition of the integers δ_{x}, r_{x}, and the genus g_{x} of the point x.

3.2. The compactification process. Let X := Spf Ô_{X,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 with group ℤ/pℤ and Y local. We assume that the special fibre of Y_{k} is reduced (this can always be achieved after a finite extension of R). We will construct a compactification of the above cover f̃ and as an application we will use this compactification in order 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. of 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 has been done in [S-1] 3.3.1 in the inequal characteristic case. we first start with the case where the formal germ X has only one boundary.

3.2.1. Proposition. Let D := Spf R < 1/T > be the formal closed disc centered at ∞ (cf. [B-L], 1, for the definition of R < 1/T >). 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 introduced in 2.5 we have the following possibilities:

a) The torsor f̃ is étale and has a reduction of type (0, −m). In this case consider the Galois cover f : Y → D given generically by the equation Z^{p} − Z = 1/T^{m}. This cover is an étale torsor and its special fibre f_{k} : Y_{k} → X_{k} is étale and Y_{k} is smooth. Moreover the genus of the smooth compactification of Y_{k} equals (m − 1)(p − 1)/2.

b) The cover f̃ is a torsor under the group scheme M_{n,R} for some positive integer n and has a reduction of type (n, m) for some integer m prime to p. Consider the following two cases:

b-1) m > 0. In this case consider the Galois cover f : Y → D given generically by the equation Z^{p} − π^{n(p−1)}Z = T^{m}. This cover is ramified above ∞ with conductor m and its special fibre f_{k} : Y_{k} → X_{k} is radicial. Moreover Y_{k} is smooth and its smooth compactification has genus 0.

b-2) m < 0. In this case consider the Galois cover f : Y → D given generically by the equation Z^{p} − π^{n(p−1)}Z = T^{m}. This cover is an étale torsor on the generic fibre and its special fibre f_{k} : Y_{k} → X_{k} is radicial. Moreover Y_{k} has a unique singular point y which is above ∞ and g_{y} = (−m − 1)(p − 1)/2.

Proof. The proof is similar to the proof of proposition 3.3.1 in [S-1]. For the convenience of the reader we treat the case b-1). In this case consider the Galois p-cover H → ℙ_{R}^{1}, with H normal, above the projective R-line with parameter T defined generically by the equation Z^{p} − π^{n(p−1)}Z = T^{m}. This cover is ramified on the generic fibre only above the point ∞ with conductor m from which we deduce that the genus of the generic fibre H_{K} of H equals (m − 1)(p − 1)/2. On the level of the special fibres the cover H_{k} → ℙ_{k}^{1} is an α_{p} torsor outside the point ∞ defined by the equation z^{p} = t^{m}. The genus of the singularity above the point t = 0 can be then easily calculated and equals (m − 1)(p − 1)/2 (cf. [Sa-1] 3.3.1). From this we deduce that H_{k} is smooth outside t = 0 since the arithmetic genus of H_{K} and H_{k} are equal.

In the next proposition we deal with the general case.

3.2.2. Proposition. Let X := Spf Ô_{x} be the formal germ of an R-curve at a closed point x and let {X_{i}}_{i=1}^{n} be the boundaries of X. Let f̃: Y → X be a Galois cover with group ℤ/pℤ and 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 Y and X, a closed point y ∈ Y and its image x = f(y), such that the formal germ of X (resp. of Y) at x (resp. at y) equals X (resp. equals 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. In particular the arithmetic genus of X_{K} equals g_{x}.

Proof. Similar to the proof of proposition 3.3.2 in [S-1].

The next result is the main one of this section. It provides an explicit formula which compares the dimensions of the spaces of vanishing cycles in a Galois cover of degree p between formal fibres of curves in equal characteristic p > 0.

3.2.3. 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 with 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 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} (cf. 2.5.1). Let (n_{i}, m_{i}) be the reduction type of f̃_{i} (cf. 2.5.2). 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 d_{η} is the degree of the divisor of ramification in the morphism f̃_{K} : Y_{K} → X_{K} induced by f̃ on the generic fibres, where X_{K} := Spec(Ô_{x} ⊗_{R} K) and Y_{K} := Spec(Ô_{Y,y} ⊗_{R} K), and d_{s} := ∑_{i∈I_{et}}(−m_{i} − 1)(p − 1) + ∑_{i∈I_{rad}}(−m_{i} − 1)(p − 1) where I^{rad} is the subset of I consisting of those i for which n_{i} ≠ 0 and I^{et} is the subset of I consisting of those i for which n_{i} = 0 and m_{i} ≠ 0.

Proof. The proof is similar, using 3.2.1, to the proof of theorem 3.4 in [S-1] with the appropriate modifications. We repeat briefly the argument for the convenience of the reader. By Proposition 3.2.2 one can compactify the above morphism f̃. More precisely there 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. of Y) at x (resp. at y) equals X (resp. equals 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 card(I)-distinct smooth projective lines which intersect at the closed point x. The formal completion of X along its special fibre has a covering which consists of card(I) formal closed unit discs which are patched with the formal fibre X along the boundaries X_{i}. The above formula follows then by comparing the arithmetic genus of the generic fibre Y_{K} of Y and the arithmetic genus of its special fibre Y_{k}. Using the precise informations given in Proposition 3.2.1 one can easily deduce that g(Y_{K}) = pg_{x} + (1 − p) + d_{η}/2 + ∑_{i∈I_{>}}(m_{i} + 1)(p − 1)/2 where I_{>} is the subset of I consisting of those i for which the degeneration type above the boundary X_{i} is (n_{i}, m_{i}) with n_{i} > 0 and m_{i} > 0. On the other hand one has g(Y_{k}) = g_{y}+∑_{i∈I_{<}}(−m_{i}−1)(p−1)/2+∑_{i∈I_{et}}(−m_{i}−1)(p−1), where I_{<} is the subset of I consisting of those i for which the degeneration type above the boundary X_{i} is (n_{i}, m_{i}) with n_{i} > 0 and m_{i} < 0, and I_{et} is the subset of I consisting of those i for which the degeneration type above the boundary X_{i} is (0, m_{i}). Now since Y is flat we obtain g(Y_{K}) = g(Y_{k}) and the above formula directly follows.

3.3. p-Cyclic covers above germs of semi-stable curves.

In what follows and as a consequence of theorem 3.2.3 we will deduce some results 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 V in order to exhibit and realise the degeneration data which describe the semi-stable reduction of Galois covers of degree p in equal characteristic p. We start with the case of a Galois cover of degree p above a germ of a smooth point.

3.3.1. Proposition. Let X := Spf R[[T]] be the formal germ of an R-curve at a smooth point x and let X_{η} := Spf R[[T]]{T^{−1}} be 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 d_{η} be the degree of the divisor of ramification in the morphism f : Y_{K} → X_{K}. Then d_{η} = r(p − 1) is divisible by p − 1. We distinguish two cases:

1) Y_{k} is unibranche at y. Let (n, m) be the degeneration type of f above the boundary X_{η} (cf. 2.5.2). Then necessarily r + m − 1 ≥ 0, and g_{y} = (r + m − 1)(p − 1)/2.

2) Y_{k} has p-branches at y. Then the cover f has an étale split reduction of type (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.

As an immediate consequence of 3.3.1 one can immediately see whether the point y is smooth or not. More precisely we have the following:

3.3.2. Corollary. We use the same notation as in 3.3.1. Then y is a smooth point which is equivalent to g_{y} = 0 if and only if r = 1 − m which implies that m ≤ 1. In particular if f has a degeneration of type (n, m) on the boundary with n > 0 and m > 0 then this happend only if r = 0 and m = 1.

Next we will give 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 3.3.3 and 3.3.4 we use the same notations as in 3.3.1. We first begin with examples with genus 0.

3.3.3. Examples. The following are examples given by explicit equations of the different cases depending on the possible degeneration type over the boundary of Galois covers f : Y → X of degree p above X = Spf R[[T]] and where g_{y} = 0 (here y denotes the closed point of Y).

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

2) For m̃ := −m a negative integer prime to p and a positive integer n consider the cover given generically by the equation X^{p} − π^{n(p−1)}X = T m̃. Here r = m + 1 and this cover has a reduction of type (n, m̃) on the boundary.

3) For a positive integer n consider the cover given generically by the equation X^{p} − π^{n(p−1)}X = T. Here r = 0 and this cover has a reduction of type (n, 1) on the boundary.

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

3.3.4. 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 ) Let m > 0 and m′ > m be integers prime to p and consider the cover given generically by the equation X^{p} − X = π/T^{m′} + 1/T^{m}. This cover has a degeneration of type (0, −m) on the boundary and the point y above x is singular and its genus equals (m′ − m)(p − 1)/2.

2 ) Let m, m′, and n be positive integers with m and m′ prime to p and consider the cover given generically by the equation X^{p} − X = T^{m}/π^{pn} + π/T^{m′}. This cover has a degeneration of type (n, m) on the boundary and the point y above x is singular and its genus equals (m′ + m)(p − 1)/2.

3 ) Let m, m′ and n be positive integers such that m and m′ are prime to p and m′ > m. Consider the cover given generically by the equation X^{p} − X = T^{−m}π^{−pn} + π/T^{m′}. This cover has a degeneration of type (n, −m) on the boundary and the point y above x is singular and its genus equals (m′ − m)(p − 1)/2.

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

3.3.5. Proposition. Let X := Spf R[[S, T]]/(ST − π^{e}) be the formal germ of an Rcurve at an ordinary double point x of thickness e. Let X_{1} := Spf R[[S]]{S^{−1}} and X_{2} := Spf R[[T]]{T^{−1}} be the boundaries of X. Let f : Y → X be a Galois cover with group ℤ/pℤ and 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 d_{η} := r(p−1) be the degree of the divisor of ramification in the morphism f : Y_{K} → X_{K}. Let (n_{i}, m_{i}) be the degeneration type on the 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.

3.3.6. Proposition. We use the same notations as in Proposition 3.3.5. We consider the remaining cases:

1) Y_{k} has p + 1 branches at y in which case we can assume that Y is completely split above X_{1}. Let (n_{2}, m_{2}) be the degeneration type on the second boundary X_{2} of X. Then necessarily r + m_{2} − 1 ≥ 0 and g_{y} = (r + m_{2} − 1)(p − 1)/2.

2) Y_{k} has 2p branches at y in which case Y is completely split above the two boundaries of X and g_{y} = (r − 2)(p − 2)/2.

With the same notations as in proposition 3.3.5 and as an immediate consequence one can recognise whether the point y is a double point or not. More precisely we have the following:

3.3.7. Corollary. We use the same notations as in 3.3.5. 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 thickness divisible by p and r = m_{1} + m_{2}. Moreover if r = 0 then g_{y} = 0 is equivalent to m_{1} + m_{2} = 0.

Next we give examples of Galois covers of degree p above the formal germ of a double point which lead to singularities with genus 0, i.e. double points, and such that r = 0. These examples will be used in V in order to realise the “degeneration data” corresponding to Galois covers of degree p in equal characteristic p > 0.

3.3.8. Examples. The following are examples given by explicit equations of the different cases, depending on the possible degeneration type 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. 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 the equation X^{p} − X = 1/T^{m} = S^{m}/π^{mpt} where m is a positive integer prime to p which leads to a reduction on the boundaries of type (0, −m) and (mt, m).

3 ) Let n and m be positive integers such that m is prime to p and n − tm > 0. Consider the cover given generically by the equation X^{p} − X = T^{m}/π^{np} = S^{−m}/π^{p(n−tm)} which leads to a reduction on the boundaries of type (n, m) and (n − tm, −m).

In fact one can describe Galois covers of degree p above formal germs of double points (in equal characteristic p) which are étale above the generic fibre and with genus 0. Namely they are all of the form given in the above examples 3.3.8. In particular these covers are uniquely determined up to isomorphism by their degeneration type on the boundaries. More precisely we have the following:

3.3.9. Proposition. Let X be the formal germ of an 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. 2.5.1). Let y be the closed point of Y and 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) one of the following holds:

a ) The cover f is generically given by the equation X^{p} − X = 1/T^{m} = S^{m}/π^{mpt} where m is a positive integer prime to p. This cover leads to a reduction on the boundaries of X of type (0, −m) and (mt, m). Here t > 0 can be any integer. In this case δ_{1} = 0 and δ_{2} = mt(p − 1).

b ) The cover f is generically given by an equation X^{p}−X = T^{m}/π^{np} = 1/π^{p(n−tm)}S^{m} where m > 0 is an integer prime to p and n > 0 such that n − tm > 0. This cover leads to a reduction on the boundaries of X of type (n, m) and (n − tm, −m). In this case δ_{1} = n(p − 1) and δ_{2} = (n − tm)(p − 1).

Proof. The proof is similar to the proof of 4.2.5 in the inequal characteristic case in [S-1].

3.3.10. variation of the different. The following result, which is a direct consequence of Proposition 3.3.9, describes the variation of the degree of the different from one boundary to another in a cover f : Y → X between formal germs at double points.

3.3.11. Proposition. Let X be the formal germ of an 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 implies necessarily that the thickness 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. 2.5.1). Also denote by X_{i} for i = 1, 2 the boundaries of X. Let f_{i} : Y_{i} → X_{i} be the torsors induced by f above X_{i}. let (n_{i}, m_{i}) be their degeneration type and let δ_{i} be the corresponding degree of the different. Say δ_{1} = δ(0) δ_{2} = δ(t) and δ(0) ≤ δ(t). We have m := −m_{1} = m_{2} say is positive. Then the following holds: 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′.

IV. Computation of vanishing cycles and examples for p^{2}-cyclic covers.

4.1. In this section we use the same notations and hypothesis as in III and we compute the dimensions of the spaces of vanishing cycles arising form a cyclic p^{2}-cover above the formal germ of an R-curve. Further we provide examples for such covers above formal germs of semi-stable curves.

4.1.1. Proposition. 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 with group ℤ/p^{2}ℤ with Y local and Y_{k} reduced. Let {℘_{i}}_{i∈I} be the minimal prime ideals of Ô_{x} which contain π 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 p^{2}-cyclic cover f̃_{i} : Y_{i} → X_{i} above the boundary X_{i} (cf. 2.5.1). Let {(n_{i,1}, m_{i,1}), (n_{i,2}, m_{i,2})} (resp. {(n_{i}, m_{i})}) be the degeneration type of f̃_{i} (cf. 2.6.2) if Y_{i,k} is irreducible (resp. if Y_{i,k} has p-components cf. 2.5.2). Let y be the closed point of Y. Then one has the following “local Riemman-Hurwitz formula”:

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

Where d_{η} = pd_{η,1} + d_{η,2} 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} := pd_{s,1} + d_{s,2} where d_{s,1} = ∑_{i∈I_{et}}(−m_{i,1} − 1)(p − 1) + ∑_{i∈I_{rad}}(−m_{i,1} − 1)(p − 1) (resp. d_{s,2} = ∑_{i∈I_{et}}(−m_{i,2} − 1)(p − 1) + ∑_{i∈Ĩ_{et}}(−m_{i} − 1)(p − 1) + ∑_{i∈I_{rad}}(−m_{i,2} − 1)(p − 1) + ∑_{i∈Ĩ_{rad}}(−m_{i} − 1)(p − 1)) where I^{rad} is the subset of I consisting of those i for which n_{i,1} ≠ 0 (resp. n_{i,2} ≠ 0) and I^{et} is the subset of I consisting of those i for which n_{i,1} = 0 (resp. n_{i,2} = 0). Here Ĩ denotes the subset of I consisting of those i for which Y_{i} has p-components.

Proof. Follows directly from 3.2.3

4.2. p^{2}-Cyclic covers above germs of semi-stable curves.

In what follows and as a consequence of 4.1.1 we will deduce some results in the case of a p^{2}-cyclic cover f : Y → X where X is the formal germ of a semi-stable R-curve at a closed point. We start with the case of a Galois cover of degree p^{2} above a germ of a smooth point.

4.2.1. Proposition. Let X := Spf R[[T]] be the germ of a formal R-curve at a smooth point x and let X_{η} := Spf R[[T]]{T^{−1}} be the boundary of X. Let f : Y → X be a Galois cover of degree p^{2} with Y local. Assume that the special fibre of Y is reduced. Let y be the unique closed point of Y_{k}. Let d_{η} be the degree of the divisor of ramification in the morphism f : Y_{K} → X_{K}. Then d_{η} = r(p − 1) is divisible by p − 1. The following cases occur:

1) Y_{k} is unibranche at y. Let {(n_{1}, m_{1}), (n_{2}, m_{2})} be the degeneration type of f above the boundary X_{η} (cf. 2.5.1). Then necessarily r + pm_{1} + m_{2} − p − 1 ≥ 0 and g_{y} = (r + pm_{1} + m_{2} − p − 1)(p − 1)/2.

2) Y_{k} has p-branches at y. Let {(n, m)} be the degeneration type of f above the boundary X_{η}. Then necessarily r + pm − p − 2 ≥ 0 and g_{y} = (r + pm − p − 2)(p − 1)/2.

3) Y_{k} has p^{2}-branches at y. Then the cover f has an étale completely split reduction on the boundary, i.e. the induced torsor above Spf R[[T]]{T^{−1}} is trivial, in which case necessarily r − 2p − 2 ≥ 0 and g_{y} = (r − 2p − 2)(p − 1)/2.

As an immediate consequence of 4.2.1 one can immediately see whether the point y is smooth or not. More precisely we have the following:

4.2.2. Corollary. We use the same notation as in 4.2.1. Then y is a smooth point which is equivalent to g_{y} = 0 if and only if we are in case 1) and r + pm_{1} + m_{2} = p + 1. In particular if f has a degeneration on the boundaries of type {(n_{1}, m_{1}), (n_{2}, m_{2})} with m_{1} and m_{2} positive then this happend only if r = 0 and m_{1} = m_{2} = 1.

Next we will give examples of cyclic covers of degree p^{2} 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 4.2.3 and 4.2.4 we use the same notations as in 4.2.1. We first begin with examples with genus 0.

4.2.3. Examples. The following are examples given by explicit equations of the different cases, depending on the possible degeneration type over the boundary, of cyclic covers f̃: Y → X of degree p^{2} above X = Spf R[[T]] and where g_{y} = 0 (here y denotes the closed point of Y).

1) Let m_{1} and m_{2} be two positive integers both prime to p and consider the cover which is generically given by the equations: (X_{1}^{p}, X_{2}^{p}) − (X_{1}, X_{2}) = (1/T^{m_{1}}, f(T)) where f(T) = ∑_{i≥−m_{2}} c_{i}T^{i} ∈ R[[T]][T^{−1}] and c_{m_{2}} is a unit in R. Assume that m_{2} ≥ pm_{1}. Then in this case r = m_{1} +1+pm_{2} +p and this cover has a degeneration on the boundary of type {(0, −m_{1}), (0, m̃_{2} := −pm_{2} + m_{1}(p − 1))}. If m_{2} < pm_{1} then r = m_{1} + 1 + p^{2}m_{1} + p and this cover has a degeneration on the boundary of type {(0, −m_{1}), (0, −m_{1}(p(p − 1) + 1))}.

2) Let m_{1} and m_{2} be two positive integers both prime to p such that m_{2} ≥ pm_{1} and consider the cover given generically by the equations: (X_{1}^{p}, X_{2}^{p}) − (X_{1}, X_{2}) = (1/T^{m_{1}}, f(T)/π^{pn}) where n is a positive integer and f(T) = ∑_{i≥−m_{2}} c_{i}T^{i} ∈ R[[T]][T^{−1}] where c_{−m_{2}} is a unit in R. Then in this case r = m_{1} + 1 + pm_{2} + p and this cover has a degeneration on the boundary of type {(0, −m_{1}), (n, m̃_{2} := −pm_{2} + m_{1}(p − 1))}. If m_{2} < pm_{1} then Y is not smooth.

3) Consider the cover given generically by the equations: (X_{1}^{p}, X_{2}^{p}) − (X_{1}, X_{2}) = (T/π^{n_{1}p}, f(T)/π^{n_{2}p}) where f(T) ∈ R[[T]][T^{−1}] has conductor m_{2} = 1 (cf. 2.6.1 for the definition of the conductor). Assume that n_{2}p ≥ n_{1}(p(p − 1) + 1). In this case r = 0 and m_{1} = m_{2} = 1 and this cover has a degeneration on the boundary of type {(n_{1}, 1), (n_{2}, 1)}.

4) Let m_{1} and m_{2} be two positive integers both prime to p. Consider the cover given generically by the equations (X_{1}^{p}, X_{2}^{p})−(X_{1}, X_{2}) = (T^{−m_{1}}/π^{n_{1}p}, f(T)/π^{pn_{2}}) where f(T) ∈ R[[T]][T^{−1}] has conductor −m_{2} (cf. 2.6.1). First assume that pn′_{2} := n_{2}p − n_{1}(p − 1) ≥ n_{1}(p(p−1)+1) and m_{2} ≥ pm_{1}. Then in this case r = m_{1} +1+pm_{2} +p and this cover has a degeneration on the boundary of type {(n_{1}, −m_{1}), (n′_{2}, m̃_{2} := −m_{2}p+m_{1}(p−1))}. Further in this case if m_{2} < pm_{1} then Y is not smooth. Second assume that n_{2}p − n_{1}(p − 1) < pn′_{2} := n_{1}(p(p−1)+1) and m_{2} ≤ pm_{1} then in this case r = m_{1}+1+m_{1}p^{2}+p and this cover has a degeneration on the boundary of type {(n_{1}, −m_{1}), ((n′_{2}, m̃_{2} := −m_{1}(p(p − 1) + 1))}. In this later case if m_{2} > pm_{1} then Y is not smooth.

4.2.4. Corollary. Let X := Spf R[[T]] be the germ of a formal R-curve at a smooth point x and let X_{η} := Spf R[[T]]{T^{−1}} be the boundary of X. Let f̃ : Y → X be a Galois cover of degree p^{2} with Y local and which ramifies above the generic fibre X_{K}. Let {(n_{1}, m_{1}), (ñ_{2}, m̃_{2})} be the degeneration type of the cover f̃ above the formal boundary X_{η}. Assume that Y is smooth. If n_{1} = 0 and n_{2} = 0 (resp. n_{1} = 0 and n_{2} ≠ 0) then necessarily m_{1} ≤ −1 and m̃_{2} = inf(m_{1}(p(p − 1) + 1), m_{2}p + m_{1}(p − 1)) for some negative integer m_{2} prime to p (resp. m̃_{2} = m_{2}p+m_{1}(p−1) for some integer m_{2} prime to p with −m_{2} ≥ pm_{1}). If n_{1} ≠ 0 then necessarily m_{1} ≤ −1 and m̃_{2} ≤ −1 and either ñ_{2} > n_{1}(p(p − 1) + 1)/p in which case m̃_{2} = pm_{2} − m_{1}(p − 1) for some integer m_{2} prime to p with −m_{2} ≥ −pm_{1} or ñ_{2} = n_{1}(p(p − 1) + 1)/p in which case m̃_{2} = m_{1}(p(p − 1) + 1) or m̃_{2} = m_{2}p − m_{1}(p − 1)) for some integer m_{2} prime to p such that m_{2} < pm_{1}.

Proof. Follows easily from 2.6.1 and the examples in 4.2.3 which cover all the possibilities for f as above.

4.2.5. Definition. An admissible pair {(n_{1}, m_{1}), (ñ_{2}, m̃_{2})} as defined in 2.6.4 is said to satisfy the condition (*) if it satisfies the numerical contraints in 4.2.4 above.

Next we give examples of p^{2}-cyclic covers above formal germs of smooth points which lead to singularities with positive genus.

4.2.6. Examples. The following are examples given by explicit equations of the different cases, depending on the possible reduction type, of cyclic covers f̃ : Y → X of degree p^{2} above X = Spf R[[T]] and where g_{y} > 0.

1 ) Let m_{1}, m′_{1}, m̃_{2} and m̃′_{2} be positive integers all prime to p such that m_{1} < m′_{1} and m̃_{2} < m̃′_{2}. Consider the cover given generically by the equations (X_{1}^{p}, X_{2}^{p}) − (X_{1}, X_{2}) = (π/T^{m′_{1}} + 1/T^{m_{1}}, f(T)) where f(T) = ∑_{i≥−m̃′_{2}} c_{i}T^{i} ∈ R[[T]][T^{−1}] with c_{−m̃′_{2}} ∈ πR and f̄(t) = ∑_{i≥−m̃_{2}} c̄_{i}t^{i} ∈ k[[T]] with c̄_{−m̃_{2}} ≠ 0 (i.e. m̃_{2} is the conductor of f_{2}). Assume that m̃′_{2} > pm′_{1}. In this case r = m′_{1} + pm′_{2} + p + 1 and this cover has a degeneration on the boundary of type {(0, −m_{1}), (0, m_{2} := −pm̃_{2} + m_{1}(p − 1))}. Moreover g_{y} = (pm̃′_{2} − pm̃_{2} + m′_{1} − m_{1})(p − 1)/2.

2) Let m_{1}, m′_{1}, m̃_{2}, and m̃′_{2} be positive integers all prime to p such that m_{1} < m′_{1} and m̃_{2} < m̃′_{2}. Consider the cover given generically by the equations (X_{1}^{p}, X_{2}^{p}) − (X_{1}, X_{2}) = (π/T^{m′_{1}} + 1/T^{m_{1}}, f(T)/π^{pn}) where n is a positive integer and f(T) is as in 1). Assume that m̃′_{2} > pm′_{1}. In this case r = m′_{1} + pm̃′_{2} + p + 1 and this cover has a degeneration on the boundary of type {(0, −m_{1}), (n, m_{2} := −pm̃_{2} + m_{1}(p − 1))}. Moreover g_{y} = (pm̃′_{2} − pm̃_{2} + m′_{1} − m_{1})(p − 1)/2.

3) Let m_{1}, m′_{1}, m̃_{2}, and m̃′_{2} be positive integers all prime to p such that m_{1} < m′_{1} and m̃_{2} < m̃′_{2}. Consider the cover given by the equations (X_{1}^{p}, X_{2}^{p}) − (X_{1}, X_{2}) = (π/T^{m′_{1}} + T^{−m_{1}}π^{n_{1}p}, f(T)/π^{pn_{2}}) where n_{1} and n_{2} are positive integers and f(T) is as in 1). Assume that m̃′_{2} > pm′_{1} in which case we have r = m′_{1} + pm̃′_{2} + p + 1. We distinguish the following cases:

3.a) Assume that pn′_{2} := n_{2}p − n_{1}(p − 1) ≥ n_{1}(p(p − 1) + 1). Then this cover has a degeneration on the boundary of type {(n_{1}, −m_{1}), (n′_{2}, m_{2} := −m̃_{2}p + m_{1}(p − 1))} and g_{y} = (pm̃′_{2} − pm̃_{2} + m′_{1} − m_{1})(p − 1)/2.

3.b) Assume that n_{2}p − n_{1}(p − 1) < pn′_{2} := n_{1}(p(p − 1) + 1) then this cover has a degeneration on the boundary of type {(n_{1}, −m_{1}), (n′_{2}, m_{2} := −m_{1}(p(p − 1) + 1))} and g_{y} = (pm̃′_{2} − p^{2}m_{1} + m′_{1} − m_{1})(p − 1)/2.

4) Let m_{1}, m′_{1}, m̃_{2}, and m̃′_{2} be positive integers all prime to p such that m_{1} < m′_{1} and m̃_{2} < m̃′_{2}. Consider the cover given generically by the equations (X_{1}^{p}, X_{2}^{p}) − (X_{1}, X_{2}) = (π/T^{m′_{1}} + T^{−m_{1}}π^{n_{1}p}, f(T)/π^{pn_{2}}) where n_{1} and n_{2} are positive integer and f(T) is as in 1). Assume that m̃′_{2} < pm′_{1} in which case we have r = m′_{1} + m′_{1}p^{2} + p + 1. We distinguish the following cases:

4.a) Assume that pn′_{2} := n_{2}p − n_{1}(p − 1) ≥ n_{1}(p(p − 1) + 1). Then this cover has a degeneration on the boundary of type {(n_{1}, −m_{1}), (n′_{2}, m_{2} := −m̃_{2}p + m_{1}(p − 1))} and g_{y} = (p^{2}m′_{1} − m̃_{2}p + m′_{1} − m_{1})(p − 1)/2.

4.b) Assume that n_{2}p − n_{1}(p − 1) < pn′_{2} := n_{1}(p(p − 1) + 1) then this cover has a degeneration on the boundary of type {(n_{1}, −m_{1}), (n′_{2}, m_{2} := −m_{1}(p(p − 1) + 1))} and g_{y} = (p^{2}m′_{1} − m_{1}p^{2} + m′_{1} − m_{1})(p − 1)/2.

Next we examine the case of cyclic covers of degree p^{2} above formal germs at double points.

4.2.7. Proposition. Let X := Spf R[[S, T]]/(ST − π^{e}) be the formal germ of an R-curve at an ordinary double point x of thickness e and let X_{1} := Spf R[[S]]{S^{−1}} and X_{2} := Spf R[[T]]{T^{−1}} be the boundaries of X. Let f : Y → X be a Galois cover with group ℤ/p^{2}ℤ and 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 d_{η} := r(p − 1) be the degree of the divisor of ramification in the morphism f : Y_{K} → X_{K}. Let {(n_{i,1}, m_{i,1}), (n_{i,2}, m_{i,2})} be the degeneration type on the boundaries of X for i = 1, 2. Then necessarily r + pm_{1,1} + pm_{2,1} + m_{1,2} + m_{2,2} ≥ 0 and g_{y} = (r + pm_{1,1} + pm_{2,1} + m_{1,2} + m_{2,2})(p − 1)/2.

Next we examine the remaining cases.

4.2.8. Proposition. We use the same notation as in Proposition 4.2.7. We consider the remaining cases:

1) Y_{k} has p+1 branches at y in which case we can assume that the degeneration type of f above the boundaries X_{1} is {(n_{1}, m_{1})}. Then necessarily r +pm_{2,1} +p+m_{1} +m_{2,2} ≥ 0 and g_{y} = (r + pm_{2,1} + p + m_{1} + m_{2,2})(p − 1)/2.

2) Y_{k} has 2p branches at y in which case we can assume that n_{1,1} = n_{2,1} = m_{1,1} = m_{2,1} = 0. Then necessarily r+m_{2,2}+m_{2,1}−2p ≥ 0 and g_{y} = (r+m_{2,2}+m_{2,1}−2p)(p−1)/2.

3) Y_{k} has p^{2} + 1 branches at y and we can assume that f is completely split above the boundary X_{1}. Then necessarily r + pm_{2,1} + m_{2,2} − p − 1 ≥ 0 and g_{y} = (r + pm_{2,1} + m_{2,2} − p − 1)(p − 1)/2.

4) Y_{k} has p^{2} + p branches at y and we can assume that f is completely split above the boundary X_{1}. Then necessarily r+m_{2,2}−2p−1 = 0 and g_{y} = (r+m_{2,2}−2p−1)(p−1)/2.

5) Y_{k} has 2p^{2} branches at y and f is completely split on both boundaries of X. In this case g_{y} = (r − 2p − 2)(p − 1)/2.

With the same notation as in proposition 4.2.7 and as an immediate consequence one can recognise whether the point y is a double point or not. More precisely we have the following:

4.2.9. Corollary. We use the same notations as in 4.2.5. 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 thickness divisible by p^{2} and r + pm_{1,1} + pm_{2,1} + m_{1,2} + m_{2,2} = 0. Moreover if r = 0 then g_{y} = 0 is equivalent to m_{1,1} + m_{2,1} = 0 and m_{1,2} + m_{2,2} = 0.

Next we give examples of p^{2}-cyclic covers above the formal germ of a double point which lead to singularities with genus 0, i.e. double points, and such that r = 0. These examples will be used in VI in order to realise the “degeneration data” corresponding to cyclic covers of degree p^{2} in equal characteristic p > 0.

4.2.10. Examples. The following are examples given by explicit equations of the different cases, depending on the possible degeneration type on the boundaries, of cyclic covers f̃: Y → X of degree p^{2} above X = Spf R[[S, T]]/(ST − π^{e}) with r = 0 and where g_{y} = 0 for a suitable choice of e. Note that e = p^{2}t must be divisible by p^{2}. In all the following examples we have r = 0.

1 ) p-Purity: if f̃ 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 the equation (X_{1}^{p}, X_{2}^{p}) − (X_{1}, X_{2}) = (1/T^{m_{1}}, f(T)) where m_{1} is a positive integer prime to p and f(T) ∈ R[[T]][T^{−1}] is such that its image f̄(t) modulo π equals ∑_{i≥−m_{2}} c_{i}t^{i} where m_{2} ≥ 0 is prime to p. This cover leads to a reduction on the boundaries of type {(0, −m_{1}), (0, m̃_{2} := −pm_{2} + m_{1}(p − 1))} and {(tm_{1}p, m_{1}), (pm_{2}t − (p − 1)m_{1}t, pm_{2} − m_{1}(p − 1))} if m_{2} > m_{1}p, and of type {(0, −m_{1}), (0, m̃_{2} := −m_{1}(p(p − 1) + 1))} and {(tm_{1}p, m_{1}), (tm_{1}(p(p − 1) + 1), m_{1}(p(p − 1) + 1))} if m_{2} ≤ m_{1}p.

3 ) Consider the cover given generically by the equation X_{1}^{p} − X_{1} = 1/T^{m_{1}} and X_{2}^{p} − X_{2} = f(T)/π^{pn} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}X_{1}^{pk}(−X_{1})^{p−k} with f(T) ∈ R[[T]][T^{−1}] is such that its image f̄(t) = ∑_{i≥m′} c_{i}t^{i} modulo π is not a p-power, and n is a positive integer such that pn − m_{2}p^{2}t > 0. Let m_{2} be the conductor of f(T). If n − m_{2}pt > m_{1}p^{2}t then Y is smooth and this cover leads to a reduction on the boundaries of type {(0, −m_{1}), (n, pm_{2} + m_{1}(p − 1))} and {(m_{1}pt, m_{1}), (n − m_{2}pt − (p − 1)m_{1}t, −pm_{2} − m_{1}(p − 1))}. Further in this case if n − m_{2}pt ≤ m_{1}p^{2}t then Y is not smooth.

4) Consider the cover given generically by the equation X_{1}^{p} − π^{n_{1}(p−1)}X_{1} = T^{m_{1}} for some integer m_{1} prime to p and:

X_{2}^{p} − X_{2} = f(T)/π^{pn_{2}} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}X_{1}^{pk}(−X_{1})^{p−k}/π^{n_{1}(p(p−k)+k)}

where f(T) = ∑_{i∈I} a_{i}T^{i} ∈ R[[T]][T^{−1}] is not a p-power and where n_{1} (resp. n_{2}) is a positive integer such that n_{1} − ptm_{1} > 0 (resp. n_{2} − ptm_{2} > 0). Here m_{2} denotes the conductor of f_{2}. We distinguish the following cases:

4-1) pn_{2} − (p − 1)n_{1} > n_{1}(p(p − 1) + 1). If pn_{2} − m_{2}p^{2}t − (p − 1)(n_{1} − m_{1}pt) > (n_{1} − m_{1}pt)(p(p−1)+1) which necessarily implies that m_{1}p > m_{2} then Y is smooth and this cover leads to a reduction on the boundaries of type {(n_{1}, m_{1}), ((n_{2}p−n_{1}(p−1))/p, m_{2}p−m_{1}(p− 1))} and {(n_{1} − m_{1}pt, −m_{1}), ((n_{2}p − m_{2}p^{2}t − (p − 1)(n_{1} − m_{1}pt))/p, −m_{2}p + m_{1}(p − 1))}. Further in this case if pn_{2} − m_{2}p^{2}t − (p − 1)(n_{1} − m_{1}pt) ≤ (n_{1} − m_{1}pt)(p(p − 1) + 1) then Y is not smooth.

4-2) pn_{2} − (p − 1)n_{1} < n_{1}(p(p − 1) + 1). If pn_{2} − m_{2}p^{2}t − (p − 1)(n_{1} − m_{1}pt) < (n_{1} − m_{1}pt)(p(p−1)+1) (resp. pn_{2}−m_{2}p^{2}t−(p−1)(n_{1}−m_{1}pt) = (n_{1}−m_{1}pt)(p(p−1)+1)) which necessarily implies that m_{1}p < m_{2} (resp. m_{1}p > m_{2}) then Y is smooth and this cover leads to a reduction on the boundaries of type {(n_{1}, m_{1}), (n_{1}(p(p − 1) + 1)/p, m_{1}(p(p − 1) + 1))} and {(n_{1} − m_{1}pt, −m_{1}), ((n_{1} − m_{1}pt)(p(p − 1) + 1)/p, −m_{1}(p(p − 1) + 1))}. Further in this case if pn_{2} − m_{2}p^{2}t − (p − 1)(n_{1} − m_{1}pt) > (n_{1} − m_{1}pt)(p(p − 1) + 1) then Y is not smooth.

4-3) pn_{2} − (p − 1)n_{1} = n_{1}(p(p − 1) + 1). If pn_{2} − m_{2}p^{2}t − (p − 1)(n_{1} − m_{1}pt) > (n_{1} − m_{1}pt)(p(p−1)+1) which is equivalent to m_{2} < m_{1}p (resp. pn_{2}−m_{2}p^{2}t−(p−1)(n_{1}−m_{1}pt) < (n_{1} −m_{1}pt)(p(p−1)+1) which is equivalent to m_{2} > m_{1}p) then Y is smooth and this cover leads to a reduction on the boundaries of type {(n_{1}, m_{1}), (n_{1}(p(p−1)+1)/p, m_{2}p−m_{1}(p− 1))} and {(n_{1} − m_{1}pt, −m_{1}), ((n_{2}p − m_{2}p^{2}t − (p − 1)(n_{1} − m_{1}pt))/p, −m_{2}p + m_{1}(p − 1))} (resp. {(n_{1}, m_{1}), (n_{1}(p(p − 1) + 1)/p, m_{1}(p(p − 1) + 1))} and {(n_{1} − m_{1}pt, −m_{1}), ((n_{1} − m_{1}pt)(p(p − 1) + 1)/p, −m_{1}(p(p − 1) + 1))}).

In fact one can describe cyclic covers of degree p^{2} above formal germs of double points (in equal characteristic p) which are étale above the generic fibre and with genus 0. Namely they are all of the form given in the above examples 4.2.10. More precisely we have the following:

4.2.11. Proposition. Let X be the formal germ of an R-curve at an ordinary double point x. Let f̃ : Y → X be a cyclic cover of degree p^{2} 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 y be the closed point of Y and assume that g_{y} = 0. Then there exists an isomorphism X ≃ Spf R[[S, T]]/(ST − π^{tp^{2}}) such that the following holds:

a) The cover f̃ is generically given by the equation:

(X_{1}^{p}, X_{2}^{p}) − (X_{1}, X_{2}) = (1/T^{m_{1}}, f(T))

where m_{1} is a positive integer prime to p and f(T) ∈ R[[T]][T^{−1}] is such that its image f̄(t) modulo π equals ∑_{i≥−m_{2}} c_{i}t^{i}, with c_{−m_{2}} ≠ 0, where m_{2} ≥ 0 is prime to p, which leads to a reduction on the boundaries of type {(0, −m_{1}), (0, m̃_{2} := −pm_{2} + m_{1}(p − 1))} and {(tm_{1}p, m_{1}), (pm_{2}t − (p − 1)m_{1}t, pm_{2} − m_{1}(p − 1))} if m_{2} > m_{1}p and of type {(0, −m_{1}), (0, m̃_{2} := −m_{1}(p(p − 1) + 1))} and {(tm_{1}p, m_{1}), (tm_{1}(p(p − 1) + 1), m_{1}(p(p − 1) + 1))} if m_{2} ≤ m_{1}p.

b ) The cover f̃ is given generically by the equations:

X_{1}^{p} − X_{1} = 1/T^{m_{1}}

and:

X_{2}^{p} − X_{2} = f(T)/π^{pn} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}X_{1}^{pk}(−X_{1})^{p−k}

with f(T) ∈ R[[T]][T^{−1}] is such that its image f̄(t) = ∑_{i≥m′} c_{i}t^{i} modulo π is not a p-power and n is a positive integer such that n − m_{2}pt > m_{1}p^{2}t. Here m_{2} denotes the conductor of f(T). this cover leads to a reduction on the boundaries of type {(0, −m_{1}), (n, pm_{2} + m_{1}(p − 1))} and {(m_{1}pt, m_{1}), (n − m_{2}pt − (p − 1)m_{1}t, −pm_{2} − m_{1}(p − 1))}.

c) The cover f̃ is given generically by the equations:

X_{1}^{p} − π^{n_{1}(p−1)}X_{1} = T^{m_{1}}

for some integer m_{1} prime to p and:

X_{2}^{p} − X_{2} = f(T)/π^{pn_{2}} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}X_{1}^{pk}(−X_{1})^{p−k}/π^{n_{1}(p(p−k)+k)}

where f(T) = ∑_{i∈I} a_{i}T^{i} ∈ R[[T]][T^{−1}] is not a p-power and where n_{1} (resp. n_{2}) is a positive integer such that n_{1} − ptm_{1} ≥ 0 (resp. n_{2} − ptm_{2} ≥ 0). Here m_{2} denotes the conductor of f_{2} and the following cases occur:

c-1) pn_{2} − (p − 1)n_{1} > n_{1}(p(p − 1) + 1) and pn_{2} − m_{2}p^{2}t − (p − 1)(n_{1} − m_{1}pt) > (n_{1} − m_{1}pt)(p(p−1)+1) (which necessarily implies that m_{1}p > m_{2}), in which case this cover leads to a reduction on the boundaries of type {(n_{1}, m_{1}), ((n_{2}p−n_{1}(p−1))/p, m_{2}p−m_{1}(p−1))} and {(n_{1} − m_{1}pt, −m_{1}), ((n_{2}p − m_{2}p^{2}t − (p − 1)(n_{1} − m_{1}pt))/p, −m_{2}p + m_{1}(p − 1))}.

c-2) pn_{2} − (p − 1)n_{1} < n_{1}(p(p − 1) + 1) and pn_{2} − m_{2}p^{2}t − (p − 1)(n_{1} − m_{1}pt) < (n_{1}−m_{1}pt)(p(p−1)+1) (resp. pn_{2}−m_{2}p^{2}t−(p−1)(n_{1}−m_{1}pt) = (n_{1}−m_{1}pt)(p(p−1)+1)) which necessarily implies that m_{1}p < m_{2} (resp. m_{1}p > m_{2}). This cover leads to a reduction on the boundaries of type {(n_{1}, m_{1}), (n_{1}(p(p − 1) + 1)/p, m_{1}(p(p − 1) + 1))} and {(n_{1} − m_{1}pt, −m_{1}), ((n_{1} − m_{1}pt)(p(p − 1) + 1)/p, −m_{1}(p(p − 1) + 1))}.

c-3) pn_{2} − (p − 1)n_{1} = n_{1}(p(p − 1) + 1) and pn_{2} − m_{2}p^{2}t − (p − 1)(n_{1} − m_{1}pt) > (n_{1}−m_{1}pt)(p(p−1)+1), which is equivalent to m_{2} < m_{1}p, (resp. pn_{2}−m_{2}p^{2}t−(p−1)(n_{1}− m_{1}pt) < (n_{1} − m_{1}pt)(p(p − 1) + 1) which is equivalent to m_{2} > m_{1}p). This cover leads to a reduction on the boundaries of type {(n_{1}, m_{1}), (n_{1}(p(p − 1) + 1)/p, m_{2}p − m_{1}(p − 1))} and {(n_{1} − m_{1}pt, −m_{1}), ((n_{2}p − m_{2}p^{2}t − (p − 1)(n_{1} − m_{1}pt))/p, −m_{2}p + m_{1}(p − 1))} (resp. {(n_{1}, m_{1}), (n_{1}(p(p−1)+1)/p, m_{1}(p(p−1)+1))} and {(n_{1} −m_{1}pt, −m_{1}), ((n_{1} −m_{1}pt)(p(p− 1) + 1)/p, −m_{1}(p(p − 1) + 1))}).

V. Semi-stable reduction of cyclic p-covers above formal germs of curves in equal characteristic p > 0.

In all this paragraph we use the following notations: R is a complete discrete valuation ring of equal characteristic p with residue field k which we assume to be algebraically closed and fraction field K := Fr R. We denote by π a uniformising parameter of R.

5.1. 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 such that Y is normal and local. In this paper we are mainly concerned with the case where G is cyclic of order p or p^{2}. It follows then easily from the theorem of semi-stable reduction for curves (cf. [De-Mu]), as well as the compactification process in 3.2, that after eventually a finite extension R′ of R with 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′

One can moreover choose the semi-stable models Ỹ and X̃ such that the set of points B_{K′} := {x_{i,K′}}_{1≤i≤s} consisting of the branch locus in the morphism f′_{K} : Y′_{K′} → X′_{K′} specialise in smooth distincts points of X′_{k} and one can choose such X̃ and Ỹ which are minimal for these properties. The fibre g̃^{−1}(x) of the closed point x in X̃ is a tree Γ of projective lines. This tree is canonically endowed with some “degeneration data” that we will exhibit below and in the next section, in the cases where G ≃ ℤ/pℤ and G ≃ ℤ/p^{2}ℤ, and which take into account the geometry of the special fibre Ỹ_{k} of Ỹ. This will follow mainly from the results in II, III and IV.

5.2. We will use the same notations as in 5.1. We assume that G ≃ ℤ/pℤ. We consider the case where X ≃ Spf A is the formal germ of a semi-stable R-curve at a smooth point x i.e. A ≃ R[[T]]. Let R′ be a finite extension of R as in 5.1 and let π′ be a uniformiser of R′. Below we exhibit the degeneration data associated to the semi-stable reduction of Y.

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 formal 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 M_{n,R′} of rank p (cf. 2.5.1) for some integer n ≥ 0. Let (n, m) be the degeneration type of the torsor f_{η} (cf. 2.5.2) which is canonically associated to f. The arithmetic genus g_{y} of the point y equals (r + m − 1)(p − 1)/2 (cf. 3.3.1) where d_{η} := r(p − 1) is the degree of the divisor of ramification in the morphism f′_{K′} : Y′_{K′} → X′_{K′}.

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

Deg.3. For each i ∈ I let {x_{i,j}}_{j∈S_{i}} be the set of points of X_{i} in which specialise some point of B_{K′} (S_{i} may be empty). Also let {z_{i,j}}_{j∈D_{i}} be the set of double points of X̃_{k} supported by X_{i}. In particular x_{i_{0},j_{0}} := x is a double point of X̃_{k}. We denote by B_{k} the set of all points ∪_{i∈I}{x_{i,j}}_{j∈S_{i}}, which is the set of specialisation of the branch locus B_{K′}, and by D_{k} the set of double points of X̃_{k}.

Deg.4. Let U := X̃−{B_{k}∪D_{k}}. Let {U_{i}}_{i∈I} be the set of connected components of U. The restriction f_{i} : V_{i} → U_{i} of f′ to U_{i} is a torsor under a commutative finite and flat R′-group scheme M_{n_{i},R′} of rank p for some integer n_{i} ≥ 0 (cf. 2.2.1) and f_{i,k} : V_{i,k} → U_{i,k} := U_{i}×_{R′}k is a torsor under the k-group scheme M_{n_{i},R′} ×_{R′} k which is either étale isomorphic to (ℤ/pℤ)_{k} or radicial isomorphic to (α_{p})_{k}. Further when we move in the graph Γ (following the above fixed orientation) from a fixed vertex X_{i} in the direction of a vertex X_{i′} such that n_{i′} = 0 then the corresponding integers n_{i} decreases strictly as follows from 3.3.10.

Deg.5. Each smooth point x_{i,j} ∈ B_{k} is endowed via f with a 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 point x_{i,j} we have the reduction type (n_{i,j} := n_{i}, m_{i,j}) on the boundary of the formal fibre at this point induced by g and such that r_{i,j} = −m_{i,j} +1 where r_{i,j}(p−1) is the contribution to d_{η} of the point which specialise into x_{i,j}. In particular m_{i,j} ≤ −1 since r_{i,j} ≠ 0.

Deg.6. Each double point z_{i,j} = z_{i′,j′} ∈ X_{i} ∩ X_{i′} of X̃ with origin vertex X_{i} and terminal vertex X_{i′} is endowed with degeneration data (n_{i,j} := n_{i}, m_{i,j}) and (n_{i′,j′} := n_{i′}, m_{i′,j′}) induced by g on the two boundaries of the formal fibre at this point and we have m_{i,j} + m_{i′,j′} = 0 (cf. 3.3.7). Let e_{i,j} be the thikness of the double point z_{i,j}. Then e_{i,j} = pt_{i,j} is necessarily divisible by p and we have n_{i} − n_{i′} = t_{i,j}m_{i,j} as follows from 3.3.10.

Deg.7. It follows 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 consisting of those i for which the torsor f_{i} is étale (i.e. n_{i} = 0).

5.2.1. 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 such that both m and m + 1 are prime to p consider the cover given generically by the equation X^{p} − X = (T^{−m} + πT^{−m−1}). Here r = m + 2 and this cover has a reduction of type (0, −m) on the boundary. In particular the arithmetic genus g_{y} of the closed point y of Y equals (p − 1)/2. The degeneration data associated to the above cover consists necessarily of a tree with only one vertex and no edges i.e. a unique projective line X_{1} with a marked 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 lead naturally to the following abstract geometric and combinatorial definition of degeneration data.

5.2.2. Definition. A simple degeneration data Deg(x) of type (r, (n, m)) and rank p consists of the following data:

Deg.1. G_{k} is a commutative finite and flat k-group scheme of rank p which is either étale if n = 0, or radicial of type α_{p} otherwise, r ≥ 0 is an integer, and m is an integer prime to p such that r + m − 1 ≥ 0.

Deg.2. Γ := X_{k} is an oriented tree of projective lines over k with vertices Vert(Γ) := {X_{i}}_{i∈I} which is endowed with an origin vertex X_{i_{0}} and a marked point x := x_{i_{0},j_{0}} on X_{i_{0}}. 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}. We further assume that the orientation of Γ is in the direction going form X_{i_{0}} towards its ends.

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

Deg.4. For each i ∈ I is given a torsor f_{i} : V_{i} → U_{i} := X_{i} − {{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, which is either étale or radicial of type α_{p}, with V_{i} smooth. Moreover for each i ∈ I is given an integer n_{i} which equals 0 if f_{i} is étale and is positive otherwise.

Deg.5. For each i ∈ I and j ∈ S_{i} are given integers m_{i,j} where m_{i,j} is the conductor of the torsor f_{i} at the point x_{i,j} (cf. [S] 1.5) with m_{i_{0},j_{0}} = −m. We assume further that m_{i,j} ≤ −1 if n_{i} > 0.

Deg.6. For each double point z_{i,j} = z_{i′,j′} ∈ X_{i} ∩X_{i′} is given an integer m_{i,j} (resp. m_{i′,j′}) prime to p where m_{i,j} (resp. m_{i′,j′}) is the conductor of the torsor f_{i} (resp. f_{i′}) at the point z_{i,j} (resp. z_{i′,j′}) (cf. [S] 1.3 and 1.5). These data must satisfy m_{i,j} + m_{i′,j′} = 0.

Deg.7. For each double point z_{i,j} = z_{i′,j′} ∈ X_{i} ∩ X_{i′} of Γ with origin vertex X_{i} is given an integer e_{i,j} = pt_{i,j} divisible by p such that with the same notations as above we have n_{i} − n_{i′} = m_{i,j}t_{i,j}. Moreover associated to x is an integer e = pt such that n − n_{i_{0}} = mt.

Deg.8. Let I_{et} denote the subset of I 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 := (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 degeneration data of a given type and rank p. We will denote by Deg_{p} the set of isomorphism classes of simple degeneration data of rank p. The discussion in 5.2 can be reinterpreted as follows:

5.2.3. Proposition. Let X be the germ of a formal R-curve at a smooth point x and let f : Y → X be a cyclic p-cover with Y normal and local. Then one can associate to f canonically a simple degeneration data Deg(x) ∈ Deg_{p} which describes the semistable reduction of Y. In other words there exists a canonical “specialisation” map Sp : H_{et}^{1}(Spec L, ℤ/pℤ) → Deg_{p} where L is the function field of the geometric fibre of X.

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

5.2.4. Theorem. The above specialisation map Sp : H_{et}^{1}(Spec L, ℤ/pℤ) → Deg_{p} defined in 5.2.3 is surjective.

Proof. Consider a degeneration data Deg(x) ∈ Deg_{p}. We have to show that Deg(x) is associated to some cyclic p-cover above the formal germ of a smooth R-curve after eventually enlarging R. We assume that the degeneration data is of type (r, (n, m)). We treat only the case n = 0, the case where n > 0 is treated in a similar way. The proof is done by induction on the length of the tree Γ of Deg(x). Assume first that the tree Γ has minimal length and consists of one irreducible component X = ℙ_{k}^{1} with one marked (double) point x and r smooth distinct marked points {x_{i}}_{i=1}^{r}. Let U := X − {x, x_{i}}_{i} and let f̄ : V → U be the torsor given by the data Deg .4 which is necessarily an α_{p}-torsor by 3.3.8 1) (i.e. the integer n_{i} := n′ associated to the vertex X in Deg .4 is non zero). First for each i ∈ {1, ..., r} consider the formal germ X_{i} := Spf R[[T_{i}]] and the cyclic pcover f_{i} : Y_{i} → X_{i} given by the equation Y_{i}^{p} − π^{n′}Y_{i} = T_{i}^{m_{i}} where m_{i} is the “conductor” associated to the point x_{i} in Deg .5 and n′ is the positive integer associated to f̄ in Deg .4. Let X be a formal projective R-line with special fibre X. Let X′ := X − {x} and let X′ (resp U) be the formal fibre of X′ (resp. of U) in X. The torsor f̄ is given by an equation t^{p} = ū where ū is a regular function on U. Let u be a regular function on U which lifts ū. Then the cover f : V → U given by the equation Y^{p} − π^{n′(p−1)}Y = u is a torsor under the R-group scheme M_{n} which lifts the α_{p}-torsor f̄. By construction the torsor f has a reduction on the formal boundary at each point x_{i} of type (n′, m_{i}) which coincides with the degeneration type of the cover f_{i} above the boundary of X_{i}. The technique of formal patching (cf. [S-1], 1) allows then one to construct a p-cyclic cover f′ : Y′ → X′ which restricted to U is isomorphic to f′ and restricted to X_{i}, for each i ∈ {i, ..., r}, is isomorphic to f_{i} (cf. loc. cit.). Let X_{1} → X be the blow up of X at the point x and let X_{1} be the exeptional fibre in X_{1} which meets X at the double point x. Let e = pt be the integer associated to the marked double point x via Deg .7. After enlarging R we can assume that the double point x of X_{1} has thikness e. We have −n′ = mt by assumption. Let X′_{1} be the formal fibre of X′_{1} := X_{1} − {x} in X_{1}. Let f′_{1} : Y′_{1} → X′_{1} be the étale ℤ/pℤtorsor given by the equation Y′^{p} − Y′ = h^{m} where h is a “parameter” on X′_{1}. Further let X_{1,x} ≃ Spf R[[S, T]]/(ST − π^{pt}) be the formal germ of X_{1} at the double point x. Consider the cover f_{1,x} : Y_{1,y} → X_{1,x} given by the equation Y^{p} −Y = S^{m} = π^{ptm}T^{−m}. Then Y_{1,y} is the formal germ of a double point of thikness t (cf. 3.3.9. a). Moreover the cover f′_{1} (resp. f′) has the same degeneration type (by construction) on the boundary corresponding to the double point x as the degeneration type of the cover f_{1,x} above the formal boundary with parameter T (resp. above the formal boundary with parameter S). A second application of the formal patching techniques allows one to construct a p-cyclic cover f_{1} : Y_{1} → X_{1} which restricted to X′ (resp. X′_{1} and X_{1,x}) is isomorphic to f′ (resp. to f′_{1} and f_{x}). Let X̃ be the R-curve obtained by contracting the irreducible component X in X_{1}. We denote the image of the double point x in X̃ simply by x. The cover f_{1} : Y_{1} → X_{1} induces canonically a p-cyclic cover f̃ : Ỹ → X̃ above X̃. Let X_{x} ≃ R[[T]] be the formal germ of X̃ at the smooth point x. Then f̃ induces canonically a p-cyclic cover f_{x} : Y_{y} → X_{x} where y is the closed point of Y above x. Now it is easy to see that the degeneration data associated to f_{x} via 5.2.3 is isomorphic to the degeneration data Deg(x) we started with. Finally the proof in the general case is very similar and is left to the reader. The only modification in the proof above is that one has to considers p-cyclic covers f_{i} : Y_{i} → X_{i} above the formal germs X_{i} := Spf R[[T_{i}]] which one obtains by induction hypothesis as realisation of the degeneration data, induced by Deg(x), on the subtrees Γ_{i} of Γ which starts from the edge x_{i} in the direction of the ends and which clearly has length smaller than the length of Γ.

5.2.5. Remarks.

1. One can also define in the same way as in 5.2.2 and using the results of II, III, and IV, the set of isomorphism classes of “double” degeneration data associated to the minimal semi-stable model of p-cyclic covers f : Y → X above the formal germ of a double point. Moreover one can prove, in a similar way as in 5.2.4, a result of realisation for such a degeneration data.

2. In [M] Maugeais proved (in equal characteristic p > 0) in theorem 5.4 a global result of lifting for finite “admissible” covers of degree p between semi-stable curves. The methods used in the proof of 5.2.4 are essentially the same as he uses but more direct in the sens that the lemmas 4.2, 4.4, and corollary 4.3 he uses are avoided and we use instead our results 3.3.3 and 3.3.8 which are a direct consequence of the computation of vanishing cycles.

3. It is easy to construct examples of p-cyclic covers f : Y → X above the forml germ of a smooth point X 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. More precisely consider the simple degeneration data Deg(x) of type (n, m), with n > 0, m > 0, and n = mt, which consists of a graph Γ with two vertices X_{1} and X_{2} linked by a unique edge x̃ with given marked points x = x_{1} on X_{1} and x_{2} on X_{2}. Further X_{1} is the original component of Γ. As part of the data are given étale torsors of rank p: f_{1} : V_{1} → U_{1} := X_{1} − {x_{1}} with conductor m at x = x_{1} and f_{2} : V_{2} → U_{2} := X_{2} − {x_{2}} with conductor m′ at x_{2}. Also is given the thikness e = pt at the “double” point x with n = tm. Then it follows from 5.2.4 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 simple 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 semi-stable reduction of Y consists of two projective curves which meet at p-double points (the above cover will be étale in reduction above the double point ỹ). In particular one has p − 1 cycles in this configuration.

VI. Semi-stable reduction of cyclic p^{2}-covers above formal germs of curves in equal characteristic p > 0.

6.1. In all this paragraph we use the same notations as in V.

6.2. Let X := Spf 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 such that Y is normal and local. We assume that G ≃ ℤ/p^{2}ℤ is cyclic of order p^{2} and we use the same notations as in 5.1 for the minimal semi-stable model Ỹ of Y. We consider the case where X ≃ Spf R[[T]] 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 5.1 and let π′ be a uniformiser of R′. Below we exhibit the degeneration data associated to the semi-stable reduction of Y and which are consequences of the results in II, III, and IV. In all what follows, and in order to make the statements of our results simpler, we make the following assumption (**): the special fibre of Y is irreducible and for each irreducible component X_{i} of the special fibre X̃ the fibre g^{−1}(X_{i}) of X_{i} in Ỹ is irreducible.

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 finite cyclic cover of degree p^{2}. Let {(n_{1}, m_{1}), (n_{2}, m_{2})} be the degeneration type of the cover f (cf. 2.6.2) which is canonically associated to f and which is an admissible pair by 2.6.4. The arithmetic genus g_{y} of the point y equals (r + pm_{1} + m_{2} − p − 1)(p − 1)/2 (cf. 4.2.1) where d_{η} := r(p − 1) is the degree of the divisor of ramification in the morphism f′_{K′} : Y′_{K′} → X′_{K′}.

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

Deg.3. For each i ∈ I let {x_{i,j}}_{j∈S_{i}} be the set of points of X_{i} in which specialise some point of B_{K′} (S_{i} may be empty). Also let {z_{i,j}}_{j∈D_{i}} be the set of double points of X̃_{k} supported by X_{i}. In particular x_{i_{0},j_{0}} := x is a double point of X̃_{k}. We denote by B_{k} the set of all points ∪_{i∈I}{x_{i,j}}_{j∈S_{i}}, which is the set of specialisation of the branch locus B_{K′}, and by D_{k} the set of double points of X̃_{k}.

Deg.4. Let U := X̃ − {B_{k} ∪ D_{k}}. Let {U_{i}}_{i∈I} be the set of connected components of U. The restriction f_{i} : V_{i} → U_{i} of f′ to U_{i} is a finite cyclic cover of degree p^{2} which factorises canonically as V_{i} ⟶^{f_{i,2}} V′_{i} ⟶^{f_{i,1}} U_{i} where f_{i,2} (resp. f_{i,1}) is a torsor under the R-group scheme M_{n_{i_{2}}} (resp. a torsor under the R-group scheme M_{n_{i_{1}}} ). Moreover to the cover f_{i} is canonically associated a degeneration data D_{i} of type A, B, or C and which describes the special fibre f_{i,k} : V_{i,k} → U_{i,k} := U_{i} ×_{R′} k of f_{i} (cf. 2.4.8). The pair (n_{i_{1}}, n_{i_{2}}) satisfies n_{i_{2}} ≥ n_{i_{1}}(p(p − 1) + 1) by 2.4.3.

Deg.5. Each smooth point x_{i,j} ∈ B_{k} is endowed via f with a 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 point x_{i,j} we have the reduction type {(n_{i_{1},j_{1}} := n_{i_{1}}, m_{i_{1},j_{1}}), (n_{i_{2},j_{2}} := n_{i_{2}}, m_{i_{2},j_{2}})} on the boundary of the formal fibre at this point induced by g, which is an admissible pair satisfying the condition (*) in 4.2.5, and such that r_{i,j} = −pm_{i_{1},j_{1}} − m_{i_{2},j_{2}} + p + 1 where r_{i,j}(p − 1) is the contribution to d_{η} of the point which specialise into x_{i,j}.

Deg.6. Each double point z_{i,j} = z_{i′,j′} ∈ X_{i} ∩ X_{i′} of Γ with origin vertex X_{i} and terminal vertex X_{i′} is endowed with degeneration data {(n_{i_{1},j_{1}} := n_{i_{1}}, m_{i_{1},j_{1}}), (n_{i_{2},j_{2}} := n_{i_{2}}, m_{i_{2},j_{2}})} and {(n_{i′_{1},j′_{1}} := n_{i′_{1}}, m_{i′_{1},j′_{1}}), (n_{i′_{2},j′_{2}} := n_{i′_{2}}, m_{i′_{2},j′_{2}})} induced by g on the two boundaries of the formal fibre at this point. These pairs are admissible. Moreover we have m_{i_{1},j_{1}} + m_{i′_{1},j′_{1}} = 0 and m_{i_{2},j_{2}} + m_{i′_{2},j′_{2}} = 0 (cf. 4.2.9). Let e_{i,j} be the thikness of the double point z_{i,j}. Then e_{i,j} = p^{2}t_{i,j} is necessarily divisible by p^{2} and we have n_{i_{1}} − n_{i′_{1}} = pt_{i,j}m_{i_{1},j_{1}} and n_{i_{2}} − n_{i′_{2}} = t_{i,j}m_{i_{2},j_{2}} as follows from 3.3.10.

Deg.7. It follows after easy calculation that: g_{y} = ∑_{i∈Ĩ}(−2 + ∑_{j∈S_{i}}(d_{i,j}) + ∑_{j∈D_{i}}(d_{i,j}))(p − 1)/2 where Ĩ denote the subset of I consisting of those i for which the cover f_{i} has a reduction of type (étale, étale) or (étale, radicial) (cf. 2.4.8). Here d_{i,j} = (m_{i_{1}} + 1) + p(pm_{i_{1}} + 1) if pm_{i_{1}} > m_{i_{2}} and d_{i,j} = (m_{i_{1}} + 1) + p(m_{i_{2}} + 1) otherwise in the case of (étale, étale) reduction type (resp. d_{i,j} = (m_{i_{1}} + 1) in the case of (étale, radicial) reduction type).

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

6.2.1. Definition. A simple irreducible degeneration data Deg(x) of type (r, {(n_{1}, m_{1}), (n_{2}, m_{2})}) and rank p^{2} consists of the following data:

Deg.1. The pair (n_{1}, n_{2}) satisfies n_{2} ≥ n_{1}(p(p − 1) + 1)/p, r ≥ 0 is an integer, and m_{1} (resp. m_{2}) is an integer prime to p such that r + pm_{1} + m_{2} − p − 1 ≥ 0. We further assume that the pair {(n_{1}, m_{1}), (n_{2}, m_{2})} is admissible (cf. 2.6.4).

Deg.2. Γ := X_{k} is an oriented tree of projective lines over k with vertices Vert(Γ) := {X_{i}}_{i∈I} which is endowed with an origin vertex X_{i_{0}} and a marked point x := x_{i_{0},j_{0}} on X_{i_{0}}. 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}. Further for each vertex X_{i} of Γ is given a set (may be empty) of smooth marked points {x_{i,j}}_{j∈S_{i}}. We further assume that the orientation of Γ is in the direction going form X_{i_{0}} towards its ends.

Deg.3. For each i ∈ I is given a degeneration data D_{i} of type A, B, or C as in 2.4.8 associated to U_{i} := X_{i} − {{x_{i,j}}_{j∈S_{i}} ∪ {z_{i,j}}_{j∈D_{i}}} which is of type (étale, étale), (étale, radicial), or (radicial, radicial) respectively. Moreover for each i ∈ I is given a pair of integers (n_{i_{1}}, n_{i_{2}}) which equals (0, 0) and (0, n_{i_{2}}) in the cases A and B respectively and which satisfies the inequality n_{i_{2}} ≥ n_{i_{1}}(p(p − 1) + 1)/p.

Deg.3’. To the degeneration data D_{i} above and the pair of integers (n_{i_{1}}, n_{i_{2}}) is associated a finite cover f̄_{i} : V_{i} → U_{i} of degeree p^{2} which factorises as V_{i} ⟶^{f̄_{i,2}} V_{i,1} ⟶^{f̄_{i,1}} U_{i} where f̄_{i,2} and f̄_{i,1} have the structure of a torsor under a finite and flat group scheme of rank p. In the case where D_{i} is of type A or B then f̄_{i} is simply the corresponding torsor. If D_{i} is of type C given by a pair (ū_{1}, ū_{2}) of regular functions on U_{i}, defined up to addition of (v̄_{1}^{p}, v̄_{2}^{p}), and a tuple (c̄_{1}, ..., c̄_{p−1}) of regular functions on U_{i}. Then the cover f̄_{i} is defined by the equations: t_{1}^{p} = ū_{1} and t_{2}^{p} = ū_{2} −c̄_{1}^{p}t_{1} −2c̄_{2}^{p}t_{1}^{p+1} −...−(p−1)c̄_{p−1}t_{1}^{p(p−2)+1} if n_{i_{2}} > n_{i_{1}}(p(p−1)+1)/p (resp. t_{2}^{p} = ū_{2}−c̄_{1}^{p}t_{1}−2c̄_{2}^{p}t_{1}^{p+1}−...−(p−1)c̄_{p−1}t_{1}^{p(p−2)+1}−t_{1}^{p(p−1)+1} if n_{i_{2}} = n_{i_{1}}(p(p − 1) + 1)/p).

Deg.4. For each i ∈ I and j ∈ S_{i} are given integers (m_{i_{1},j_{1}}, m_{i_{2},j_{2}}) such that m_{i_{1},j_{1}} (resp. m_{i_{2},j_{2}}) is the conductor of the torsor f̄_{i,1} (resp. the torsor f̄_{i,2}) above in Deg 3′ at the point x_{i,j} (resp. the point of V_{i,1} above x_{i,j}). we assume that the pair {(n_{i_{1}}, m_{i_{1},j_{1}}), (n_{i_{2}}, m_{i_{2},j_{2}})} satisfies the condition (*) in 4.2.5. Moreover we assume that (m_{i_{0}_{1},j_{0}_{1}}, m_{i_{0}_{2},j_{0}_{2}}) = (−m_{1}, −m_{2}).

Deg.5. For each double point z_{i,j} = z_{i′,j′} ∈ X_{i} ∩ X_{i′} is given a pair of integers (m_{i_{1},j_{1}}, m_{i_{2},j_{2}}) (resp. (m_{i′_{1},j′_{1}}, m_{i′_{2},j′_{2}})) prime to p such that m_{i_{1},j_{1}} and m_{i_{2},j_{2}} are the conductors of the torsor f̄_{i,1} (resp. the torsor f̄_{i,2}) above at the point z_{i,j} (resp. m_{i′_{1},j′_{1}} and m_{i′_{2},j′_{2}} are the conductors of the torsor f̄_{i′,1} (resp. the torsor f̄_{i′,2}) above at the point z_{i,j}). We assume further that the pair {(n_{i_{1}}, m_{i_{1},j_{1}}), (n_{i_{2}}, m_{i_{2},j_{2}})} (resp.{(n_{i′_{1}}, m_{i′_{1},j′_{1}}), (n_{i′_{2}}, m_{i′_{2},j′_{2}})}) is admissible. These data must satisfy m_{i_{1},j_{1}} + m_{i′_{1},j′_{1}} = m_{i_{2},j_{2}} + m_{i′_{2},j′_{2}} = 0.

Deg.6. For each double point z_{i,j} = z_{i′,j′} ∈ X_{i} ∩ X_{i′} of Γ with origin vertex X_{i} is given an integer e_{i,j} = p^{2}t_{i,j} divisible by p^{2} such that with the same notations as above we have n_{i_{1}} − n_{i′_{1}} = pt_{i,j}m_{i_{1},j_{1}} and n_{i_{2}} − n_{i′_{2}} = t_{i,j}m_{i_{2},j_{2}}.

Deg.7. Let Ĩ denote the subset of I consisting of those i for which the degeneration data D_{i} f_{i} is of type (étale, étale) or (étale, radicial). Then the following equality should hold: (r + pm_{1} + m_{2} − p − 1)(p − 1)/2 = ∑_{i∈I_{et}}(−2 + ∑_{j∈S_{i}}(d_{i,j}) + ∑_{j∈D_{i}}(d_{i,j}))(p − 1)/2 where d_{i,j} = (m_{i_{1}} + 1) + p(pm_{i_{1}} + 1) if pm_{i_{1}} > m_{i_{2}} and d_{i,j} = (m_{i_{1}} + 1) + p(m_{i_{2}} + 1) otherwise in the case of (étale, étale) type (resp. d_{i,j} = (m_{i_{1}} + 1) in the case of (étale, radicial) type).

We further make the following important assumption as part of the degeneration data:

Deg.8. For each i ∈ I let U_{i} be as in Deg .3 and let U_{i} be a formal affine R-scheme with special fibre U_{i}. We assume that for each i ∈ I there exists a lifting of the degeneration data D_{i} as in 2.4.9 to a p^{2}-cyclic cover f_{i} : V_{i} → U_{i}, which is an étale torsor above the generic fibre of U_{i}, such that the following “compatibility” condition holds: for each double point z := z_{i,j} = z_{i′,j′} ∈ X_{i} ∩ X_{i′} the restriction of f_{i} (resp. restriction of f_{i′}) to the boundary X_{z,i} ≃ Spf R[[T]]{T^{−1}} corresponding to the double point z (resp. the boundary X_{z,i′} ≃ Spf R[[S]]{S^{−1}} corresponding to the point z) are Galois isomorphic.

There is a natural notion of isomorphism of simple irreducible degeneration data of a given type and rank p^{2}. We will denote by Deg_{p^{2}} the set of isomorphism classes of simple irreducible degeneration data of rank p^{2}. The discussion in 6.2 can be reinterpreted as follows:

6.2.2. Proposition. Let X be the germ of a formal R-curve at a smooth point x and let f : Y → X be a cyclic p^{2}-cover with Y normal and local. We assume that the condition (∗∗) in 6.2 holds. Then one can associate to f canonically a simple irreducible degeneration data Deg(x) ∈ Deg_{p^{2}}. In other words there exists a canonical “specialisation” map Sp : H_{et}^{1}(Spec L, ℤ/p^{2}ℤ)^{ir} → Deg_{p^{2}} where L is the function field of the geometric fibre of X and H_{et}^{1}(Spec L, ℤ/p^{2}ℤ)^{ir} is the subset of H_{et}^{1}(Spec L, ℤ/p^{2}ℤ) consisting of those covers satisfying the condition (∗∗).

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

6.2.3. Theorem. The above specialisation map Sp : H_{et}^{1}(Spec L, ℤ/p^{2}ℤ)^{ir} → Deg_{p^{2}} defined in 6.2.2 is surjective.

Proof. Consider a simple irreducible degeneration data Deg(x) ∈ Deg_{p^{2}}. We have to show that Deg(x) is associated to some cyclic p^{2}-cover above the formal germ of a smooth R-curve after eventually enlarging R. We assume that the degeneration data is of type (r, {(n′_{1}, m_{1}), (n′_{2}, m_{2})}). We treat only the case where n′_{1} = n′_{2} = 0. The remaining cases are treated in a similar way. As in the proof of 5.2.4 we proceed by induction on the length of the tree Γ of Deg(x). Assume first that the tree Γ has minimal length and consists of one irreducible component X = ℙ_{k}^{1} with one marked (double) point x = x_{i_{0}} and r smooth marked points {x_{i}}_{i=1}^{r}. Let X be a formal projective R-line with special fibre X. Let X′ := X − {x} and U := X − {x, x_{i}}_{i}. Further let X′ (resp U) be the formal fibre of X′ (resp. of U) in X. Let D be the degeneration data given by Deg .3. This data is necessarily of type C (i.e (radicial, radicial)), since n′_{1} = n′_{2} = 0, hence is given by an element of H_{fppf}^{1}(U, H_{k}) ⊕ Γ(U, O_{X_{k}})^{p−1} where H_{k} is the finite commutative group scheme extension of α_{p} by α_{p} defined in 2.3.1. The degeneration data D is thus given by a pair of functions (ū_{1}, ū_{2}) on U, defined up to addition of (v̄_{1}^{p}, v̄_{2}^{p}), and a tuple (c̄_{1}, ..., c̄_{p−1}) of regular functions on U. Also let (n_{1}, n_{2}) be the pair of integers associated to the vertex X via Deg .3 and which satisfy n_{2} ≥ n_{1}(p(p − 1) + 1)/p. We assume for simplicity that n_{1} = pñ_{1} and that n_{2} = ñ_{1}(p(p − 1) + 1). By 2.4.9 there exists a p^{2}-cyclic cover f′ : V → U which gives rise to the degeneration data D by 2.4.8. For example we can choose the cover f′ to be generically given by the equations: (T_{1}^{p}, T_{2}^{p}) − (T_{1}, T_{2}) = (u_{1}π^{−ñ_{1}p}, f(u_{1})π^{−p^{2}ñ_{1}} + u_{2}π^{−p^{2}ñ_{1}+ñ(p−1)}) where u_{1} (resp. u_{2}) is a regular function on U which lifts ū_{1} (resp. ū_{2}) and f(u_{1}) = c_{1}^{p}u_{1} + c_{2}^{p}u_{1}^{2} + ... + c_{p−1}u_{1}^{p−1} where c_{i} is a regular function on U which lifts c̄_{i}.

For each i ∈ {1, ..., r} let X′_{i} ≃ R[[T_{i}]]{T_{i}^{−1}} be the formal boundary of U at the point x_{i}. The cover f′ induces canonically a cover f′_{i} : Y′_{i} → X′_{i} which by construction has a reduction of type {(n_{1}, m_{i,1}), (n_{2}, m_{i,2})}:= the admissible pair satisfying (*) which is associated to the point x_{i} via Deg .4. Note that we are necessarily in the case c-3) of 2.6.1 and the cover f_{i} is generically given by the equations:

X_{1}^{p} − X_{1} = T_{i}^{m_{i,1}}/π^{ñ_{1}p}

and:

X_{2}^{p} − X_{2} = f(T_{i})/π^{p^{2}ñ_{1}} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}X_{1}^{pk}(−X_{1})^{p−k}

where f(T_{i}) = ∑_{j∈I} a_{j}T_{i}^{j} ∈ R[[T_{i}]]{T_{i}^{−1}} is not a p-power modulo π. Let m_{2} be the conductor of f then m_{i,2} = inf(m_{i,1}(p(p−1)+1), m_{2}p−m_{i,1}(p−1)) (cf. loc. cit.). Further we can assume after some transformation which eliminate the p-powers that f(T_{i}) = ∑_{j≥m_{2}} a_{j}T_{i}^{j}. Let X_{i} ≃ R[[T_{i}]] be the formal open disc with parameter T_{i} and consider the p^{2}-cyclic cover f_{i} : Y_{i} → X_{i} which is generically given by the equations:

X_{1}^{p} − X_{1} = T_{i}^{m_{i,1}}/π^{ñ_{1}p}

and:

X_{2}^{p} − X_{2} = f(T_{i})/π^{p^{2}ñ_{1}} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}X_{1}^{pk}(−X_{1})^{p−k}

Then Y_{i} is smooth (cf. 4.2.3 4)). Now by construction the covers f_{i} and f′ coincide when restricted to the formal fibre X′_{i}. The technique of formal patching (cf. [S-1], 1) allows then one to construct a p^{2}-cyclic cover f̃: Y′ → X′ which restricted to U is isomorphic to f′ and restricted to X_{i}, for each i ∈ {i, ..., r}, is isomorphic to f_{i} (cf. loc. cit.).

Let X_{i_{0}} ≃ Spf R[[T]]{T^{−1}} be the formal boundary of U at the point x = x_{i_{0}}. Then the cover f̃ induces canonically a p^{2}-cyclic cover f_{i_{0}} : Y_{i_{0}} → X_{i,0} which by construction has a reduction of type {(n_{1}, −m_{1} = m_{i_{0,1},j_{0,1}}), (n_{2}, −m_{2} := m_{i_{0,2},j_{0,2}})}:=the pair associated to the point x_{i_{0}} via Deg .4. Note that m_{1} and m_{2} must be negative since we assumed that n_{1} = n_{2} = 0. Moreover because −n_{1} = m_{1}pt and −n_{2} = m_{2}t by Deg .6 and because we assumed n_{2} = n_{1}(p(p − 1) + 1) we see that necessarily m_{2} = m_{1}(p(p − 1) + 1). The cover f_{i_{0}} is then generically given by 2.6.1 c) by equations:

X_{1}^{p} − X_{1} = T^{−m_{1}}/π^{n_{1}p}

and:

X_{2}^{p} − X_{2} = f(T)/π^{pn′′_{2}} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}X_{1}^{pk}(−X_{1})^{p−k}

where f(T) = ∑_{j∈J} a_{j}T^{j} ∈ R[[T]]{T^{−1}} has conductor m′_{2} and the following holds: either pn′′_{2} − n_{1}(p − 1) < n_{1}(p(p − 1) + 1) or pn′′_{2} − n_{1}(p − 1) = n_{1}(p(p − 1) + 1) and m′_{2} ≥ m_{1}p. Moreover we can assume after some transformation which eliminate the ppowers that f(T) = ∑_{j≥m′_{2}} a_{j}T^{j}. Let X_{1} → X be the blow up of X at the point x and let X_{1} be the exeptional fibre in X_{1} which meets X at the double point x = x_{i_{0}}. Let e = p^{2}t be the integer associated to the marked double point x via Deg .6. After enlarging R we can assume that the double point x of X_{1} has thikness e. Let X′_{1} ≃ Spf R < S^{−1} > be the formal fibre of X′_{1} := X_{1} − {x} in X_{1}. Let f′_{1} : Y′_{1} → X′_{1} be the étale ℤ/p^{2}ℤ-torsor given by the equation:

X_{1}^{p} − X_{1} = S^{m_{1}}

and

X_{2}^{p} − X_{2} = h′(S) + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}X_{1}^{pk}(−X_{1})^{p−k}

where h′(S) = f(π^{p^{2}t}S^{−1})/π^{pn′′_{2}} ∈ R < S^{−1} >.

Further let X_{1,x} ≃ Spf R[[S, T]]/(ST − π^{p^{2}t}) be the formal germ of X_{1} at the double point x. Consider the cover f_{1,x} : Y_{1,y} → X_{1,x} given by the equation:

X_{1}^{p} − X_{1} = T^{−m_{1}}/π^{n_{1}p}

and:

X_{2}^{p} − X_{2} = f(T)/π^{pn′′_{2}} + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}X_{1}^{pk}(−X_{1})^{p−k}

hen Y_{1,y} is the formal germ of a double point of thikness t (cf. 4.2.8). Moreover the cover f_{1,x} restricted to the boundary Spf R[[S]]{S^{−1}} is given by the equations:

X_{1}^{p} − X_{1} = S^{−m_{1}}

and

X_{2}^{p} − X_{2} = h′(S) + p^{−1} ∑_{k=1}^{p−1} \binom{p}{k}X_{1}^{pk}(−X_{1})^{p−k}

In particular the cover f′_{1} (resp. f′) has the same degeneration type, by construction, on the boundary corresponding to the double point x as the degeneration type of the cover f_{1,x} above the formal boundary with parameter T (resp. above the formal boundary with parameter S). A second application of the formal patching techniques allows one to construct a p^{2}-cyclic cover f_{1} : Y_{1} → X_{1} which restricted to X′ (resp. X′_{1} and X_{1,x}) is isomorphic to f′ (resp. to f′_{1} and f_{x}). Let X̃ be the R-curve obtained by contracting the irreducible component X in X_{1}. We denote the image of the double point x in X̃ simply by x. The cover f_{1} : Y_{1} → X_{1} induces canonically a p^{2}-cyclic cover f̃: Ỹ → X̃ above X̃. Let X_{x} ≃ R[[T]] be the formal germ of X̃ at the smooth point x. Then f̃ induces canonically a p^{2}-cyclic cover f_{x} : Y_{y} → X_{x} where y is the closed point of Y above x. Now it is easy to see that the degeneration data associated to f_{x} via 6.2.2 is isomorphic to the degeneration data Deg(x) we started with. Now the proof in the general case is very similar and is left to the reader. The only modification is that one has to consider p^{2}-cyclic covers f_{i} : Y_{i} → X_{i} above the formal germs X_{i} := Spf R[[T_{i}]] which one obtains by induction hypothesis as realisation of the degeneration data, induced by Deg(x), on the subtrees Γ_{i} of Γ which starts from the edge x_{i} going to the ends and which have clearly length strictly less than the length of Γ. For this one still has to use the data in Deg .8 to insure that there exists a lifting f′ as above of the degeneration data D above U which restricted to the formal boundary corresponding to the point x_{i} is Galois isomorphic to the restriction of f_{i} to the formal boundary Spf R[[T_{i}]]{T_{i}^{−1}}.

6.2.4. Remarks.

1. The data Deg .8 in 6.2.1 seems necessary for the proof of 6.2.3. Indeed given two p^{2}- cyclic covers above the two boundaries of the formal germ X of an R-curve at a double point with “compatible” degeneration type as in 4.2.11 it is not always possible to find a p^{2}-cyclic cover f : Y → X where Y is the forml germ of a double point and f is étale above X_{K}. Note that the later is possible for p-cyclic covers.

2. Using the same techniques as in 6.2.3 it is possible to prove global results of lifting for finite covers between proper and semi-stable k-curves of degree p^{2} to a p^{2}-cyclic cover between R′-curve over some finite extension R′ of R.

References

[B-L] S. Bosch and W. Lütkebohmert, Formal and rigid geometry, Math. Ann, 295, 291-317, (1993).

[De-Mu] P. Deligne and D. Munford, The irreducibility of the space of curves with given genus, Publ. Math. IHES. 36, 75-109, (1969).

[D-G] M. Demazure and P. Gabriel, Groupes Algébriques, Tome 1, Masson and CIE Éditeur, Paris, North-Holland Publishing Company, Amsterdam, (1970).

[E] H. Epp, Eliminating wild ramification, Invent. Math. 19, 235-249, (1973).

[G-Ma] B. Green and M. Matignon, Order p-automorphisms of the open disc of a p-adic field, J. Amer. Math. Soc. 12, 269-303, (1999).

[H] Y. Henrio, arbres de Hurwitz et automorphismes d’ordre p de disques et couronnes p-adique formels, PHD Thesis, university of Bordeaux, (1999).

[L] C. Lehr, Reduction of wildly ramified covers of curves, PHD Thesis, University of Pennsylvania, (2001).

[Ma] M. Matignon, Vers un algorithme pour la réduction stable des revêtements p-cycliques de la droite projective sur un corps p-adique., Math. Ann. 325, 323-354, (2003).

[Ma-Y] M. Matignon and T. Youssefi, Prolongements de morphismes entre fibres formelles et cycles evanecsents, preprint, (1992).

[M] S. Maugeais, Relèvement des revêtements p-cycliques des courbes rationnelles semistable, Math. Ann., 327, 365-393, (2003).

[R] M. Raynaud, p-groupes et réduction semi-stable des courbes, The Grothendieck Festschrift, Vol III, Progress in Mathematics 88 (1990), Birkhäuser, 179-197.

[R-1] M. Raynaud, Spécialisation des revêtements en caractértistique p > 0, Ann. Scient. Éc. Norm. Sup. 32, 87-126, (1999).

[S] M. Saïdi, Torsors under finite and flat group schemes of rank p with Galois action, Math. Z., 245, 695-710, (2003).

[S-1] M. Saïdi, Wild ramification and a vanishing cycles formula, J. of Algebra, 273, 1, 108-128, (2004).

[S-2] M. Saïdi, Revêtements modérés et groupe fondamental de graphe de groupes, Compositio Math., 107, 319-338, (1997).

[S-3] M. Saïdi, Galois covers of degrre p: Galois action and semi-stable reduction, math.AG/0106249.

[SGA-1] A. Grothedieck, Revêtements étales et groupe fondamental, Lect. Notes in Math. 224, Springer-Verlag, (1971).

[We] S. Wewers, Three point covers with bad reduction, J. Amer. Math. Soc., 16, 991-1032, (2003).

[W] E. Witt, Zyclische Körper und algebren der characteristic p vom grad p^{n}. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Characteristic p, J. Für die reine und angewandte Mathematik (Crelle), 176, 126-140, (1937).