LINEAR SYSTEMS ATTACHED TO CYCLIC INERTIA

MARCO A. GARUTI

Abstract. We construct inductively an equivariant compactification of the algebraic group 𝕎_{n} of Witt vectors of finite length over a field of characteristic p > 0. We obtain smooth projective rational varieties \overline{𝕎}_{n}, defined over F_{p}; the boundary is a divisor whose reduced subscheme has normal crossings.

The Artin-Schreier-Witt isogeny F − 1 : 𝕎_{n} → 𝕎_{n} extends to a finite cyclic cover Ψ_{n} : \overline{𝕎}_{n} → \overline{𝕎}_{n} of degree p^{n} ramified at the boundary. This is used to give an extrinsic description of the local behavior of a separable cover of curves in char. p at a wildly ramified point whose inertia group is cyclic.

In an appendix, we give an elementary computation of the conductor of such a covering, which can otherwise be determined using class field theory.

Let f : C → D be a finite separable cover of (germs of) curves over a perfect field k of characteristic p > 0. Suppose that f is Galois of group G and let y_{0} ∈ C be a ramification point at which the residue field extension k(y_{0})/k(x_{0}) is separable, where x_{0} = f(y_{0}). Recall ([5], chap. IV), that the higher ramification groups G_{i} (in lower numbering) are defined as

G_{i} = {g ∈ G| v_{y_{0}} (g(t) − t) ≥ i + 1}

where t is a local parameter at y_{0}; they do not depend on the choice of t. The inertia group G_{0} is an extension of a p-group by a cyclic group of order prime to p and, for i > 0, the G_{i} are p-groups; they are trivial for large values of i. The degree (at y_{0}) of the different 𝔇_{C/D} = Ann (Ω_{C/D}^{1}) of the extension C/D can be expressed as

deg 𝔇_{C/D,y_{0}} = ∑_{i=0}^{∞} (|G_{i}| − 1)

Using Herbrand’s φ and ψ functions, one can introduce (loc.cit., §3) higher ramification groups in upper numbering G^{φ(r)} = G_{r}, indexed over R_{≥0}. Hence, if e is the largest integer such that G_{e} is nontrivial, the real number

m = φ(e)

is the largest for which G^{m} is nontrivial. When G is abelian, the classical Hasse-Arf theorem (loc.cit.) guarantees that this number is in fact an integer. We shall write m(f) and e(f) to emphasize that these integers are related to the cover f : C → D. The conductor (at y_{0}) of the extension C/D is defined as

cond (C/D, y_{0}) = m(f) + 1.

Geometrically, the conductor can be characterized as follows ([3], chap. VI, 2.12 and [4], 3.6): m + 1 = deg_{y_{0}} 𝔪, where 𝔪 is the smallest modulus such that the cover C/D is isomorphic to a pull-back from an isogeny of the generalized jacobian J_{𝔪} of D.

Research at MSRI is supported in part by NSF grant DMS-9701755.

We shall be concerned with the situation where G_{0} is cyclic of order p^{n}. By Artin-Schreier-Witt theory, there are étale neighborhoods V of y_{0} and U of x_{0} and a map \underline{u}, from U′ = U − \{x_{0}\} to the group 𝕎_{n} of Witt vectors of length n, such that V′ = V − {y_{0}} is the fibred product

V′ ⟶ 𝕎_{n}
f|_{V′} ↓  ↓^{F −1}
U′ ⟶^{\underline{u}} 𝕎_{n}

The maps \underline{u} and \underline{u} + F \underline{w} − \underline{w}, define isomorphic covers for any Witt-vector valued function \underline{w}.

We will make the following assumptions: f itself is cyclic of degree p^{n}, totally ramified at y_{0}; furthermore, U = D and V = C and \underline{u} is defined over k. Working locally for the étale topology, this is no great loss of generality.

We are going to show that, for every positive integer n, there is a smooth projective rational variety \overline{𝕎}_{n} over the prime field F_{p}, equipped with a tautological line bundle O_{\overline{𝕎}_{n}}(1) such that:

1. \overline{𝕎}_{n} contains 𝕎_{n} as an affine open dense subvariety; the boundary B_{n} = \overline{𝕎}_{n}\𝕎_{n} is a divisor such that L(B_{n}) = O_{\overline{𝕎}_{n}}(1) and (B_{n})_{red} has normal crossings.

2. The isogeny F − 1 extends to a separable cyclic cover Ψ_{n} : \overline{𝕎}_{n} → \overline{𝕎}_{n} branched

along B_{n}. Our main result is then:

Theorem 1. Let f : C_{n} → D be a separable cyclic cover of degree p^{n} of curves over a perfect field k, totally branched above x_{0} ∈ D(k). Let \underline{u} = (u_{0}, …, u_{n−1}) be a Witt vector of rational functions on D, with poles of order ν_{i}, prime to p, at x_{0}, such that k(C_{n})/k(D) is defined by F (\underline{Y}) − \underline{Y} = \underline{u}.

Then \underline{u} extends to a morphism \underline{u} : D → \overline{𝕎}_{n} × k such that C_{n} is isomorphic to the normalization of the fibred product D ×_{\overline{𝕎}_{n}} \overline{𝕎}_{n} and

cond (C_{n}/D, y_{0}) − 1 = deg \underline{u}^{∗}O_{\overline{𝕎}_{n}}(1) = max_{0≤i≤n−1}\{p^{n−i−1}ν_{i}\}.

This work was completed while visiting the M.S.R.I. (october-december 1999) during the program on Galois Groups and Fundamental Groups. I would like to thank the participants for many stimulating conversations and in particular Ahmed Abbes and Mohamed Saïdi for pointing out to me the references [1] and [2] respectively.

We begin with the following Definition 1. For every positive integer n, define inductively a variety \overline{𝕎}_{n} over the prime field F_{p}, equipped with a tautological line bundle O_{\overline{𝕎}_{n}}(1), by letting

\begin{aligned}(\overline{𝕎}_{1}, O_{\overline{𝕎}_{1}}(1)) & = (P^{1}, O_{P^{1}}(1)) \\ (\overline{𝕎}_{n+1}, O_{\overline{𝕎}_{n+1}}(1)) & = (P (O_{\overline{𝕎}_{n}} ⊕ O_{\overline{𝕎}_{n}}(p)) , O_{P}(1)) \end{aligned}

(as usual, P(E) denotes the projective bundle associated to a locally free module E and O_{P}(1) its tautological line bundle cf.[E.G.A. II], 4.1.1; for any m ∈ Z, put O(m) = O(1)^{⊗m}).

By definition, there is a natural projection

r : \overline{𝕎}_{n+1} ⟶ \overline{𝕎}_{n} (1)

which is a fibration by projective lines; it follows immediately that \overline{𝕎}_{n} is a smooth projective rational variety of dimension n. Note that O_{\overline{𝕎}_{n}}(1) is not ample (for n ≥ 2); it is only ample relative to r.

On the other hand, \overline{𝕎}_{n+1} is the projective closure of the vector bundle V(O_{\overline{𝕎}_{n}}(p)) over \overline{𝕎}_{n} (cf. [E.G.A. II], 8.4) and there are two canonical sections of r. The zero section, whose divisor we will denote Z_{n+1}, arises from the exact sequence

0 ⟶ O_{\overline{𝕎}_{n}} ⟶ O_{\overline{𝕎}_{n}} ⊕ O_{\overline{𝕎}_{n}}(p) ⟶ O_{\overline{𝕎}_{n}}(p) ⟶ 0

which gives a global section of O_{\overline{𝕎}_{n}} ⊕ O_{\overline{𝕎}_{n}}(p) = r_{∗}O_{\overline{𝕎}_{n+1}}(1), whence a global section of O_{\overline{𝕎}_{n+1}}(1) (the latter restricts to O_{P^{1}}(1) on the fibres of r, so its higher direct images vanish and H^{i}(\overline{𝕎}_{n}, r_{∗}O_{\overline{𝕎}_{n+1}}(1)) = H^{i}(\overline{𝕎}_{n+1}, O_{\overline{𝕎}_{n+1}}(1)) for all i).

The infinity section is obtained by the same construction composed with the isomorphism

P (O_{\overline{𝕎}_{n}} ⊕ O_{\overline{𝕎}_{n}}(p)) ≃ P (O_{\overline{𝕎}_{n}}(−p) ⊕ O_{\overline{𝕎}_{n}}) .

We denote Σ_{n+1} the corresponding divisor in H^{0}(\overline{𝕎}_{n+1}, O_{\overline{𝕎}_{n+1}}(1) ⊗ r^{∗}O_{\overline{𝕎}_{n}}(−p)).

As suggested by the notation, \overline{𝕎}_{n} is a compactification of the Witt group scheme 𝕎_{n}; more precisely:

Proposition 1. There is a natural system of open immersions j_{n} : 𝕎_{n} ⊂ \overline{𝕎}_{n}, compatible w.r.t. the projections r and such that O_{\overline{𝕎}_{n}}(1) trivializes over j_{n}(𝕎_{n}).

The closed subscheme B_{n} = \overline{𝕎}_{n} \ j_{n}(𝕎_{n}) is a divisor linearly equivalent to Σ_{n} + p r^{∗}B_{n−1} (and hence to Z_{n}) and (B_{n})_{red} has normal crossings.

Proof: Indeed, we have a canonical embedding j_{1} : 𝔾_{a} ≅ A^{1} ⊂ P^{1}; by induction, assume O_{\overline{𝕎}_{n}}(1) trivializes over the affine open subset j_{n}(𝕎_{n}) = U_{n}: then, by definition (cf.[E.G.A. II], 3.1), r^{−1}(U_{n}) = P_{U_{n}}^{1}. Clearly, O_{\overline{𝕎}_{n+1}}(1) trivializes over the complement U_{n+1} of the infinity section, and, identifying r^{−1}(0) ∩ U_{n+1} with A^{1}, we can choose an isomorphism j_{n+1} : 𝕎_{n+1} ≅ U_{n+1} fitting in the diagram

A^{1} ⟶ U_{n+1} ⟶^{r|_{U_{n+1}}} U_{n}
j_{1}↑  ↑  ↑^{j_{n}}
𝔾_{a} ⟶^{V^{n}} 𝕎_{n+1} ⟶^{r} 𝕎_{n}

where the bottom one is the exact sequence of group schemes defining 𝕎_{n+1}. The second statement follows easily, as j_{n+1}(𝕎_{n+1}) has been defined as the intersection of r^{−1}(U_{n}), which is the complement of the vertical divisor r^{∗}(pB_{n}), with the complement of the horizontal infinity divisor Σ_{n+1}.

Corollary 1. For all 1 ≤ i ≤ n, let B_{n,i} = (r^{n−i})^{∗}(Σ_{i}) be the pull back of the infinity divisor on \overline{𝕎}_{i} via the iteration of the projection (1). The boundary divisor is then

B_{n} = ∑_{i=1}^{n} p^{n−i}B_{n,i}

From now on, we identify 𝕎_{n} with its image in \overline{𝕎}_{n}. Choose a system of variables Y_{0}, Y_{1}, …, Y_{n}, … such that 𝕎_{n} = Spec F_{p}[Y_{0}, …, Y_{n−1}] and the projection r corresponds to the inclusion F_{p}[Y_{0}, …, Y_{n−1}] ⊂ F_{p}[Y_{0}, …, Y_{n}]. On the ring of polynomials

H = F_{p}[T, Y_{0}, …, Y_{n}, …]

we define a grading by deg T = 1 and deg Y_{i} = p^{i} (recall [AC IX] that with this choice for the degrees, the polynomials defining the sum, product etc. of Witt vectors are isobaric). As usual, we denote by H_{d} the d-th graded piece, consisting of homogeneous elements of degree d.

Lemma 1. The tautological line bundle O_{\overline{𝕎}_{n}}(1) is generated by global sections; there is a canonical isomorphism H^{0}(\overline{𝕎}_{n}, O_{\overline{𝕎}_{n}}(1)) = H_{p^{n−1}} under which Z_{n} = (Y_{n−1})_{0} and B_{n} = (T^{p^{n−1}})_{0}.

Proof: The assertions being clear for n = 1, the proof is by induction. We need a classical explicit description of the projective bundle \overline{𝕎}_{n+1}. By inductive assumption, the complete linear series |pZ_{n}| defines a morphism

ϕ_{|pZ_{n}|} : \overline{𝕎}_{n} ⟶ P^{m} = PH^{0}(\overline{𝕎}_{n}, O_{\overline{𝕎}_{n}}(p)).

The vector space decomposition H_{p^{n}} = H^{0}(\overline{𝕎}_{n}, O_{\overline{𝕎}_{n}}(p)) ⊕ ⟨Y_{n}⟩ induces a rational map

π : P^{m+1} = PH_{p^{n}} −−→ P^{m}

defined outside [0 : ⋯: 0 : 1]. The indeterminacy is solved on the blow-up of this point, P̃^{m+1} = P(O_{P^{m}} ⊕ O_{P^{m}}(1)), and \overline{𝕎}_{n+1} is canonically isomorphic to the fibred product:

\overline{𝕎}_{n+1} ⟶ P̃^{m+1}
r↓  ↓π̃
\overline{𝕎}_{n} ⟶^{ϕ_{|pZn|}} P^{m}

To complete the proof of the lemma it suffices to remark that the composite morphism \overline{𝕎}_{n+1} → P̃^{m+1} → P^{m+1} is given by the complete linear system |Z_{n+1}|; its image is the projective cone over ϕ_{|pZ_{n}|}(\overline{𝕎}_{n}).

Proposition 2. The action of 𝕎_{n} on itself extends to an action of 𝕎_{n} on \overline{𝕎}_{n} and the line bundle O_{\overline{𝕎}_{n}}(1) is stable under this action.

In particular, Z/p^{n}Z = 𝕎_{n}(F_{p}) acts on \overline{𝕎}_{n} and the the Artin-Schreier-Witt isogeny (F − 1) : 𝕎_{n} → 𝕎_{n} extends to a cyclic cover

Ψ_{n} : \overline{𝕎}_{n} ⟶ \overline{𝕎}_{n}

of degree p^{n}, defined over F_{p}, commuting with the projection r. It is branched along the boundary divisor B_{n} and the inertia subgroup of the infinity section Σ_{n} has order p.

The induced map Ψ_{n}^{∗} on the group of k-cycles modulo rational equivalence is multiplication by p^{n−k}.

Proof: Clearly, 𝔾_{a} acts on P^{1} and the isogeny ℘ : 𝔾_{a} → 𝔾_{a} extends to a cyclic cover of degree p of P^{1} branched at ∞; we proceed by induction. The group 𝕎_{n+1} acts on itself by translations; recall (e.g. [AC IX]) that the sum of two Witt vectors \underline{X}, \underline{Y} is expressed by polynomials S_{i}(\underline{X}; \underline{Y}) ∈ Z [X_{0}, …, X_{i}, Y_{0}, …, Y_{i}] characterized by the relations

Φ_{n}(S_{0}, …, S_{n}) = Φ_{n}(X_{0}, …, X_{n}) + Φ_{n}(Y_{0}, …, X_{n})

If the variables X_{i} and Y_{i} are given the weight p^{i}, then S_{i} is isobaric of weight p^{i}. Moreover:

S_{i}(\underline{X}; \underline{Y}) = X_{i} + Y_{i} + c_{i}(\underline{X}; \underline{Y}), c_{i} ∈ Z [X_{0}, …, X_{i−1}; Y_{0}, …, Y_{i−1}] (2)

Therefore, given an F_{p}-algebra A, for any \underline{a} ∈ 𝕎_{n+1}(A), the polynomial

T^{p^{n}}S_{n}(\frac{Y_{0}}{T} , …, \frac{Y_{n}}{T^{p^{n}}}; \underline{a}) = Y_{n} + a_{n}T^{p^{n}} + T^{p^{n}}c_{n}(\frac{Y_{0}}{T} , …, \frac{Y_{n−1}}{T^{p^{n−1}}} ; \underline{a})

is a homogeneous element in H_{p^{n}} ⊗ A. As O_{\overline{𝕎}_{n}}(1) is stable under the action, 𝕎_{n}(A) acts A-linearly on H^{0}(\overline{𝕎}_{n,A}, O_{\overline{𝕎}_{n}}(p)): we extend it to an action of 𝕎_{n+1}(A) on H_{p^{n}} ⊗ A by

\underline{a} · Y_{n} = Y_{n} + a_{n}T^{p^{n}} + T^{p^{n}}c_{n}(\frac{Y_{0}}{T} , …, \frac{Y_{n−1}}{T^{p^{n−1}}} ; \underline{a}) (3)

The point [0 : ⋯: 0 : 1] ∈ P_{A}^{m+1} = PH_{p^{n}} ⊗ A is fixed and one easily checks that the action extends on the blow-up P̃^{m+1} ⊂ P^{m+1} × P^{m}; in particular, the inertia subgroup of the exceptional divisor for the action of Z/p^{n+1}Z = 𝕎_{n+1}(F_{p}) has order p. As \overline{𝕎}_{n+1} = P̃^{m+1} ×_{P^{m}} \overline{𝕎}_{n}, we have the obtained the desired action; moreover, the Z/p^{n+1}Z-inertia of the infinity section Σ_{n+1} is of order p.

Let π : P → \overline{𝕎}_{n+1} be the cyclic cover of degree p^{n} given by the fibred product

P ⟶^{π} \overline{𝕎}_{n+1}
q↓  ↓^{r}
\overline{𝕎}_{n} ⟶^{Ψ_{n}} \overline{𝕎}_{n}

As Ψ_{n}^{∗} is multiplication by p, we have P = P (O_{\overline{𝕎}_{n}} ⊕ O_{\overline{𝕎}_{n}}(p^{2})). We have a sheaf of graded O_{\overline{𝕎}_{n}}-algebras

S = ⨁_{d≥0} O_{\overline{𝕎}_{n}}(pd)

and, with the notations as in [E.G.A. II] §2, \overline{𝕎}_{n+1} = Proj S[Y_{n}] and P = Proj S^{(p)}[X_{n}]. Define a morphism of graded O_{\overline{𝕎}_{n}}-algebras:

φ^{#} : S^{(p)}[X_{n}] ⟶ (S[Y_{n}])^{(p)}

which is the identity on S^{(p)} and sends X_{n} to

T^{p^{n+1}}S_{n}(\frac{Y_{0}^{p}}{T^{p}} , …, \frac{Y_{n}^{p}}{T^{p^{n+1}}} ; −(\frac{Y_{0}}{T} , …, \frac{Y_{n}}{T^{p^{n}}})) = Y_{n}^{p}−Y_{n}T^{p^{n}(p−1)}+T^{p^{n+1}}c_{n}(\frac{Y_{0}^{p}}{T^{p}} , …; −\frac{Y_{0}}{T} , …).

This defines a \overline{𝕎}_{n}-morphism φ : G(φ^{#}) → P from an open subset of Proj (S[Y_{n}])^{(p)} = \overline{𝕎}_{n+1} (loc.cit, prop. 3.1.8). The only homogeneous prime ideal containing S^{(p)} and Y_{n} is the irrelevant ideal, hence G(φ^{#}) = \overline{𝕎}_{n+1}. As φ is equivariant under the action of Z/pZ given by equation (3) (via the n-th iteration of the Verschiebung), the composite morphism

Ψ_{n+1} : \overline{𝕎}_{n+1} ⟶^{φ} P ⟶^{π} \overline{𝕎}_{n+1}

extends the Artin-Schreier-Witt isogeny and is equivariant under the Z/p^{n+1}Z-action defined above (it suffices to check it over the dense open subset 𝕎_{n+1}) and satisfies

Ψ_{n} ∘ r = r ∘ Ψ_{n+1}.

To prove the last statement, recall that, via r^{∗}, the cohomology group A^{∗}(\overline{𝕎}_{n+1}) is a free A^{∗}(\overline{𝕎}_{n})-module generated by the classes 1 and ξ = [Z_{n+1}]. As Ψ commutes with r, it suffices to prove that Ψ_{n+1}^{∗}ξ = p ξ. Write:

Ψ_{n+1}^{∗}ξ = aξ + β a ∈ Z, β ∈ r^{∗}A^{1}(\overline{𝕎}_{n}). (4)

Intersecting with a fibre of r, we deduce a = p. On the other hand, let η = r^{∗}[Z_{n}]; since c_{t}(O_{\overline{𝕎}_{n}} ⊕ O_{\overline{𝕎}_{n}}(p)) = c_{t}(O_{\overline{𝕎}_{n}}(p)) = 1 + p [Z_{n}]t, we have

ξ^{2} = p ξη.

As Ψ_{n+1}^{∗}η = p η, taking self intersection on both sides of (4) we get:

p^{2}ξ^{2} + p^{2}βη = p(pξ + β)pη = Ψ_{n+1}^{∗}(pξη) = Ψ_{n+1}^{∗}(ξ^{2}) = p^{2}ξ^{2} + 2pξβ + β^{2}

whence (2pβ) ξ +(β^{2} −p^{2}βη) = 0. Since 1, ξ form a A^{∗}(\overline{𝕎}_{n})-basis, this implies β = 0.

Corollary 2. With the notation as in Corollary 1, the inertia subgroup at the i-th component B_{n,i} of the boundary divisor B_{n} on \overline{𝕎}_{n} is cyclic of order p^{n−i}.

Let now f_{n} : C_{n} → D be a cyclic cover as in Theorem 1 and \underline{u} : D − \{x_{0}\} → 𝕎_{n} the corresponding Witt representative; clearly, \underline{u} extends to

\underline{u} : D ⟶ \overline{𝕎}_{n}.

Lemma 2. With the notation as above, deg \underline{u}^{∗}O_{\overline{𝕎}_{n}}(1) = max_{0≤i≤n−1}\{p^{n−i−1}ν_{i}\}.

Proof: It suffices to consider the composite map (cf. lemma 1)

D ⟶^{\underline{u}} \overline{𝕎}_{n} ⟶^{ϕ_{|Zn|}} PH_{p^{n−1}}.

In a neighborhood of x_{0}, it is defined by [s^{M} : s^{M}u_{0}^{p^{n−1}} : …, s^{M}u_{n−1}], where s is a local parameter at x_{0} and

M = max {i_{0}ν_{0} + ⋯ + i_{n−1}ν_{n−1}| 0 ≤ i_{h} ≤ p^{n−1−h} and ∑_{h=0}^{n−1} p^{h}i_{h} = p^{n−1}}
= max_{0≤i≤n−1}{p^{n−i−1}ν_{i}}.

Proposition 3. Let f_{n} : C_{n} → D be as in Theorem 1 and let M(f_{n}) = deg \underline{u}^{∗}O_{\overline{𝕎}_{n}}(1). If m(f_{n}) + 1 is the degree of the conductor, then

m(f_{n}) ≤ M(f_{n}).

Proof: We show that 𝔲 = (1 + M(f_{n})).∞ is a modulus, in the sense of [3], chap. III, for the rational map \underline{u} : D − \{∞\} → 𝕎_{n}, with the smallest possible degree. By the universal property of the generalized jacobians, \underline{u} factors through J_{𝔲} and, by the minimality of the conductor, 𝔪 ≤ 𝔲.

We have to show that (\underline{u}, α)_{∞} = \underline{0} ∈ 𝕎_{n}(k) for any rational function α on D such that α ≡ 1 mod 𝔲, where (\underline{u}, −)_{∞} denotes the local symbol associated to \underline{u}: it is the ”Residuenvektor” defined in [6], §2, whose j-th phantom component is

Φ_{j} ((\underline{u}, α)_{∞}) = Res (Φ_{j}(u_{0}, …, u_{j})\frac{dα}{α})

Let s be a local parameter on D at ∞; if A is any complete discrete valuation ring of characteristic zero with residue field k, we can lift α ∈ k[[s]] and u_{0}, …, u_{n−1} ∈ k((s)) to formal power series α̃ ∈ A[[s]] and ũ_{0}, …, ũ_{n−1} ∈ A((s)), with the ũ_{i} having a pole at s = 0 of the same order as u_{i}: by definition, (\underline{u}, α) is the reduction modulo pA of (\underline{ũ}, α̃). Recall that u_{i} has a pole of order ν_{i} and that M = max {p^{n−i−1}ν_{i} : 0 ≤ i ≤ n − 1}. It is then clear that M is the smallest integer ρ for which

Φ_{j} ((\underline{ũ}, α̃)) = Res (Φ_{j}(ũ_{0}, …, ũ_{j})\frac{dα̃}{α̃})
= Res ((ũ_{0}^{p^{j}} + ⋯ + p^{j}ũ_{j}) \frac{dα̃}{α̃})

vanishes for any 0 ≤ j ≤ n − 1 and for any α̃ such that 1 − α̃ has a zero of order ρ + 1 at s = 0.

To complete the proof of the Theorem, we need to show that the bound provided by Proposition 3 is attained. When k is finite, this was done by H.L. Schmid, [2], §3: if χ : G_{k(D)}^{ab} → Z/p^{n}Z is the character corresponding to the cover f_{n}, then the residue vector is related to the reciprocity map by the formula

χ(rec(α)) = Tr ((\underline{u}, α)_{∞}) ∀α ∈ k(D)^{×},

where Tr : 𝕎_{n}(k) → 𝕎_{n}(F_{p}) is the trace map. The computation of the conductor follows at once from its classical description in terms of norms of units (e.g. [5], chap. XV, §2).

For an arbitrary field k, this classical approach no longer holds, but it is still possible to prove the desired equality using higher class field theory: see [1], §2.

In the appendix we give an elementary proof of this formula; the reader who is willing to venture through it will certainly appreciate the difference between the clean and conceptual approach via class field theory and the following tedious but elementary computations.

Appendix: Elementary computation of the conductor of a cyclic extension.

Our goal is to show that the inequality in Proposition 3 is indeed an equality. Again, our proof is by induction on n, the case n = 1 being classical (e.g. [4], 4.4). We shall therefore assume that the Theorem is true for cyclic extensions of degree p^{l}, for l ≤ n and prove that it is true for l = n + 1 by showing that, with the notations of Proposition 3, m(f_{n+1}) = M(f_{n+1}).

To this end, we need to compute the conductor of C_{n+1}/D or, which amounts to the same, that of C_{n+1}/C_{n}. The latter is a cyclic extension of degree p and hence it is described by an Artin-Schreier equation y^{p} − y = z, for a suitable rational function z on C_{n} which is not, however, in standard form i.e. z has a pole of order divisible by p, and we cannot read off the conductor immediately.

Some work is needed in order to bring the Artin-Schreier equation describing the cover C_{n+1}/C_{n} in standard form. The key to all our computations is the following, elementary

Lemma 3. (Adjustment lemma). Let f : Y → X be a finite separable cover of curves of degree p^{n}, totally ramified at y_{0}. Let t (resp. s) be local parameters at y_{0} (resp. x_{0} = f(y_{0})). Then, in a neighborhood of y_{0}

s = t^{p^{n}} (α_{1}^{p} + t^{μ(f)}α_{0})

for some α_{0}, α_{1} ∈ O_{Y,y_{0}}^{×}, with

μ(f) = deg 𝔇_{Y/X,y_{0}} − p^{n} + 1

Proof: Let s = αt^{p^{n}} for some unit α ∈ O_{Y,y_{0}}^{×} and expand α = ∑_{r≥0} a_{r}t^{r} with a_{r} ∈ k and a_{0} ∈ k^{×}. If μ is the smallest positive integer prime to p such that a_{μ} ≠ 0, then

deg 𝔇_{Y/X,y_{0}} = v_{y_{0}} (\frac{ds}{dt}) = p^{n} + v_{y_{0}} (\frac{dα}{dt}) = p^{n} + μ − 1

We can therefore collect: α = ∑_{0≤r< \frac{μ}{p}} a_{pr}t^{pr} + t^{μ} ∑_{r≥μ} a_{r}t^{r} = α_{1}^{p} + t^{μ}α_{0}.

Corollary 3. With the notations above, let g be a rational function on X. Then f^{∗}(g) = g_{1}^{p} +g_{0} for suitable rational functions g_{i} on Y with v_{y_{0}}(g_{0}) = p^{n}v_{x_{0}}(g)+μ(f).

Proof: Expanding g as a power series in s, it is sufficient to prove it when g(s) = as^{r} is a monomial of degree prime to p:

as^{r} = at^{p^{n}r} (α_{1}^{p} + t^{μ}α_{0})^{r} = at^{p^{n}r} (α_{1}^{pr} + t^{μ}α′_{0}) = (a^{\frac{1}{p}} t^{p^{n−1}}α_{1}^{r})^{p} + at^{p^{n}r+μ}α′_{0}

When the cover is Galois, it is easy to relate the number μ(f) to the conductor of the extension and how it varies in towers:

Proposition 4. With the notation as in Lemma 3, suppose moreover that f is Galois of group G and conductor m(f) + 1.

1. If e(f) is the largest integer such that G_{e} ≠ 1, then

μ(f) = p^{n}m(f) − e(f)

2. Let g : Z → Y → X be a second separable Galois cover, totally ramified above x_{0} whose group G̃ is an extension of G by H = Z/pZ. Let e(g) be the largest integer such that G̃_{e} ≠ 1 and suppose that G̃_{e(g)} ∩ H ≠ 1. If m(g) + 1 denotes the conductor of g, then

e(g) = p^{n}m(g) − μ(f)

Proof: (1). Denote, as usual, by g_{i} = |G_{i}|; as f is totally ramified, G_{0} = G. By[5], chap. IV, proposition 12, m = \frac{1}{g_{0}} ∑_{i=0}^{e} g_{i} − 1, hence

μ(f) = deg 𝔇_{Y/X,y_{0}} − p^{n} + 1
= ∑_{i=0}^{e}(g_{i} − 1) − p^{n} + 1
= ∑_{i=0}^{e} g_{i} − e − p^{n}
= p^{n} (\frac{1}{p^{n}} ∑_{i=0}^{e} g_{i} − 1) − e.

(2). Denote by φ the Herbrand function of the cover g and by g̃_{i} = |G̃_{i}|. The hypotheses imply that g̃_{i} = pg_{i} for 0 ≤ i ≤ e(f), hence

m(g) = φ(e(g)) = \frac{1}{p^{n+1}} (g̃_{1} + ⋯ + g̃_{e(f)} + p [e(g) − e(f)])
= \frac{1}{p}φ(e(f)) + \frac{1}{p^{n}} [(e(g) − e(f)]
= m(f) + \frac{1}{p^{n}} [e(g) − e(f)].

Corollary 4. Let f_{n} : C_{n} → D be a separable cyclic cover of degree p^{n}, totally ramified above ∞. For any 0 ≤ i ≤ n, let f_{i} : C_{i} → D be the quotient cover of degree p^{i} and by g_{i} : C_{n} → C_{i} the subcover of degree p^{n−i}. Put m_{i} = m(f_{i}) (resp. e_{i} = e(f_{i}), resp. μ_{i} = μ(f_{i})). Then:

e_{n} = ∑_{i=1}^{n} p^{i−1}(m_{i} − m_{i−1});  μ_{n} = ∑_{i=1}^{n}(p^{i} − p^{i−1})m_{i};  μ(g_{i}) = μ_{n} − p^{n−i}μ_{i}.

Proof: From Proposition 4.2, we deduce the inductive relations

e_{n} − e_{n−1} = p^{n−1}(m_{n} − m_{n−1}); μ_{n} − μ_{n−1} = (p^{n} − p^{n−1})m_{n}

from which the first two formulas in the claim follow at once. As for the third, it is a consequence of the transitivity of the different of a composite extension:

μ(g_{i}) = deg 𝔇_{C_{n}/C_{i}} − p^{n−i} + 1 = deg 𝔇_{C_{n}/D} − p^{n} + 1 − p^{n−i} (deg 𝔇_{C_{i}/D} − p^{i} + 1) .

We are now ready to resume the discussion interrupted before Lemma 3. We are given a separable cyclic cover f_{n+1} : C_{n+1} → D of degree p^{n+1}, as in the theorem, and we want to compute its conductor, to show that the inequality provided by Proposition 3 is indeed an equality.

Equivalently, we can compute the conductor of the degree p subextension C_{n+1}/C_{n}. With the notations as in Proposition 4, let G̃ = Z/p^{n+1}Z be the group of C_{n+1}/D and H = Z/pZ that of C_{n+1}/C_{n}: by [5], IV, prop. 2, H_{i} = H ∩ G̃_{i} and it is clear that H = H_{e_{n+1}} = G̃_{e_{n+1}} is the last nontrivial subgroup for both extensions.

We begin by making the Artin-Schreier equation defining the cover C_{n+1}/C_{n} more explicit. Away from ∞, the cover C_{n+1}/D is described by the Witt vector equation

F (\underline{Y}) − \underline{Y} = (u_{0}, …, u_{n}) (5)

where the left hand side should be understood as a difference of two vectors, while the equation defining C_{n}/D is the image of (5) under the restriction homomorphism r : 𝕎_{n+1} → 𝕎_{n}. Therefore, the n-th component of (5) is

Y_{n}^{p} − Y_{n} + c_{n} (F (\underline{Y}), −\underline{Y}) = u_{n}

where the polynomial c_{n}(\underline{X}, \underline{Y}) is the one defined by equation (2). If \underline{y} = (y_{0}, …, y_{n−1}) denotes any solution of the Artin-Schreier-Witt equation defining C_{n}/D, the cover C_{n+1}/C_{n} is described by the Artin-Schreier equation

Y_{n}^{p} − Y_{n} = u_{n} − c_{n} (\underline{y}^{p}, −\underline{y}) . (6)

As remarked, this equation is not in standard form: the datum on the right hand side has, in general, poles of order divisible by p. We can get rid of them by adding terms of the type ℘(h) = h^{p} − h for suitable rational functions h on C_{n}; the result will be a rational function with a pole at ∞ of order e_{n+1}. By Proposition 4.2, e_{n+1} = p^{n}m_{n+1} − μ_{n}.

To complete the proof of the theorem, we should then prove the following:

Claim 1. The right hand side of equation (6) is congruent mod ℘ (k(C_{n})) to a function with a pole at ∞ of order p^{n}M − μ_{n}, where M = max {p^{n−i}ν_{i} : 0 ≤ i ≤ n}.

It is easy to handle the term u_{n} in (6) with the adjustment lemma: u_{n} is a rational function on D with a pole of order ν_{n} at ∞, hence we can find rational functions u′_{n}, u′′_{n} on C_{n} such that:

u_{n} = u′_{n}^{p} + u′′_{n} with v_{n}(u′′_{n}) = −p^{n}ν_{n} + μ_{n} (7)

We can then rewrite u_{n} = u′′_{n} + u′_{n} + ℘(u′_{n}). When M = ν_{n}, the function u′′_{n} has the predicted valuation and in any event the contribution of u′_{n} is negligible, as v_{n}(u′_{n}) = −p^{n−1}ν_{n} ≥ −p^{n−1}M, hence

\begin{aligned}−p^{n}M + μ_{n} + p^{n−1}ν_{n} & ≤ −(p^{n} − p^{n−1})M + ∑_{i=1}^{n}(p^{i} − p^{i−1})m_{i} \\ & < −m_{n}(p^{n+1} − 2p^{n} + 1) < 0 \end{aligned} (8)

as m_{i} < m_{n} for i < n and M ≥ pm_{n} (by our inductive hypothesis, m_{j+1} = max {p^{j−i}ν_{i} : i ≤ j} for j ≤ n − 1).

We must then analyze the contribution of the second term on the right hand side of (6); according to the claim, it should be dominant when ν_{n} < M. Indeed, we will prove the following:

Claim 2. The function c_{n}(\underline{y}^{p}, −\underline{y}) is congruent mod ℘ (k(C_{n})) to a function with a pole at ∞ of order p^{n+1}m_{n} − μ_{n}.

By the induction hypothesis, M = max {ν_{n}, pm_{n}}, and these two numbers cannot be equal; hence, the latter claim implies the former.

The following lemma gathers the information we shall need about the polynomials c_{i}.

Lemma 4. Fix integers n and i ≤ n − 1. Then

c_{n}(\underline{X}, \underline{Y}) = −X_{i}^{p^{n−i}−1} (Y_{i} + c_{i}(\underline{X}, \underline{Y})) + R_{i,n}

where R_{i,n} is an isobaric polynomial of weight p^{n} in which the variable X_{i} appears with degree strictly smaller than p^{n−i} − 1.

Proof: It is well known that c_{0} = 0 and c_{1} = − ∑_{r=1}^{p−1} \frac{1}{p} \binom{p}{r}X_{0}^{r}Y_{0}^{p−r}. We proceed by induction. Using the recursive relation Φ_{n}(X_{0}, …, X_{n}) = Φ_{n−1}(X_{0}^{p}, …, X_{n−1}^{p}) + p^{n}X_{n}, we see that

c_{n} = S_{n} − X_{n} − Y_{n} = \frac{1}{p^{n}} [∑_{h=0}^{n−1} p^{h} (X_{h}^{p^{n−h}} + Y_{h}^{p^{n−h}} − S_{h}^{p^{n−h}})] .

Fix 0 ≤ h ≤ n − 1; applying the binomial formula twice:

\begin{matrix}X_{h}^{p^{n−h}} + Y_{h}^{p^{n−h}} − (X_{h} + Y_{h} + c_{h})^{p^{n−h}} = \\ −c_{h}^{p^{n−h}} − ∑_{s=1}^{p^{n−h}−1} \binom{p^{n−h}}{s}X_{h}^{s}Y_{h}^{p^{n−h}−s} − ∑_{r=1}^{p^{n−h}−1} ∑_{s=0}^{r} \binom{p^{n−h}}{r}\binom{r}{s}X_{h}^{s}Y_{h}^{r−s}c_{h}^{p^{n−h}−r}. \end{matrix} (9)

The variable X_{i} appears in the expression above only if h ≥ i; suppose h > i, then by the induction hypothesis we can write

c_{h} = −X_{i}^{p^{h−i}−1} (Y_{i} + c_{i}) + R_{i,h}

hence the highest power of X_{i} in (9) appears in the term c_{h}^{p^{n−h}} and its contribution in c_{n} is of degree at most (p^{h−i} − 1)p^{n−h}.

For h = i, the variable X_{i} does not appear in c_{i} hence the term containing the highest power of X_{i} in (9) is

−\binom{p^{n−i}}{1}X_{i}^{p^{n−i}−1}Y_{i} − \binom{p^{n−i}}{p^{n−i} − 1}\binom{p^{n−i} − 1}{p^{n−i} − 1}X_{i}^{p^{n−i}−1}c_{i} = −p^{n−i}X_{i}^{p^{n−i}−1} (Y_{i} + c_{i}) .

Our task is now to compute the valuation at ∞ of a rational function on C_{n}, congruent to c_{n}(\underline{y}^{p}, −\underline{y}) mod ℘(k(C_{n})) and having a pole of order prime to p, where \underline{y} = (y_{0}, …, y_{n−1}) is a vector of solutions for the Artin-Schreier-Witt equation defining C_{n}/D.

Lest the reader should get lost or depressed (or both) by the following computations, let us state immediately that we are going to show that, for any i ≤ n − 1, only the monomials

y_{i}^{p(p^{n−i}−1)+1}

are relevant. More precisely, if j ≤ n − 1 is the (necessarily unique) integer for which

m_{n} = p^{n−j−1}ν_{j} = max {p^{n−i−1}ν_{i} : 0 ≤ i ≤ n − 1}

then y_{j}^{p(p^{n−j}−1)+1} is congruent mod ℘ (k(C_{n})) to a function with a pole at ∞ of order p^{2n−j}ν_{j} − μ_{n} and that c_{n}(\underline{y}^{p}, −\underline{y}) − y_{j}^{p(p^{n−j}−1)+1} is congruent to a function with a pole of order strictly smaller.

The approach is in two steps: first, we obtain an estimate of the valuation of c_{n}(\underline{y}^{p}, −\underline{y}) and then we apply the adjustment lemma to the functions y_{i} to get rid of the terms in ℘(k(C_{n})). In both steps, the problem to sort out the relevant monomials is reduced to a linear optimization.

The proof is only a tedious exercise in linear programming, and the reader would loose very little, should he indulge in the temptation to skip the rest of this section.

When the variables X_{i} and Y_{i} are given the weight p^{i}, the polynomial c_{n}(\underline{X}, \underline{Y}) is isobaric of weight p^{n}; evaluating at \underline{X} = \underline{y}^{p} and \underline{Y} = −\underline{y}, our task is to compute the valuation of a sum of monomials ∏_{i=0}^{n−1} y_{i}^{pa_{i}+b_{i}}.

Put v_{n}(y_{i}) = w_{i}; regarding a_{i} and b_{i} ad variables, the question can be rephrased as an optimization problem for a linear function under a linear constraint:

(10) Minimize : ∑_{i=0}^{n−1} w_{i}(pa_{i} + b_{i}) under : ∑_{i=0}^{n−1} p^{i}(a_{i} + b_{i}) = p^{n}

with a_{i} ∈ [0, p^{n−i}−1] and b_{i} ∈ [0, p^{n−i}]. The range of the a_{i}’s is restricted by Lemma 4; we do not apply it to the b_{i}’s because of the irregular behavior in characteristic 2 of the polynomials I_{i}(\underline{Y}) expressing the opposite of a Witt vector.

Actually, it is more convenient to rephrase the problem slightly: putting α_{i} = pa_{i} + b_{i} and taking α_{i}, b_{i} as variables, it becomes:

(11) Minimize : ∑_{i=0}^{n−1} w_{i}α_{i} under : ∑_{i=0}^{n−1} p^{i}α_{i} + ∑_{i=0}^{n−1}(p^{i+1} − p^{i})b_{i} = p^{n+1}

with α_{i} ∈ [0, p^{n−i+1} − p + 1] and b_{i} ∈ [0, p^{n−i}].

Extremal values of a linear function on a convex polytope are attained at the vertices; as b_{i} = p^{n−i} forces a_{i} = 0, hence α_{i} = p^{n−i}, best admissible solutions are for α_{i} = p^{n−i+1} − p + 1, b_{i} = 1 and and α_{l} = b_{l} = 0 for l ≠ i. We have thus:

v_{n}(c_{n}(\underline{y}^{p}, −\underline{y})) ≥ min \{(p^{n−i+1} − p + 1)w_{i} | 0 ≤ i ≤ n − 1\}

and this in turn implies our first estimate:

Lemma 5. 1. v_{n}(c_{n}(\underline{y}^{p}, −\underline{y})) ≥ −(p^{n+1} − p + 1)m_{n}.

2. v_{n+1}(y_{n}) ≥ −p^{n}m_{n+1}, with an equality if and only if ν_{n} = m_{n+1}.

Proof: Induction on n: the case n = 0 being clear, assume (1) and (2) hold for c_{i} and y_{i} for all i ≤ n − 1. Then, since m_{n} ≥ p^{n−i−1}m_{i+1}, we have

v_{n}(y_{i}^{p(p^{n−i}−1)+1}) = p^{n−i−1}v_{i+1}(y_{i}^{p(p^{n−i}−1)+1})
≥ −p^{n−i−1}(p^{n+1} − p^{i+1} + p^{i})m_{i+1}
≥ −(p^{n+1} − p^{i+1} + p^{i})m_{n}

Finally, as y_{n} is a solution of the equation y_{n}^{p} − y_{n} = u_{n} − c_{n} and clearly v_{n}(u_{n}) = −p^{n}ν_{n} ≥ −p^{n}m_{n+1}, with equality holding if and only if ν_{i} = m_{i+1}, (2) follows at once.

To compute the valuation of c_{n}(\underline{y}^{p}, −\underline{y}) mod ℘(k(C_{n})), we have to apply the adjustment lemma twice, first on C_{i} and then on C_{n}.

Fix i ≤ n − 1 and denote by v_{i} the discrete valuation of O_{C_{i},∞}. By definition, y_{i} is the solution (on C_{i+1}) of the Artin-Schreier equation y_{i}^{p} − y_{i} = u_{i} − c_{i}(\underline{y}^{p}, −\underline{y}); applying the adjustment lemma to u_{i} and the induction hypothesis to c_{i}, we can find rational functions g_{i} and ũ_{i} on C_{i} such that

u_{i} − c_{i}(\underline{y}^{p}, −\underline{y}) = ũ_{i} + g_{i}^{p} − g_{i}

with v_{i}(ũ_{i}) = p^{i}m_{i+1} − μ_{i} and, by the lemma above, v_{i}(g_{i}^{p}) ≥ −p^{n}m_{n+1}, equality holding iff ν_{i} = m_{i+1}.

On C_{i+1} we have thus y_{i} = ỹ_{i} + g_{i} with

v_{i+1}(ỹ_{i}) = −p^{i}m_{i+1} + μ_{i}; v_{i+1}(y_{i}) = v_{i+1}(g_{i}) ≥ p^{i}m_{i+1}

with an equality iff ν_{i} = m_{i+1}.

We now apply the adjustment lemma on C_{n} both to ỹ_{i} and g_{i}: we can write

ỹ_{i} = γ̃_{i}^{p} + η̃_{i} g_{i} = γ′_{i}^{p} + η′_{i}

with v_{n}(γ̃_{i}) = p^{−1}v_{n}(y_{i}) = −p^{n−2}m_{i+1} + p^{n−i−1}μ_{i+1} and, applying Corollary 4,

v_{n}(η̃_{i}) = v_{n}(ỹ_{i}) + μ(C_{n}/C_{i+1})
= −p^{n−i−1}(p^{i}m_{i+1} − μ_{i}) + μ_{n} − p^{n−i−1}μ_{i+1}
= −p^{n−1}m_{i+1} − p^{n−i−1}(p^{i+1} − p^{i})m_{i+1} + μ_{n}
= −p^{n}m_{i+1} + μ_{n}.

As g_{i} comes from C_{i}, we have v_{n}(γ′_{i}) = p^{−1}v_{n}(g_{i}) ≥ −p^{n−2}m_{i+1}, with equality iff ν_{i} = m_{i+1} and

v_{n}(η′_{i}) = v_{n}(g_{i}) + μ(C_{n}/C_{i})
= p^{n−i}v_{i}(g_{i}) + μ_{n} − p^{n−i}μ_{i}
≥ −p^{n−1}m_{i+1} + μ_{n} − p^{n−i}μ_{i}

Trivially, v_{n}(γ′_{i}) ⪇ v_{n}(γ̃_{i}); the same rough estimates as in equation (8) yield v_{n}(η̃_{i}) ⪇ v_{n}(η′_{i}). Putting γ_{i} = γ̃_{i} + γ′_{i} and η_{i} = η̃_{i} + η′_{i}, we can write

y_{i} = γ_{i}^{p} + η_{i} with: v_{n}(η_{i}) = −p^{n}m_{i+1} + μ_{n}; v_{n}(γ_{i}) ≥ −p^{n−2}m_{i+1}

again the last inequality being an equality iff ν_{i} = m_{i+1}.

Substituting in ∏_{i=0}^{n−1} y_{i}^{α_{i}} we get:

∏_{i=0}^{n−1}(γ_{i}^{p} + η_{i})^{α_{i}} = ∏_{i=0}^{n−1} γ_{i}^{pα_{i}} + ∑_{i=0}^{n−1} (α_{i}γ_{i}^{pα_{i}−1}η_{i} ∏_{h≠i} γ_{h}^{pα_{h}}) + …

The first term is a p-power and, by Lemma 5, its valuation is bounded from below by −(p^{n+1} − p + 1)m_{n}; the same estimates as those of formula (8) show that its p-th root has a pole of order strictly smaller than p^{n+1}m_{n} − μ_{n} and we can get rid of it mod℘(k(C_{n})).

To prove the claim, we should therefore compute, for all i, h ≤ n − 1, the valuation of the monomials γ_{i}^{pα_{i}−1}η_{i} ∏_{h≠i} γ_{h}^{pα_{h}}.

Our optimization problem (11) becomes then:

Minimize : v_{n}(η_{i}) + w_{i}α_{i} + ∑_{h≠i} w_{h}α_{h} under : ∑_{l=0}^{n−1} p^{l}α_{l} + ∑_{l=0}^{n−1}(p^{l+1} − p^{l})b_{l} = p^{n+1}

with α_{l} ∈ [0, p^{n−l+1} − p + 1] and b_{l} ∈ [0, p^{n−l}].

Proof of Claim 2. All we have to do is check the extremal values of the function above, for all i. For α_{i} = p^{n−i+1} − p + 1 and b_{i} = 1 we get

v_{n}(γ_{i}^{p^{n−i+2}−p^{2}}η_{i}) ≥ −(p^{n−i+2} − p^{2})p^{n−2}m_{i+1} − p^{n}m_{i+1} + μ_{n} = −p^{2n−i}m_{i+1} + μ_{n}

with an equality iff ν_{i} = m_{i+1}. Hence, if j ≤ n − 1 is the (necessarily unique) integer for which

m_{n} = p^{n−j−1}ν_{j} = max {p^{n−i−1}ν_{i} : 0 ≤ i ≤ n − 1}

then y_{j}^{p(p^{n−j}−1)+1} is congruent mod ℘ (k(C_{n})) to a function with a pole at ∞ of order p^{2n−j}ν_{j} −μ_{n}. If i ≠ j then p^{n−i−1}ν_{i} ⪇ m_{n} and the contribution of y_{i} is strictly smaller.

Finally, consider c_{n}(\underline{y}^{p}, −\underline{y}) − y_{j}^{p(p^{n−j}−1)+1}: repeating the argument above, to show that it is congruent mod℘(k(C_{n})) to a function with a pole of order strictly smaller than p^{2n−j}ν_{j} − μ_{n}, it suffices to show that the term containing the highest power of y_{j}, namely

y_{j}^{p(p^{n−j}−1)}c_{j}(\underline{y}^{p}, −\underline{y})

(cf. Lemma 4) satisfies this property. As above, after a double application of the adjustment lemma on C_{n}, we can write

c_{j}(\underline{y}^{p}, −\underline{y}) = θ_{j}^{p} + ξ_{j}

with v_{n}(θ_{j}^{p}) = p^{n−j}v_{j}(c_{j}) ≥ −p^{n−j}(p^{j+1} − p + 1)m_{j} (cf. Lemma 5) and

v_{n}(ξ_{j}) = −p^{n−j}(p^{j+1}m_{j} − μ_{j}) + μ_{n} − p^{n−j}μ_{j} = −p^{n+1}m_{j} + μ_{n}.

Therefore, mod℘(k(C_{n})) we can get rid of y_{j}^{p(p^{n−j}−1)}θ_{j}^{p} and, by Lemma 5.2,

v_{n}(y_{j}^{p(p^{n−j}−1)}ξ_{j}) = p(p^{n−j} − 1)p^{n−j−1}v_{j+1}(y_{j}) − p^{n+1}m_{j} + μ_{n}
= −(p^{n+1} − p^{j+1})p^{n−j−1}ν_{j} − p^{n+1}m_{j} + μ_{n}
= −p^{2n−j}ν_{j} + p^{j}(ν_{j} − pm_{j}) + μ_{n}

and the assumption on j implies that ν_{j} − pm_{j} is strictly positive.

References

[AC IX] N. Bourbaki, Algèbre Commutative, Chap IX, Masson, Paris (1983). [E.G.A. II] A. Grothendieck et J. Dieudonné, Eléments de Géométrie Algébrique, Chap II,

Publications Mathématiques de l’I.H.E.S. 8 (1961).

[1] J.-L. Brylinski, Théorie du corps de classes de Kato et revêtements abéliens de surfaces, Annales de l’Institut Fourier 33 (1983), 23-38.

[2] H.L. Schmid, Zur Arithmetik der zyklische p-Körper, Journal für die reine und angewandte Mathematik 176 (1936), 161-167.

[3] J.-P. Serre, Groupes algébriques et corps de classes, Hermann, Paris (1961).

[4] J.-P. Serre, Sur les corps locaux à corps résiduel algébriquement clos, Bulletin de la SMF, 89 (1961), 105-154.

[5] J.-P. Serre, Corps locaux, Hermann, Paris (1968).

[6] E. Witt, Zyklische Körper und Algebren der Charakteristik p vom Grad p^{n}, Journal für die reine und angewandte Mathematik 176 (1936), 126-140.

Marco A. Garuti, Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, Via Belzoni 7, 35131, Padova - ITALY

E-mail address: mgaruti@galileo.math.unipd.it