A STRONG PROPERTY OF RAMIFICATION FILTRATION

Victor A. Abrashkin

Under some natural assumptions about the structure of a finite extension K of ℚ_{p} we prove that it can be uniquely recovered from its Galois group with filtration by higher ramification subgroups. Bibl. — 9 issues.

0. Introduction.

Throught all this paper p is a prime number, K is a finite extension of the padic field ℚ_{p}, Γ_{K} = Gal(K̄/K), where K̄ is some fixed algebraic closure of K, k is the residue field and e_{K} is the absolute ramification index of K. We also use permanently the notation ℤ^{+}(p) for the set of natural numbers prime to p.

0.1. The group theoretic structure of the Galois group Γ_{K} depends only on simplest invariants of the field K and was completely described in [J-W]. This description shows that the field K can not be recovered by its Galois group (in the case of global fields the situation is completely different by the Neukirch-Uchida-Iwasawa-Ikeda Theorem, cf. [Po] for this and another results related to Grothendieck’s anabelian philosophy). Consider an additional structure on Γ_{K} given by decreasing filtration of its ramification subgroups in upper numbering {Γ_{K}^{(v)}}_{v⩾−1} (cf. [Se] for the definition). We note that Γ_{K}^{(−1)} = Γ_{K}; for −1 < v ⩽ 0, Γ_{K}^{(v)} = Gal(K̄/K_{ur}); the group Γ^{(0+)} := ⋃_{v>0} Γ_{K}^{(v)} coincides with Gal(K̄/K_{tr}) (here K_{ur} (resp., K_{tr}) is the maximal unramified (resp., tamely ramified) extension of K in K̄), and all rational positive numbers appear as jumps of this filtration.

The following questions appear very naturally:

a) Is the field K uniquely determined by the filtered group {Γ_{K}^{(v)}}_{v⩾−1}?

b) Choose an algebraic closure ℚ̄_{p} and consider two finite extensions K_{1} and K_{2} in ℚ̄_{p}. Does any compatible with ramification filtrations isomorphism from Γ_{K_{1}} to Γ_{K_{2}} comes from an inner automorphism of Γ_{ℚ_{p}}?

c) Let K = k((t)), where k is a finite field. Does any compatible with ramification filtration automorphism of the Galois group G_{K} = Gal(K_{sep}/K) comes from conjugation by an automorphism of K_{sep}?

0.2. The above questions seem to be very difficult to answer, because our knowledge of ramification filtration is far from to be complete. We are going to apply

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The author had a nice opportunity to work on this subject during his stay at the Sydney University (the end of 1997) and at the MPI, Bonn (the spring of 1998), and expresses his gratitude to these organisations for hospitality. He is also very grateful to Dr. K.F.Lai (Sydney University)

for helpful and stimulating discussions.

our recent results about this filtration from [Ab1,2] to study the following slightly modified question.

What properties of the field K can be recovered from the structure of the filtered group {Γ_{K}^{(v)}/Γ_{K}^{(a)}}_{−1⩽v⩽a}, where a > 0 is some fixed rational number?

In the paper [De] it was shown that the group Γ_{K}/Γ_{K}^{(a)} together with its ramification filtration depends roughly speaking only on the structure of the artinian algebra O_{K} mod π_{K}^{a+1}, where O_{K} is the valuation ring of K and π_{K} is its uniformising element.

So, the field K can not be reconstructed completely in the general case by the knowledge only of some segment of the ramification filtration. Nevertheless, one can use the structure of O_{K} mod π_{K}^{a+1} to recover the field K under some natural additional assumptions. Namely, let E be a complete subfield of K. Define the rational positive number

v(K/E) = max{v | Γ_{E}^{(v)} acts nontrivially on K}.

We note that such number exists because of the left-continuity property of ramification filtration. If we assume the field E to be given, then Krasner’s lemma implies that the extension K/E is uniquely determined by the O_{E}-algebra O_{K} mod π_{E}^{v(K/E)+1+ε}, where ε is any positive number (for any b ∈ ℝ, the ideal (π_{E}^{b}) is the ideal of elements α ∈ K having the valuation v(α) ⩾ bv(π_{E})). We apply this argument below to recover uniquely the field K as an extension of the cyclotomic field ℚ_{p}(ζ_{M}), where ζ_{M} is a primitive root of unity of degree p^{M}, M ∈ ℕ, in the case when this extension is not “very wildly” ramified.

0.3. In fact, the main result of this paper deals with more simple filtered group {Γ_{K}(p)^{(v)}/Γ_{K}(p)^{(a)}}_{−1⩽v⩽a}, where Γ_{K}(p) is the Galois group of the maximal pextension K(p) of K in K̄. The following properties show, nevertheless, that this quotient still has sufficiently rich structure:

a) the group Γ_{K}(p)/Γ_{K}(p)^{(a)} has finitely many generators but infinitely many relations, cf. [Go] (also [Ma]);

b) any finitely generated, closed but not open subgroup of Γ_{K}(p)/Γ_{K}(p)^{(a)} is a free pro-p-group, cf. [Ab3]. Roughly speaking, this group does not have “simple” relations, i.e. any relation can not be expressed in terms of any proper subset of any minimal system of generators. In particular, it does not have torsion and nontrivially commuting elements.

0.4. In order to formulate our main result, assume that p ⩾ 5, a primitive root ζ_{M} of degree p^{M}, where M ∈ ℕ, belongs to K, and the degree f_{K} = [k : 𝔽_{p}] of the residue field k of K is sufficiently large, or more precisely, f_{K} ⩾ 2 log_{p}(p^{2}v_{K}), where v_{K} = p^{M+1}e_{K}/(p − 1).

Theorem. Under the above assumptions let a_{K} = e_{K} (p/(p − 1) + M) and assume that for a finite extension K′ of ℚ_{p} in K̄ there exists a compatible with ramification filtration group isomorphism

G : Γ_{K}(p)/Γ_{K}(p)^{(a_{K})} ⟶ Γ_{K′}(p)/Γ_{K′}(p)^{(a_{K})}.

Then ζ_{M} ∈ K′, e_{K′} = e_{K} and for some ϕ ∈ Gal(ℚ_{p}(ζ_{M})/ℚ_{p}), there exists ϕ-linear isomorphism of ℤ_{p}[ζ_{M}]-algebras

O_{K} mod p^{1/(p−1)} ≃ O_{K′} mod p^{1/(p−1)}.

The existence of an isomorphism of the above two ℤ_{p}[ζ_{M}]-algebras is equivalent to the fact, that truncated decompositions of primitive roots of unity ζ_{M} ∈ K and ϕ(ζ_{M}) ∈ K′ in powers of some uniformizing elements π ∈ K and π′ ∈ K′ modulo p^{1/(p−1)} have “the same ” coefficients under some isomorphism of residue fields.

This result can be considered as an evidence to believe that the answer to the question a) of n.0.1 should be positive. It becomes clear from the proof of the theorem that the condition for an automorphism of the group Γ_{K} (and the group G_{K} in the finite characteristics case) to be compatible with ramification filtration is very restrictive. This can be considered as an argument for expecting the positive answer to the questions b) and c) of n.0.1.

0.5. As we already have mentioned in n.0.2, the field K can be recovered from the filtered group {Γ_{K}(p)^{(v)}/Γ_{K}(p)^{(a_{K})}}_{−1⩽v⩽a_{K}} under some additional assumptions. We apply this argument for the following statement.

Corollary. If v(K′/ℚ_{p}(ζ_{M})) < p^{M−1} − 1, then such field K′ is unique up to isomorphism. In particular, if v(K/ℚ_{p}(ζ_{M})) < p^{M−1} − 1, then K ≃ K′.

This follows immediately from above mentioned Krasner’s lemma (cf. n.0.2). Indeed, we can take E = ℚ_{p}(ζ_{M}), π_{E} = ζ_{M} − 1. Then it is sufficient to note that for some ε > 0, there is the following equality of ideals in O_{K}

(p^{1/(p−1)}) = (π_{E}^{p^{M−1}}) = (π^{v(K/ℚ_{p}(ζ_{M}))+1+ε}) .

Denote by A_{M} the set of finite extensions K of ℚ_{p}(ζ_{M}) in K̄ such that v(K/ℚ_{p}(ζ_{M})) < p^{M−1}−1. Then elementary calculations with Herbrandt’s function give that K ∈ A_{M} iff K = K_{1}ℚ_{p}(ζ_{M}), where [K_{1} : ℚ_{p}] < ∞ and v(K_{1}/ℚ_{p}) < M−1.

This alows us to reformulate the main result of this paper in the following form.

Corollary. If p ⩾ 5 and K ∈ A_{M} has sufficiently large residue field, then it can be uniquely recovered from the filtered group {Γ_{K}(p)/Γ_{K}(p)^{(a_{K})}}_{−1⩽v⩽a_{K}}.

0.6. We outline briefly the basic ideas and constructions used in the proof of our theorem. In n.4 we construct the fundamental exact sequence to express the group Γ_{K}(p)/Γ_{K}(p)^{(a_{K})} as an extension of the cyclic group Gal(K(π_{K}^{1/p^{M}})/K) ≃ ℤ/p^{M}ℤ by the quotient G(p)/G(p)^{(v_{K})}, where G(p) is the Galois group of the maximal pextension of a complete discrete valuation field K = k((t)) of characteristic p. This construction is a suitable application of the Fontaine-Wintenberger field-of-norms functor, [Wi].

In Section 5 we prove that the isomorphism G relates fundamental exact sequences for the fields K and K′. This gives automorphism of the group G(p)/G(p)^{(v_{K})}, which is compatible with ramification filtration and relates actions of ℤ/p^{M}ℤ by conjugation on this group coming from the fields K and K′.

In Section 3 we use our general result about the structure of ramification filtration modulo commutators of order ⩾ p in the characteristic p case, [Ab1,2]. This description provides us with generators of ramification subgroups explicitly given in terms of so-called special basises of the W -module G(p)^{ab} ⊗ W (W is the Witt vectors ring with coefficients in k). In fact, we apply this description to a considerably more simple group Ḡ/Ḡ^{(v_{K})}, where Ḡ = G(p)/G(p)^{p^{3}}C_{3}(G(p)) and C_{3}(G(p)) is the subgroup of commutators of order ⩾ 3, and use only the “modulo t^{v_{K}} ”-part of the special basis. The main lemma of n.3 states that any compatible with ramification filtration automorphism of the group Ḡ “almost ” preserves the “modulo t^{v_{K}/p}”-part of special basises.

On the other hand, the actions of ℤ/p^{M}ℤ on the quotient (Ḡ/Ḡ^{p})^{ab} of the group Ḡ coming from the fundamental exact sequence related to the field K (resp., K′) can be expressed in terms of expansions of some roots of degree p^{M} from unity in power series in uniformiser π_{K} (resp., π_{K′}) by the use of special basises. This alows us to relate in Section 5 these expansions modulo p^{1/(p−1)}, what gives finally the proof of our theorem.

1. Preliminaries: special basises.

In this section K is a complete discrete valuation field of characteristic p with finite residue field k ≃ 𝔽_{p^{N_{0}}} . We assume that a separable closure K_{sep} of K is chosen and set G = Gal(K_{sep}/K).

1.1. Consider the p-completion K̂^{∗} = \underleftarrow{lim} K^{∗}/K^{∗p^{n}} of the multiplicative group K^{∗} with its natural ℤ_{p}-algebra structure. Fix a uniformising element t of the field K and consider the ℤ_{p}-algebra isomorphism

χ_{t} : K̂^{∗} ⟶ ℤ_{p} × ∏_{a∈ℤ^{+}(p)} W_{a},

where for all a ∈ ℤ^{+}(p), W_{a} = W is the ℤ_{p}-algebra of Witt vectors with coefficients in k. The isomorphism χ_{t} is induced by the decomposition of any element α ∈ K^{∗} in the form

α = t^{a_{0}(α)} ∏_{(a,p)=1} E(w_{a}(α), t^{a}),

where a_{0}(α) ∈ ℤ, w_{a}(α) ∈ W for all (a, p) = 1, and for any w ∈ W,

E(w, X) = exp (wX + ⋯ + σ^{n}(w)X^{p^{n}}/p^{n} + …) ∈ W [[X]]

(as usually, σ is the Frobenius morphism of Witt vectors).

Consider extension of scalars of χ_{t}

χ_{t} ⊗ W : K̂^{∗} ⊗ W ⟶ W_{0} × ∏_{\begin{matrix}(a,p)=1 \\ n∈ℤ mod N_{0} \end{matrix}} W_{an}.

Here W_{0} = W and the W -algebras W_{an} consist of elements of W with the W - algebra structure coming from the morphism σ^{−n} : W ⟶ W. The action of σ on scalars gives σ-linear automorphism of K̂^{∗} ⊗ W. We shall denote this morphism also by σ. We note that (K̂^{∗} ⊗ W)|_{σ=id} = K̂^{∗}.

The algebras W_{0} and W_{an} have standard generators D_{0} and D_{an} coming from 1 ∈ W. The system

D(t) = {D_{an} | (a, p) = 1, n ∈ ℤ mod N_{0}} ∪ {D_{0}}

gives a topological basis of the W -algebra K̂^{∗} ⊗ W. This basis depends only on the choice of uniformiser in K. We note that σ acts on its elements D_{an} by the shift σ : D_{an} ↦ D_{a,n+1}, and σ(D_{0}) = D_{0}, i.e. D_{0} ∈ K̂^{∗}. On the other hand, if fr_{t} is an automorphism of the field K such that fr_{t}(t) = t and fr_{t} |_{k} = σ, then fr_{t}(D_{an}) = D_{a,n−1} for all a ∈ ℤ^{+}(p) and n ∈ ℤ mod N_{0}.

If we take elements from D(t) modulo p, we obtain a basis of the k-module K^{∗}/K^{∗p} ⊗ k. The basises D(t) and D(t) mod p will be called special basises.

1.2. Consider the quotient M = G/G^{p}C_{2}(G) as a profinite 𝔽_{p}-module, and introduce 𝔽_{p}-modules M_{k} = M ˆ⊗k, M_{K} = M ˆ⊗K, and M_{sep} = M ˆ⊗K_{sep}. All these modules have the natural action of the Frobenius morphism coming from extension of scalars. We shall denote this action also by σ. The module M_{sep} has also natural action of the Galois group G.

The Artin-Schreier theory gives the identification M = Hom (K/(σ − id)K, 𝔽_{p}). Its equivalent version can be given as follows. If ẽ ∈ M ⊗ K/(σ − id)K, take its lifting e ∈ M_{K} with respect to the natural projection to M ⊗ K/(σ − id)K, and take F ∈ M_{sep} such that σF − F = e. Then for any τ ∈ G, the correspondence

τ ↦ τF − F ∈ M_{sep}|_{σ=id} = M

gives an element of Hom(G, M). So, we have the map

ψ_{1} : M ⊗ K/(σ − id)K ⟶ Hom(G, M)

and this map is isomorphism.

We note that

M ⊗ K/(σ − id)K = End_{𝔽_{p}}(K/(σ − id)K)

and if ē ∈ M ⊗ K/(σ − id)K corresponds to the identity endomorphism of K/(σ − id)K, then ψ_{1}(ē) is nothing but the natural projection π : G ⟶ M.

1.3. Take an uniformiser t of K and an element α_{0} ∈ k such that Tr_{k/𝔽_{p}} α_{0} = 1. Then the natural projection from K to K/(σ − id)K has the unique section with values in the 𝔽_{p}-submodule 𝔽_{p}α_{0} ⊕ (⊕_{a∈ℤ^{+}(p)}t^{−a}k) of K. We use this section to obtain the lifting e_{α_{0},t} ∈ M_{K} of the element ē ∈ M ⊗ K/(σ − id)K defined in above n.1.2.

Lemma. Under the class field theory identification M = K^{∗}/K^{∗p}, we have

e_{α_{0},t_{0}} = α_{0}D_{0} + ∑_{a∈ℤ^{+}(p)} t^{−a}D_{a0},

where D_{0} and D_{a0} for all a ∈ ℤ^{+}(p), are elements from the special basis D(t) mod p, cf. n.1.1.

The proof is an easy application of the Artin-Schreier explicit reciprocity law.

1.4. It can be verified easily that the condition Tr_{k/𝔽_{p}} α_{0} = 1 implies that D_{0} induces the Frobenius automorphism on the maximal unramified extension K_{ur}^{(1)}/K inside K^{(1)} = K_{sep}^{G^{p}C_{2}(G)}. Set M^{(1)} = Gal(K^{(1)}/K_{ur}^{(1)}) ⊂ M = Gal(K^{(1)}/K). Then M^{(1)} is the inertia subgroup of M, and the elements of the set D^{(1)}(t) mod p, where

D^{(1)}(t) := {D_{an} | (a, p) = 1, n ∈ ℤ mod N_{0}} = D(t) \ {D_{0}},

give a special basis of the k-submodule M_{k}^{(1)} := M^{(1)} ⊗ k of M_{k}. We agree to use also the notation D_{an} for elements of the set D^{(1)}(t) mod p.

If t′ is another uniformiser of the field K, then the elements of

D^{(1)}(t′) mod p = {D′_{an} | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}},

give another special k-basis of M_{k}^{(1)}. Therefore, for a ∈ ℤ^{+}(p) we have

D′_{a0} = ∑ α_{abm}D_{bm},

where the sum is taken for b ∈ ℤ^{+}(p) and m ∈ ℤ mod N_{0}, and all coefficients α_{abm} belong to k. It can be immediately seen from definitions that α_{abm} = 0 if b < a. We also note that , if a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}, and

D′_{an} = ∑_{\begin{matrix}(b,p)=1 \\ m∈ℤ mod N_{0} \end{matrix}} α_{anbm}D_{bm}

with all α_{anbm} ∈ k, then α_{anbm} = σ^{n}(α_{a,b,m−n}). In particular, α_{anbm} = 0 for b < a.

1.5. We need more precise information concerning the relation between elements of two different special basises of M_{k}^{(1)}. Lemma of n.1.3 and the formalism of Artin-Schreier theory from n.1.2 imply the following property.

Lemma. If t and t′ are uniformising elements of K, then the elements of special basises D^{(1)}(t) mod p and D^{(1)}(t′) mod p are related by the equality

∑_{a∈ℤ^{+}(p)} t^{−a}D_{a0} ≡ ∑_{a∈ℤ^{+}(p)} t′^{−a}D′_{a0} mod(σ − id)M_{K} + M ⊗ O_{K}.

For a ∈ ℕ, denote by M^{(a)} the submodule of M, which consists of elements coming from principal units congruent to unity modulo a^{th} power of the maximal ideal of the valuation ring O_{K} of the field K. Set M_{k}^{(a)} = M^{(a)} ⊗ k, and apply the above lemma to prove the following proposition.

Proposition. For any a ∈ ℤ^{+}(p), we have:

a) if t′ ≡ αt mod t^{2}, where α ∈ k, then

D′_{a0} ≡ α^{a}D_{a0} mod M_{k}^{(a+1)};

b) let t′ ≡ t + αt^{b+1} mod t^{b+2}, where b ⩾ 1 and α ∈ k; then for a ≢ −b mod p,

D′_{a0} ≡ D_{a0} + (a + b)αD_{a+b,0} mod M_{k}^{(a+b+1)};

and for a ≡ −b mod p,

D′_{a0} ≡ D_{a0} mod M_{k}^{(a+b+1)}.

Proof. From the congruence t′ ≡ αt mod t^{2} it follows that for any a ∈ ℤ^{+}(p) and some coefficients β_{ab} ∈ k defined for 1 ⩽ b < a, we have

t^{′−a} ≡ α^{−a}t^{−a} + ∑_{1⩽b<a} β_{ab}t^{−b} mod O_{K}.

Then

∑_{a∈ℤ^{+}(p)} t^{′−a}D′_{a0} ≡ ∑_{a∈ℤ^{+}(p)} α^{−a}t^{−a}D′_{a0} + ∑_{\begin{matrix}a∈ℤ^{+}(p) \\ 1⩽b<a \end{matrix}} β_{ab}t^{−b}D′_{a0} ≡

∑_{a∈ℤ^{+}(p)} t^{−a} (α^{−a}D′_{a0} + ∑ σ^{−s}(β_{a′,ap^{s}})D′_{a′,−s}) mod(σ − id)M_{K} + M ⊗ O_{K},

where the last sum is taken for a′ ∈ ℤ^{+}(p) and s ∈ ℤ_{⩾0}, such that a′ > ap^{s}. This gives D_{a0} ≡ α^{−a}D′_{a0} mod M_{k}^{(a+1)}. The part a) is proved.

In order to prove the part b) we note that for some β_{ac} ∈ k, where a ∈ ℤ^{+}(p) and 1 ⩽ c < a − b, we have

t^{′−a} ≡ t^{−a} − aαt^{−a+b} + ∑_{1⩽c<a−b} β_{ac}t^{−c} mod O_{K},

and the proof can be finished similarly as in the previous case.

1.6. Consider a k-basis {C_{an} | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}} of M_{k}^{(1)} satisfying for all a ∈ ℤ^{+}(p) and n ∈ ℤ mod N_{0} the following conditions:

a) σC_{an} = C_{a,n+1};

b) the elements of the set {C_{bm} | b ∈ ℤ^{+}(p), b ⩾ a, m ∈ ℤ mod N_{0}} generate the k-module M_{k}^{(a)}.

Proposition. If the system of elements {C_{an} ∈ M_{k}^{(1)} | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}} satisfies the above conditions a) and b), then there exists n_{0} ∈ ℤ mod N_{0} and a special basis {D_{an} | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}} such that if

σ^{n_{0}}C_{a0} = ∑_{\begin{matrix}b∈ℤ^{+}(p) \\ m∈ℤ mod N_{0} \end{matrix}} α_{abm}D_{bm},

where all coefficients α_{abm} belong to k, then

1) α_{110} = 1;

2) for all b ∈ ℤ^{+}(p) and b > 1, we have α_{1b0} = 0;

3) for all b ∈ ℕ such that b ≡ 1 mod p, we have α_{2b0} = 0.

Proof. Choose a uniformiser t of the field K, consider corresponding special basis

D^{(1)}(t) mod p = {D_{an} | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}},

and set

C_{10} = ∑_{b,m} α_{1bm}D_{bm}, C_{20} = ∑_{b,m} α_{2bm}D_{bm}.

By changing, if necessary, the system C_{an} to the system σ^{n_{0}}C_{an} for some n_{0} ∈ ℤ mod N_{0}, we can assume that α_{110} ≠ 0.

Now we reformulate our proposition in the following form

there exists a special basis D^{(1)}(t) mod p such that α_{110} = 1 and for all natural numbers n > 1, we have α_{1n0} = 0 if n ∈ ℤ^{+}(p), and α_{2,n+1,0} = 0 if n ≡ 0 mod p.

We prove it by induction on n.

If n = 1, then we apply the part a) of Proposition n.1.5 to make a suitable change of the uniformiser t modulo t^{2} to obtain the coefficient α_{110} = 1 in the new special basis.

Now we assume that if n ⩾ 2, then α_{1b0} = 0 for all 1 < b < n, b ∈ ℤ^{+}(p), and α_{2b0} = 0 for all b ≡ 1 mod p, b < n + 1.

If n ≢ 0 mod p, then by the part b) of the proposition of n.1.5, we can chose a new uniformiser t′ = t + αt^{n} for a suitable α ∈ k, to annihilate α_{1n0}. We note that this procedure does not disturb the coefficients α_{2b0} for all 2 ⩽ b ⩽ n.

If n ≡ 0 mod p, then we apply again the above proposition for a suitable uniformiser t′ = t + αt^{n}, α ∈ k, to annihilate α_{2,n+1,0}. Again this operation does not disturb the coefficients α_{1b0} and α_{2b0} for 1 ⩽ b ⩽ n.

1.7. Consider the group Aut K of all continuous automorphisms of the field K. If ϕ ∈ Aut K, consider ϕ̂ ∈ Aut K_{sep} such that ϕ̂|_{K} = ϕ and for τ ∈ G, set ϕ ∗ τ = ϕ̂^{−1}τϕ̂. This induces some action of ϕ̂ on M, which does not depend on the choice of lifting ϕ̂ of ϕ. So, we can set ϕ̂ ∗ m = ϕ ∗ m for any m ∈ M. We use the same symbol for extension of scalars of this action to M_{k}.

Denote by Aut^{1} K the sugroup of Aut(K) consisting of automorphisms inducing identity on the residue field k of K. Take uniformising element t of the field K, and consider corresponding special basis of M_{k}^{(1)}

D^{(1)}(t) mod p = {D_{an} | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}}.

Proposition 1. If ϕ ∈ Aut^{1}(K), then

∑_{a∈ℤ^{+}(p)} ϕ^{−1}(t)^{−a}D_{a0} ≡ ∑_{a∈ℤ^{+}(p)} t^{−a}(ϕ ∗ D_{a0}) mod(σ − id)M_{K} + M ⊗ O_{K}.

Proof. Take F ∈ M_{sep} such that

σF − F = ∑_{a∈ℤ^{+}(p)} t^{−a}D_{a0} + α_{0}D_{0},

then τF − F = π(τ), where τ ∈ G and π : G ⟶ M is the natural projection. Take a lifting ϕ̂ ∈ Aut^{1} K_{sep} and set F_{1} = ϕ̂^{−1}(F). Then

σF_{1} − F_{1} = ∑_{a∈ℤ^{+}(p)} ϕ^{−1}(t)^{−a}D_{a0} + α_{0}D_{0}.

If τ ∈ G, then

τF_{1} − F_{1} = ϕ̂^{−1} ((ϕ̂ ∗ τ)F − F ) = (ϕ̂ ∗ τ)F − F = π(ϕ̂ ∗ τ) = ϕ ∗ π(τ).

But the homomorphism τ ↦ ϕ ∗ π(τ) can be obtained by the correspondence τ ↦ τF_{2} − F_{2}, where

σF_{2} − F_{2} = ∑_{a∈ℤ^{+}(p)} t^{−a}(ϕ ∗ D_{a0}) + α_{0}(ϕ ∗ D_{0}).

Therefore,

∑_{a∈ℤ^{+}(p)} t^{−a}(ϕ ∗ D_{a0}) + α_{0}(ϕ ∗ D_{0}) ≡ ∑_{a∈ℤ^{+}(p)} ϕ^{−1}(t)^{−a}D_{a0} + α_{0}D_{0} mod(σ − id)M_{K}.

and the proposition is proved.

Consider fr_{t} ∈ Aut K such that fr_{t}(t) = t and fr_{t} |_{k} = σ.

Proposition 2. For all a ∈ ℤ^{+}(p) and n ∈ ℤ mod N_{0}, we have

(fr_{t}) ∗ D_{an} = D_{a,n−1}.

Proof. Let ϕ̂ be a lifting of fr_{t} to Aut K_{sep}. Set F_{1} = ϕ̂^{−1}(F), then

σF_{1} − F_{1} = ∑_{a∈ℤ^{+}(p)} t^{−a}D_{a,−1} + (σ^{−1}α_{0})D_{0}.

Similarly to the previous proposition we obtain

∑_{a∈ℤ^{+}(p)} t^{−a}D_{a,−1}+(σ^{−1}α_{0})D_{0} ≡ ∑_{a∈ℤ^{+}(p)} t^{−a}(fr_{t} ∗D_{a0})+α_{0}(fr_{t} ∗D_{0}) mod(σ−id)M_{K}.

This gives fr_{t} ∗D_{a0} = D_{a,−1} and the proposition follows from commutativity of fr_{t} and σ.

2. Preliminaries: ramification filtration modulo 3^{rd} commutators.

In this section K = k((t)), where k ≃ 𝔽_{p^{N_{0}}} and t is a fixed uniformiser of K. As earlier, G = Gal(K_{sep}/K), and G(p) is the Galois group of the maximal p-extension of K in K_{sep}. We shall prove some auxiliary facts about structure of the ramification filtration {G(p)^{(v)} mod C_{3}(G(p))}_{1⩽v⩽v_{0}}, where v_{0} is some fixed rational number and C_{3}(G(p)) is the closure of the subgroup of commutators of order ⩾ 3. This description comes as a special case of the general result from [Ab1,2]. We assume that N_{0} ⩾ 2 log_{p}(pv_{0}).

2.1. Consider the profinite ℤ_{p}-module K̂^{∗} from Section 1, and denote by L̃ the profree Lie algebra over ℤ_{p} with this module of generators. Denote by L the quotient of L̃ by the ideal of commutators of order ⩾ p, and consider the group G(L) of its elements with composition law given by the Campbel-Hausdorff formula. Choose α_{0} ∈ W with trace 1 and use the uniformiser t of the field K. The formalism of Artin-Schreier theory from n.1.2 can be generalised, cf. [Ab1], to construct the isomorphism

ψ : G(p)/C_{p}(G(p)) ⟶ G(L)

(its construction depends on the above choice of α_{0} ∈ W and t ∈ K). This isomorphism transfers the ramification filtration of the Galois group G to the filtration {L^{(v)}}_{v⩾−1} of the Lie algebra L by its “ramification ideals” L^{(v)}, v ⩾ −1.

Consider corresponding extensions of scalars L_{W} = L⊗W and L_{W}^{(v)} = L^{(v)} ⊗W. For a given v ⩾ 1, explicit description of the ideal L_{W}^{(v)} can be given as follows.

Consider the W -basis D(t) = {D_{an} | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}}∪{D_{0}} of K̂^{∗}⊗W from n.1.1. Set D_{0n} = (σ^{n}α_{0})D_{0} for all n ∈ ℤ mod N_{0}. It will be convienient to use the notation ℤ^{0}(p) = ℤ^{+}(p) ∪ {0} and for all a ∈ ℤ^{0}(p) and n ∈ ℤ, the notation D_{an} as well as the notation D_{a,n mod N_{0}}. If s ⩾ 1 and (n_{1}, …, n_{s}) ∈ ℤ^{s}, we set by definition

η(n_{1}, …, n_{s}) = \frac{1}{s_{1}!(s_{2} − s_{1})! … (s_{l} − s_{l−1})!},

if for some l ⩾ 1 and 1 ⩽ s_{1} < ⋯ < s_{l} = s, one has n_{1} = ⋯ = n_{s_{1}} > ⋯ > n_{s_{l−1}} = ⋯ = n_{s_{l}}, and we set

η(n_{1}, …, n_{s}) = 0,

otherwise.

If γ ∈ ℚ, γ > 0 and N ∈ ℕ, set

F_{γ,−N} = ∑(−1)^{s}η(n_{1}, …, n_{s})a_{1}p^{n_{1}} [… [D_{a_{1}n_{1}}, D_{a_{2}n_{2}}] , …,] ,

where the sum is taken for all 1 ⩽ s < p, n_{1} ⩾ 0, n_{2}, …, n_{s} ⩾ −N and a_{1}, …, a_{s} ∈ ℤ^{0}(p) such that p^{n_{1}}a_{1} + ⋯ + p^{n_{s}}a_{s} = γ.

Then we define the ideal L_{W}^{(v)}(N) as the minimal σ-invariant ideal of L_{W} which contains all F_{γ,−N} with γ ⩾ v. Finally, in the above notation the description of the ramification filtration modulo commutators of order ⩾ p is given by the following statement, cf. [Ab1,2]

Theorem. There exists N(v) ∈ ℕ such that the ideals L_{W}^{(v)}(N) do not depend on N if N ⩾ N(v), and are equal to the ramification ideal L_{W}^{(v)}.

2.2. Consider the special case of the above general construction modulo the ideal C_{2}(L_{W}) of second commutators of the Lie algebra L_{W}. Note that for any given v ⩾ 1, the ideals L_{W}^{(v)}(N) mod C_{2}(L_{W}) coincide for all N ⩾ 0. So, the ideal L_{W}^{(v)} mod C_{2}(L_{W}) appears as the W -submodule of K̂^{∗}⊗W generated by the elements p^{s}aD_{an}, where a ∈ ℤ^{+}(p), s ∈ ℤ_{⩾0} and p^{s}a ⩾ v. Clearly, this module comes from the subgroup of principal units of the valuation ring O_{K} of K congruent to 1 modulo t^{v} (what is very well-known from class field theory).

2.3. We need to simplify the general construction from n.2.1 to work with ramification filtration modulo commutators of order ⩾ 3.

First, we simplify the notation by setting L = L mod C_{3}(L), L_{W} = L_{W} mod C_{3}(L_{W}), L^{(v)} = L^{(v)} mod C_{3}(L) and L_{W}^{(v)} = L_{W}^{(v)} mod C_{3}(L_{W}) for all v ⩾ 1. Introduce new generators of the Lie algebra L_{W} related more closely to the ramification filtration. Note that the elements F_{γ,−N} mod C_{3}(L_{W}) does not depend on N ∈ ℤ_{⩾0} if γ ∈ ℕ. We use these elements to set

Δ_{a0} = (1/a)F_{a0} mod C_{3}(L_{W})

for all a ∈ ℤ^{+}(p). We set also Δ_{an} := σ^{n}Δ_{a0} and note that

Δ_{an} mod C_{2}(L_{W}) = D_{an} mod C_{2}(L_{W}),

for all a ∈ ℤ^{+}(p) and n ∈ ℤ mod N_{0}. So, if we define Δ_{0} = D_{0}, then we obtain the system of generators of the Lie W -algebra L_{W}

{Δ_{an} | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}} ∪ {Δ_{0}}.

Clearly, the description of the ramification filtration L_{W}^{(v)} mod C_{2}(L_{W}) in new generators coincide with its description in terms of generators D_{an} from n.2.2.

In terms of new generators we have also:

— if γ = ap^{m} with a ∈ ℤ^{+}(p) and m ∈ ℤ_{⩾0}, then

F_{γ0} mod C_{3}(L_{W}) = ap^{m}Δ_{am} − \frac{1}{2} ∑ p^{s}a_{1}[Δ_{a_{1}s}, Δ_{a_{2}s}],

where the sum is taken for all a_{1}, a_{2} ∈ ℤ^{+}(p) and 0 ⩽ s < m under the condition p^{s}(a_{1} + a_{2}) = γ;

— if γ ∉ ℕ, then

F_{γ,−N} mod C_{3}(L_{W}) = − ∑ p^{n_{1}}a_{1}[Δ_{a_{1}n_{1}}, Δ_{a_{2}n_{2}}],

where the sum is taken for all a_{1}, a_{2} ∈ ℤ^{+}(p) and n_{1} ⩾ 0 and n_{2} ⩾ −N under the condition p^{n_{1}}a_{1} + p^{n_{2}}a_{2} = γ.

Take some v ⩾ 1. If γ ⩾ v and γ = ap^{m} ∈ ℕ with a ∈ ℤ^{+}(p) and m ∈ ℤ_{⩾0}, set

F_{γ}^{(v)} = ap^{m}Δ_{am} − \frac{1}{2} ∑ p^{s}a_{1}[Δ_{a_{1}s}, Δ_{a_{2}s}],

where the sum is taken for a_{1}, a_{2} ∈ ℤ^{+}(p) and 0 ⩽ s < m such that p^{s}a_{1}, p^{s}a_{2} < v and p^{s}(a_{1} + a_{2}) = γ.

If γ ∉ ℕ and γ ⩾ v, set

F_{γ}^{(v)} = − ∑ p^{n_{1}}a_{1}[Δ_{a_{1}n_{1}}, Δ_{a_{2}n_{2}}],

where the sum is taken for a_{1}, a_{2} ∈ ℤ^{+}(p) and n_{1} ⩾ 0 such that p^{n_{1}}a_{1}, p^{n_{1}}a_{2} < v and p^{n_{1}}a_{1} + p^{n_{2}}a_{2} = γ.

Proposition. In the above notation for any v ⩾ 1, L_{W}^{(v)} is the minimal σ-invariant ideal of L_{W} containing the elements F_{γ}^{(v)} for all γ ⩾ v.

Proof. Let n^{∗}(v) be the maximal integer < log_{p} (v/{v]), where {v] denotes the usual fractional part of v if v ∉ ℤ and is equal to 1, if v ∈ ℤ. Then we have the following property:

if γ = p^{n_{1}}a_{1} + p^{n_{2}}a_{2} ⩾ v with p^{n_{1}}a_{1}, p^{n_{1}}a_{2} < v, then n_{2} ⩾ n_{1} − n^{∗}(v).

Indeed, this follows from the inequality

p^{n_{1}−n_{2}} = \frac{p^{n_{1}}a_{2}}{γ − p^{n_{1}}a_{1}} < \frac{v}{\{v]}.

Therefore, for any γ ⩾ v and N ⩾ n^{∗}(v), the above element F_{γ}^{(v)} differs from the element F_{γ,−N} mod C_{3}(L_{W}) by an element from the ideal [L_{W}^{(v)}, L_{W}]. The proposition is proved.

2.4. Denote by L_{triv}^{(v)} the W -submodule of the W -module L_{W} containing the elements p^{s}Δ_{an} such that p^{s}a ⩾ pv, and the elements p^{s}[Δ_{a_{1}n_{1}}, Δ_{a_{2}n_{2}}], such that max{p^{s}a_{1}, p^{s}a_{2}} ⩾ v (here a, a_{1}, a_{2} ∈ ℤ^{+}(p), n, n_{1}, n_{2} ∈ ℤ mod N_{0} and s ∈ ℤ_{⩾0}). It is easy to see that L_{triv}^{(v)} is generated by elements σ^{n}F_{γ}^{(v)} with γ ⩾ pv and elements from [L_{W}^{(v)}, L_{W}]. Therefore, L_{triv}^{(v)} ⊂ L_{W}^{(v)}.

Let

S_{1} = {(a, n) | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}},
S_{2} = {(a_{1}, n_{1}, a_{2}, n_{2}) | a_{1}, a_{2} ∈ ℤ^{+}(p), n_{1}, n_{2} ∈ ℤ mod N_{0}}.

If (a, n) ∈ S_{1}, set s = s(a) for the integer s such that v ⩽ p^{s}a < pv, and set

γ(a, n) = p^{s(a)}a.

Denote by S_{1,adm}^{(v)} the subset of elements (a, n) ∈ S_{1} such that s(a) ∈ ℤ_{⩾0}.

If (a_{1}, n_{1}, a_{2}, n_{2}) ∈ S_{2}, set s = s(a_{1}, a_{2}) for the maximal integer s such that max{p^{s}a_{1}, p^{s}a_{2}} < v. If n_{12}^{∗} ∈ ℤ is such that n_{12}^{∗} mod N_{0} = n_{1} − n_{2} and 0 ⩽ n_{12}^{∗} < N_{0}, we set

γ(a_{1}, n_{1}, a_{2}, n_{2}) = p^{s(a_{1},a_{2})} (a_{1} + \frac{a_{2}}{p^{n_{12}^{∗}}} ) .

Denote by S_{2,adm}^{(v)} the subset of (a_{1}, n_{1}, a_{2}, n_{2}) ∈ S_{2} such that s(a_{1}, a_{2}) ∈ ℤ_{⩾0}, γ(a_{1}, n_{1}, a_{2}, n_{2}) ⩾ v, and if n_{1} = n_{2} then a_{1} > a_{2} and a_{1} + a_{2} ≡ 0 mod p.

Denote by L_{adm}^{(v)} the submodule of L_{W} generated by elements from L_{triv}^{(v)} and by elements p^{s(a)}Δ_{an} and p^{s(a_{1},a_{2})}[Δ_{a_{1}n_{1}}, Δ_{a_{2}n_{2}}], where (a, n) ∈ S_{1,adm}^{(v)} and

(a_{1}, n_{1}, a_{2}, n_{2}) ∈ S_{2,adm}^{(v)}.

Proposition 1. L_{adm}^{(v)} ⊃ L_{W}^{(v)} ⊃ L_{triv}^{(v)} ⊃ pL_{adm}^{(v)}.

Proof. The inclusion pL_{adm}^{(v)} ⊂ L_{triv}^{(v)} follows directly from definitions. The inclusion L_{triv}^{(v)} ⊂ L_{W}^{(v)} was explained in the beginning of this subsection.

In order to prove that L_{W}^{(v)} ⊂ L_{adm}^{(v)}, it is sufficient to verify that for any γ ⩾ v, we have F_{γ}^{(v)} ∈ L_{adm}^{(v)}.

This follows directly from the definition of S_{2,adm}^{(v)} in the case γ ∉ ℕ.

If γ = ap^{m} ∈ ℕ with a ∈ ℤ^{+}(p) and m ∈ ℤ_{⩾0}, we note that

F_{γ}^{(v)} ≡ ap^{m}Δ_{am} − ∑ p^{s}a_{1}[Δ_{a_{1}s}, Δ_{a_{2}s}] mod L_{triv}^{(v)},

where the sum is taken for all a_{1}, a_{2} ∈ ℤ^{+}(p) and 0 ⩽ s < m, such that a_{1} > a_{2} and p^{s}(a_{1} + a_{2}) = ap^{m} (one has a_{1} + a_{2} ≡ 0 mod p because s < m). The proposition is proved.

Proposition 2. Let 1 ⩽ v ⩽ v_{0} and pv ∈ ℕ. Then images of elements from the sets

{p^{s(a)}Δ_{an} | (a, n) ∈ S_{1,adm}^{(v)}}

and

{p^{s(a_{1},a_{2})}[Δ_{a_{1}n_{1}}, Δ_{a_{2}n_{2}}] | (a_{1}, n_{1}, a_{2}, n_{2}) ∈ S_{2,adm}^{(v)}}

in the k-module L_{adm}^{(v)}/L_{triv}^{(v)} are linearly independent (and therefore, give its kbasis).

Proof. It is sufficient to verify the following implication

(a_{1}, n_{1}, a_{2}, n_{2}) ∈ S_{2,adm}^{(v)} ⇒ (a_{2}, n_{2}, a_{1}, n_{1}) ∉ S_{2,adm}^{(v)}.

Indeed, if n_{1} = n_{2}, then a_{1} > a_{2} by definition, and therefore, in this case (a_{2}, n_{2}, a_{1}, n_{1}) ∉ S_{2,adm}^{(v)}.

Assume that n_{1} ≠ n_{2}. If s = s(a_{1}, a_{2}), then

p^{s} (a_{1} + \frac{a_{2}}{p^{n_{12}^{∗}}} ) ⩾ v,

because (a_{1}, n_{1}, a_{2}, n_{2}) ∈ S_{2,adm}^{(v)}. Because pv ∈ ℕ and p^{s}a_{1} < v ⩽ v_{0}, we have v − p^{s}a_{1} ⩾ 1/p, and therefore, p^{n_{12}^{∗}} ⩽ p^{s}a_{2}(1/p)^{−1} < pv_{0}, i.e. n_{12}^{∗} < log_{p}(pv_{0}).

Similarly, if we assume that (a_{2}, n_{2}, a_{1}, n_{1}) ∈ S_{2,adm}, then n_{21}^{∗} < log_{p}(pv_{0}). But this gives the contradiction

2 log_{p}(pv_{0}) ⩽ N_{0} = n_{12}^{∗} + n_{21}^{∗} < 2 log_{p}(pv_{0}).

The proposition is proved.

Consider an arbitrary element l ∈ L_{adm}^{(v)} mod L_{triv}^{(v)}. Then it can be presented in the following form

∑_{(a,n)∈S_{1,adm}^{(v)}} α_{an}p^{s(a)}Δ_{an} + ∑_{(a_{1},n_{1},a_{2},n_{2})∈S_{2,adm}^{(v)}} α_{a_{1}n_{1}a_{2}n_{2}}p^{s(a_{1},a_{2})}[Δ_{a_{1}n_{1}}, Δ_{a_{2}n_{2}}]

with some coefficients α_{an} and α_{a_{1}n_{1}a_{2}n_{2}} from the residue field k.

Introduce the following subset of rational numbers

Γ_{adm}(v) = {γ(a, n) | (a, n) ∈ S_{1,adm}^{(v)}}∪{γ(a_{1}, n_{1}, a_{2}, n_{2}) | (a_{1}, n_{1}, a_{2}, n_{2}) ∈ S_{2,adm}^{(v)}}.

Clearly, the elements from the set

S(v) = {σ^{n}F_{γ}^{(v)} | γ ∈ Γ_{adm}(v), n ∈ ℤ mod N_{0}}

generate the k-module L_{W}^{(v)}. But expressions of any two different elements of this set in terms of the k-basis from above Proposition 2 do not have common terms (with non-zero coefficients). Therefore, elements of the set S(v) give a k-basis of L_{W}^{(v)} modulo L_{triv}^{(v)}. We state this property as the following proposition.

Proposition 3. In the above notation the element l ∈ L_{adm}^{(v)} mod L_{triv}^{(v)} comes from the ramification ideal L_{W}^{(v)} if and only if for all γ ∈ Γ_{adm}(v) and m ∈ ℤ mod N_{0}, there exist A_{γ,m} ∈ k, such that

1) if (a, n) ∈ S_{1,adm}^{(v)}, γ = γ(a, n) and n = s(a) + m, then a^{−1}α_{an} = A_{γ,m};

2) if (a_{1}, n_{1}, a_{2}, n_{2}) ∈ S_{2,adm}^{(v)}, γ = γ(a_{1}, n_{1}, a_{2}, n_{2}) and n_{1} = s(a_{1}, a_{2}) + m, then a_{1}^{−1}α_{a_{1},n_{1},a_{2},n_{2}} = −A_{γ,m}.

3. The main lemma.

In this section K is a complete discrete valuation field of characteristic p with finite residue field k ≃ 𝔽_{p^{N_{0}}} , G = Gal(K_{sep}/K) and {G^{(v)}}_{v⩾−1} — the ramification filtration of G in upper numbering. We also use all notation introduced in Sections 1 and 2.

3.1. Assume that p ⩾ 5 and v_{0} is a natural number such that v_{0} ≡ 0 mod p^{2}. We assume that N_{0} ⩾ 2 log_{p}(p^{2}v_{0}). We note that N_{0} certainly satisfies the assumption N_{0} ⩾ 2 log_{p}(pv_{0}) from Section 2.

Consider the quotient

Ḡ = G/G^{p^{3}}C_{3}(G)

with filtration {Ḡ^{(v)}}_{v⩾−1} induced by the ramification filtration of the group G. As in Section 1 we set M = G/G^{p}C_{2}(G). Then the ramification filtration of M coincides, in fact, with its filtration by submodules M^{(a)}, a ∈ ℕ, cf. n.1.5.

Assume F̄ : Ḡ ⟶ Ḡ is a group automorphism such that for any rational number v such that 1 ⩽ v ⩽ v_{0}, we have F̄(Ḡ^{(v)}) = Ḡ^{(v)}. We note, the existence of such morphism F̄ is equivalent to the existence of a compatible with ramification filtration group automorphism of the quotient Ḡ/Ḡ^{(v_{0})}.

Consider extension of scalars f_{k} : M_{k} ⟶ M_{k} of the morphism f : M ⟶ M induced by F̄. Clearly, for any 1 ⩽ a ⩽ v_{0}, we have f_{k}(M_{k}^{(a)}) = M_{k}^{(a)}. Take uniformising element t ∈ K and consider corresponding special basis of M_{k}^{(1)}

D^{(1)}(t) mod p = {D_{an} | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}}.

Set

f_{k}(D_{a0}) = ∑_{\begin{matrix}b∈ℤ^{+}(p) \\ m∈ℤ mod N_{0} \end{matrix}} α_{abm}D_{bm},

where all α_{abm} ∈ k. Clearly, α_{abm} = 0 if b < a < v_{0}.

If m ∈ ℤ mod N_{0} we use the notation m̂ for the unique natural number satisfying the conditions 0 ⩽ m̂ < N_{0} and m̂ mod N_{0} = m.

With the above notation and assumptions we have the following statement.

Main Lemma. Assume that:

— α_{110} = 1;

— α_{1b0} = 0 for all b ∈ ℤ^{+}(p), 1 < b < v_{0};

— α_{2b0} = 0 for all b ≡ 1 mod p, 1 < b < v_{0}. Then for all 1 ⩽ a < b < v_{0}/p such that a, b ∈ ℤ^{+}(p) and m ≠ 0 mod N_{0}, we have:

a) α_{aa0} = 1 and α_{ab0} = 0;

b) if b/pm̂ < a, then α_{a,b,−m} = 0;

c) there exist elements β_{c,m} ∈ k, where c ∈ ℤ^{+}(p), such that if c = b − apm̂ then α_{a,b,−m} = β_{c,−m}.

Remark. Finally, the morphism f_{k} is given by the correspondences

D_{a0} ↦ D_{a0} + ∑_{\begin{matrix}m≠0 mod N_{0} \\ c∈ℤ^{+}(p) \end{matrix}} β_{c,−m}D_{apm̂+c,−m} mod M_{k}^{(v_{0}/p)},

where a ∈ ℤ^{+}(p) and 1 ⩽ a < v_{0}/p. It can be also shown that any such morphism comes from some automorphism of the group G(p)/C_{3}(G(p)) respecting ramification filtration for all rational upper numbers v such that 1 ⩽ v ⩽ v_{0}/p.

Proof of main lemma.

3.2. Choose α_{0} ∈ W with trace 1, and use the uniformizer t to construct the identification ψ : G(p)/C_{p}(G(p)) ≃ G(L) from n.2.1. Then in the notation of n.2, the group Ḡ is identified with G(L̄), where L̄ = L/p^{3}L is a Lie algebra over ℤ_{p}, and the ramification filtration of the group Ḡ corresponds to the filtration of L̄ by its ideals L̄^{(v)} = L^{(v)} mod p^{3}L, v ⩾ 1. Under this identification, the group automorphism F̄ gives rise to the automorphism F̄ of the Lie algebra L̄. Consider extensions of scalars L̄_{W} = L̄ ⊗ W = L_{W} mod p^{3}L_{W} and L̄_{W}^{(v)} = L̄^{(v)} ⊗ W = L_{W}^{(v)} mod p^{3}L_{W}, where v ⩾ 1.

Without special mentioning we shall put a bar to any notation related to the Lie algebra L̄ if it came from the similar notation introduced earlier for the algebra L.

The automorphism F̄_{W} of the Lie algebra L̄_{W} commutes with Frobenius and satisfies the condition

F̄_{W} (L̄_{W}^{(v)}) = L̄_{W}^{(v)}

for all 1 ⩽ v ⩽ v_{0}. We can use the description of the filtration {L_{W}^{(v)}}_{v⩾1} from n.2 to obtain the description of the induced filtration {L̄_{W}^{(v)}}_{v⩾1} in terms of generators Δ̄_{an} = Δ_{an} mod p^{3} (where a ∈ ℤ^{+}(p) and n ∈ ℤ mod N_{0}) and Δ̄_{0} = Δ_{0} mod p^{3}. Note that for any a ∈ ℤ^{+}(p) and n ∈ ℤ mod N_{0},

F̄_{W} (Δ̄_{an}) ≡ ∑_{\begin{matrix}b∈ℤ^{+}(p) \\ m∈ℤ mod N_{0} \end{matrix}} σ^{n}(α_{abm})Δ̄_{b,m+n} mod (C_{2}(L̄_{W}) + pL̄_{W} ) .

Proposition 1. If n ≠ 0 mod N_{0}, then α_{11n} = 0.

Proof. Let v = 2 + 1/p. Then (1/2)F̄_{v}^{(v)} = [Δ̄_{2,0}, Δ̄_{1,−1}] ∈ L̄_{W}^{(v)} is mapped by F̄_{W} to

∑_{n,m∈ℤ mod N_{0}} α_{22m}σ^{−1}(α_{11n})[Δ̄_{2,−m}, Δ̄_{1,−n−1}]+

∑_{n,m∈ℤ mod N_{0}} α_{22m}σ^{−1}(α_{12n})[Δ̄_{2,−m}, Δ̄_{2,−n−1}] mod (L̄_{triv}^{(v)} + pL̄_{W} ) .

It is easy to see that L̄_{adm}^{(v)} mod L̄_{triv}^{(v)} has the following k-basis

{σ^{n}Δ_{10}, σ^{n}Δ_{20}, σ^{n}[Δ_{20}, Δ_{1,−1}], σ^{n}[Δ_{20}, Δ_{2,−1}] | n ∈ ℤ mod N_{0}}.

From α_{110} ≠ 0, we obtain α_{22m} = 0 for all m ≠ 0, but this implies α_{220} ≠ 0 and, therefore, α_{11n} = 0, if n ≠ 0 mod N_{0}. The proposition is proved.

Proposition 2. If a ∈ ℤ^{+}(p), 1 < a < v_{0} and n ≠ 0 mod N_{0}, then α_{aan} = 0.

Proof. Assume that for some n ≠ 0 mod N_{0}, we have α_{aan} ≠ 0.

Take v = a + 1/p, and consider

F̄_{v}^{(v)} = ∑ a_{1}[Δ̄_{a_{1}0}, Δ̄_{a_{2},−1}] = a[Δ̄_{a0}, Δ̄_{1,−1}] + ⋯ ∈ L̄_{W}^{(v)} mod pL̄_{W} ,

where the sum is taken for all a_{1}, a_{2} ∈ ℤ^{+}(p) such that a_{1}+a_{2}/p = v and a_{1}, a_{2} ⩽ a. The expression for F̄_{W} (F̄_{v}^{(v)}) contains the commutator [Δ̄_{an}, Δ̄_{1,−1}] with non-zero coefficient aα_{aan}. This commutator belongs to L̄_{adm}^{(v)} mod p only if γ(a, n, 1, −1) ⩾ v.

Because n ≠ 0 mod N_{0}, this implies that n = −1 mod N_{0} and a + 1 ≡ 0 mod p. The commutator [Δ̄_{a,−1}, Δ̄_{1,−1}] appears in the expression of σ^{−1}F̄_{a+1}^{(v)} with non-zero coefficient, therefore,

F̄_{W} (F̄_{v}^{(v)}) mod L̄_{triv}^{(v)} + pL̄_{W}

must contain another term [Δ̄_{a−1,0}, Δ̄_{2,−1}] of σ^{−1}F̄_{a+1}^{(v)} also with non-zero coefficient (we use that p ⩾ 5 and therefore, a ⩾ 4).

This is impossible, because

F̄_{W} (F̄_{v}^{(v)}) mod L̄_{triv}^{(v)} + pL̄_{W}

is a linear combination of commutators of the form [Δ̄_{b_{1}n_{1}}, Δ̄_{b_{2}n_{2}}], where b_{1} ⩾ a_{1}, b_{2} ⩾ a_{2}, and a_{1} + (a_{2}/p) = v, with 1 ⩽ a_{1}, a_{2} ⩽ a (indeed, if b_{1} = a − 1, then 2 = b_{2} ⩾ p + 1 gives contradiction; if b_{1} = 2, then a_{1} ⩽ 2, but then a ⩾ a_{2} ⩾ p(a − 2) + 1/p gives contradiction). The proposition is proved.

3.3. For a ∈ ℤ^{+}(p), a < v_{0}, set for simplicity α_{aa0} = α_{a}.

Proposition. There exists λ ∈ k such that for all a ∈ ℤ^{+}(p), one has α_{a} = λ^{a−1}.

Proof.

Lemma 1. If a > p, a ≢ 1 mod p, then α_{a} = α_{a−1}σ^{−1}(α_{p+1}).

Proof. Take v = a + 1/p. Then

F̄_{v}^{(v)} = a[Δ̄_{a0}, Δ̄_{1,−1}] + (a − 1)[Δ̄_{a−1,0}, Δ̄_{p+1,−1}] + …

The image of F̄_{v}^{(v)} modulo L̄_{triv}^{(v)} + pL̄_{W} contains the term a[Δ̄_{a0}, Δ̄_{1,−1}] with coefficient α_{a}, and the term (a − 1)[Δ̄_{a−1,0}, Δ̄_{p+1,−1}] with coefficient α_{a−1}σ^{−1}(α_{p+1}). So, the lemma follows from Proposition 3 of n.2.4.

We note, the above lemma implies that α_{a} = λ^{a−1} for p < a < 2p with λ := σ^{−1}(α_{p+1}).

Lemma 2. α_{a} = λ^{a−1} for 1 ⩽ a < p.

Proof. Take v = 2p−1+a/p for 1 ⩽ a < p. The image of F̄_{v}^{(v)} modulo L̄_{triv}^{(v)} +pL̄_{W} contains the term (2p − 1)[Δ̄_{2p−1,0}, Δ̄_{a,−1}] with coefficient α_{2p−1}σ^{−1}(α_{a}), and the term (2p − 2)[Δ̄_{2p−2,0}, Δ̄_{p+a,−1}] with coefficient α_{2p−2}σ^{−1}(α_{p+a}). This gives by Proposition 3 of n.2.4, that α_{2p−1}σ^{−1}(α_{a}) = α_{2p−2}σ^{−1}(α_{p+a}), and therefore, α_{a} = σ(λ)^{−1}α_{a+p} = λ^{a−1}. The lemma is proved.

Lemma 3. α_{2p+1} = λ^{2p}.

Proof. Take v = 2p + 3 + 1/p. The image of F̄_{v}^{(v)} contains the term

(2p + 3)[Δ̄_{2p+3,0}, Δ̄_{1,−1}]

with coefficient α_{2p+3}, and the term

(2p + 1)[Δ̄_{2p+1,0}, Δ̄_{2p+1,−1}]

with coefficient α_{2p+1}σ^{−1}(α_{2p+1}). So, α_{2p+3} = α_{2p+1}σ^{−1}(α_{2p+1}) by the proposition 3 of n.2.4. From the above lemma 1 we obtain α_{2p+3} = λ^{2}α_{2p+1}, and therefore, α_{2p+1} = λ^{2p}.

We finish the proof of proposition by induction on a. It has been proved already, that α_{a} = λ^{a−1} for a ⩽ 2p − 1.

Assume that a ∈ ℤ^{+}(p), v_{0} > a > 2p + 1 and for any b ∈ ℤ^{+}(p), b < a, we have α_{b} = λ^{b−1}.

If a ≢ 1 mod p, then α_{a} = λ^{a−1} by lemma 1. If a ≡ 1 mod p, then a ⩾ 3p + 1. Take v = a + 1/p. Then the image of F̄_{v}^{(v)} contains the term a[Δ̄_{a0}, Δ̄_{1,−1}] with coefficient α_{a}, and the term (a − 2)[Δ̄_{a−2,0}, Δ̄_{2p+1,−1}] with coefficient α_{a−2}σ^{−1}α_{2p+1}. Therefore, α_{a} = α_{a−2}σ^{−1}α_{2p+1} = λ^{a−1}.

The proposition is proved.

3.4. Consider λ ∈ k from above n.3.3.

Proposition. λ = 1.

Proof. Consider the image F̄_{W} (F̄_{p^{2}}^{(p^{2})}) mod L̄_{triv}^{(p^{2})} of

F̄_{p^{2}}^{(p^{2})} = p^{2}Δ̄_{1,2} − ∑_{\begin{matrix}a_{1}+a_{2}=p \\ a_{1}>a_{2} \\ a_{1},a_{2}∈ℤ^{+}(p) \end{matrix}} pa_{1}[Δ̄_{a_{1},1}, Δ̄_{a_{2},1}] − ∑_{\begin{matrix}a_{1}+a_{2}=p^{2} \\ a_{1}>a_{2} \\ a_{1},a_{2}∈ℤ^{+}(p) \end{matrix}} a_{1}[Δ̄_{a_{1},0}, Δ̄_{a_{2},0}] ∈ L̄_{W}^{(p^{2})}.

Following the terms of F̄_{p^{2}}^{(p^{2})} in this image, we obtain by Proposition 3 of n.2.4 the

equalities 1 = λ^{p(p−2)} = λ^{p^{2}−2}, and they imply that λ = 1.

3.5. We need the following lemma in the nn.3.6-3.7 below.

Lemma. If a, b ∈ ℤ^{+}(p), a < b < v_{0} and b/p < a, then α_{a,b,−1} = 0.

Proof. Assume that α_{a,b,−1} ≠ 0. Take v = b + a/p^{m}, where m is the minimal integer such that p^{m} ⩾ v_{0}. Then b < v < b + 1 ⩽ v_{0} and, therefore,

F̄_{v}^{(v)} ≡ b[Δ̄_{b0}, Δ̄_{a,−m}] mod L̄_{triv}^{(v)} + pL̄_{W} .

By Proposition of n.3.4, the image

F̄_{W} (F̄_{v}^{(v)}) mod L̄_{triv}^{(v)} + pL̄_{W}

contains the term [Δ̄_{b0}, Δ̄_{b,−(m+1)}] with non-zero coefficient σ^{−m}(α_{a,b,−1}). Therefore,

max{γ(b, 0, b, −(m + 1)), γ(b, −(m + 1), b, 0)} ⩾ v,

cf. notation n.2.5. Clearly, 1 ⩽ m + 1 < log_{p}(v_{0}) + 2 < N_{0}, therefore,

γ(b, 0, b, −(m + 1)) = b (1 + \frac{1}{p^{m+1}} ) , γ(b, −(m + 1), b, 0) = b (1 + \frac{1}{p^{N_{0}−(m+1)}} ) .

Because γ(b, 0, b, −(m + 1)) < b + a/p^{m} = v, it is sufficient to verify that γ(b, −(m + 1), b, 0) ⩽ γ(b, 0, b, −(m + 1)) or, equivalently, N_{0} ⩾ 2(m + 1). Indeed,

2(m + 1) < 2(log_{p}(v_{0}) + 2) = 2 log_{p}(p^{2}v_{0}) ⩽ N_{0}.

The lemma is proved.

3.6. For any a, b ∈ ℤ^{+}(p) such that 1 ⩽ a < b < v_{0}, we shall use the simpler notation α_{ab} = α_{ab0}.

Proposition. Assume that for some r ∈ ℤ_{⩾0} and all l, l_{1} ∈ ℤ^{+}(p) such that

l < l_{1} ⩽ min{l + r, v_{0} − 1},

we have α_{ll_{1}} = 0. Then for all b, b_{1} ∈ ℤ^{+}(p) such that

p < b < b_{1} < min{(r + 1)p, v_{0}},

we have α_{bb_{1}} = α_{b−p,b_{1}−p}.

Proof. Let a = v_{0} − 1 and v = a − 1 + b/p. Then

F̄_{v}^{(v_{0})} = ∑_{\begin{matrix}a′+b′/p=v \\ a′,b′<v_{0} \end{matrix}} a′[Δ̄_{a′0}, Δ̄_{b′,−1}] = a[Δ̄_{a0}, Δ̄_{b−p,−1}] + (a − 1)[Δ̄_{a−1,0}, Δ̄_{b,−1}] + …

Note that for any (a′, b′), such that a′ + b′/p = v, we have

pa′ = p(v_{0} − 1) + (b − p) − b′ > (p − 1)(v_{0} − 1) ⩾ v_{0}.

Therefore, by Lemma of n.3.5, it holds α_{a′,b_{1},−1} = 0. So, the image of F̄_{v}^{(v_{0})} mod L̄_{triv}^{(v_{0})} + pL̄_{W} contains the term [Δ̄_{a−1,0}, Δ̄_{b_{1},−1}] with coefficient

∑_{\begin{matrix}a′,b′<v_{0} \\ a′+b′/p=v \end{matrix}} a′α_{a′,a−1}σ^{−1}(α_{b′b_{1}}).

But, if a′ ≠ a − 1 and α_{a′,a−1} ≠ 0, then a′ ⩽ (a − 1) − (r + 1), i.e. a − 1 − a′ ⩾ r + 1. Because a′ + b′/p = a − 1 + b/p, we have

b′ ⩾ b + p(a − 1 − a′) ⩾ b + p(r + 1) > b_{1},

and therefore, α_{b′b_{1}} = 0. This proves that the coefficient for the term [Δ̄_{a−1,0}, Δ̄_{b_{1},−1}] equals (a − 1)σ^{−1}(α_{bb_{1}}).

Similarly, we obtain that the coefficient for the term [Δ_{a0}, Δ_{b_{1}−p,−1}] is equal to aσ^{−1}(α_{b−p,b_{1}−p}), and our proposition follows from Proposition 3 of n.2.4.

3.7. The following proposition proves the part a) of the Main Lemma.

Proposition. Let r ∈ ℤ_{⩾0} be such that (r + 2)p ⩽ v_{0}. Then for any l, l_{1} ∈ ℤ^{+}(p) such that l < l_{1} ⩽ min{l + r, v_{0}}, we have α_{ll_{1}} = 0.

Proof. We apply induction on r. So, we can assume that α_{ll_{1}} = 0 for l < l_{1} < l + r. By Proposition of n.3.6, this gives that α_{bb_{1}} = α_{b−p,b_{1}−p} for b < b_{1} < b + rp (all indices are assumed to belong to the segment [1, v_{0})).

In particular, we have α_{sp+1,b} = α_{1,b−sp} = 0 for b < 1 + (s + r)p and s ∈ ℕ. . We shall use the agreement Δ̄_{a0} = 0 and α_{ab} = 0, if one of indices a or b is not

prime to p. We introduce also the new notation α_{m}(n) := α_{m,m+n}, where m, n ∈ ℕ. For j = 1, …, p − 1, set c_{j} = p(r + 1) + j and v_{j} = c_{j} + 1/p, and consider

F̄_{v_{j}}^{(v_{j})} ≡ c_{j}[Δ_{c_{j},0}, Δ_{1,−1}] + ⋯ + (c_{j} − r − 1)[Δ_{c_{j}−r−1,0}, Δ_{p(r+1)+1,−1}] mod pL̄_{W} .

Note that α_{c,c′,−1} = 0, where c ⩾ c_{j} − r − 1 and c′ < v_{j}. Indeed, this follows from Lemma of n.3.5, because

pc ⩾ p(c_{j} − r − 1) ⩾ p(p(r + 1) − r) ⩾ p(r + 2) > v_{j} > c′.

Therefore, all terms of the kind [Δ̄_{∗,0}, Δ̄_{∗,−1}] appearing in the image of F̄_{v_{j}}^{(v_{j})} modulo L̄_{triv}^{(v_{j})} + pL̄_{W}, can be written in the following way

c_{j} [Δ̄_{c_{j},0}, Δ̄_{1,−1}] +
(c_{j} − 1) [Δ̄_{c_{j}−1,0}, Δ̄_{p+1,−1} + α_{p+1}(rp)Δ̄_{(r+1)p+1,−1} + …] +
(c_{j} − 2) [Δ̄_{c_{j}−2,0}, Δ̄_{2p+1,−1}] + ⋯ +
(c_{j} − r) [Δ̄_{c_{j}−r,0} + α_{c_{j}−r}(r)Δ̄_{c_{j},0}, Δ̄_{rp+1,−1}] +

(c_{j} − r − 1) [Δ̄_{c_{j}−r−1,0} + α_{c_{j}−r−1}(r)Δ̄_{c_{j}−1,0} + α_{c_{j}−r}(r)Δ̄_{c_{j},0}, Δ̄_{(r+1)p+1,−1}] .

Consider the terms coming from F̄_{γ}^{(v_{j})}, where

γ = c_{j} + \frac{rp + 1}{p} = c_{j} − 1 + \frac{(r + 1)p + 1}{p}.

These terms are

(c_{j} − r)α_{c_{j}−r}(r) [Δ̄_{c_{j},0}, Δ̄_{rp+1,−1}] +
(c_{j} − 1)σ^{−1}(α_{p+1}(rp)) [Δ̄_{c_{j}−1,0}, Δ̄_{(r+1)p+1,−1}] +
(c_{j} − r − 1)α_{c_{j}−r−1}(r) [Δ̄_{c_{j}−1,0}, Δ̄_{(r+1)p+1,−1}] .

Therefore, for j = 2, …, p − 1, we obtain the equalities

\frac{(j − r)α_{c_{j}−r}(r)}{j} = \frac{(j − r − 1)α_{c_{j}−r−1}(r)}{j − 1} + σ^{−1}(α_{p+1}(rp))

(we use that c_{j} ≡ j mod p).

Set X = σ^{−1}α_{p+1}(rp) and β_{j} = (j − r)α_{c_{j}−r}(r)/j for j = 1, …, p − 1. Then the above equalities can be rewritten in the form

β_{2} = β_{1} + X, …, β_{p−1} = β_{p−2} + X.

But now we use that:

a) β_{r} = 0, if r ≢ 0 mod p;

b) if r ≢ −1 mod p, then β_{r+1} = 0 (this follows from the property α_{1}(r) = 0, cf. the statement of main lemma), and if r ≡ −1 mod p, then β_{1} = 0 (this follows from the property α_{2}(r) = 0, cf. loc.cit).

In the case r ≢ 0 mod p, this gives X = β_{1} = ⋯ = β_{p−1} = 0. Therefore, for all (r + 1)p < a < (r + 2)p, we have α_{a−r,a} = 0. So, in this case our proposition follows from Proposition of n.3.6.

Consider the case r ≡ 0 mod p. Here we have only that α_{c_{j}−r}(r) = (j − 1)X for all 1 ⩽ j < p. By Proposition of n.3.6 this gives

α_{a}(r) = (a − 1)X

for all a ∈ ℤ^{+}(p) such that a < (r + 2)p − r.

Consider the natural number b such that b = r/p + 1, if r ≢ −p mod p^{2}, and b = r/p + 2, if r ≡ −p mod p^{2}.

Then b ∈ ℤ^{+}(p) and pb ⩽ r + 2p ⩽ (r + 2)p ⩽ v_{0}.

Take v = pb and consider

F̄_{v}^{(v)} = pbΔ̄_{b1} − ∑ a_{1}[Δ̄_{a_{1}0}, Δ̄_{a_{2}0}] mod(L̄_{triv}^{(v)} + p^{2}L̄_{W}),

where the sum is taken for all a_{1}, a_{2} ∈ ℤ^{+}(p) such that a_{1} + a_{2} = pb and a_{1} > a_{2}. Consider

F̄_{W} (F̄_{v}^{(v)}) mod L̄_{triv}^{(v)} + p^{2}L̄_{W} .

By inductive assumption, the expression of

F̄_{W} (pΔ̄_{b1}) mod L̄_{triv}^{(v)} + p^{2}L̄_{W}

does not contain (with non-zero coefficient) the term pΔ̄_{b+r/p,1} (because r/p < r and α_{b}(r/p) = 0). Therefore, the terms of F̄_{v+r}^{(v)} does not enter into the expression of the element

F̄_{W} (F̄_{v}^{(v)}) mod L̄_{triv}^{(v)} + p^{2}L̄_{W} .

On the other hand, if we follow the terms of this image in the form [Δ̄_{b_{1}0}, Δ̄_{b_{2}0}], where b_{1} + b_{2} = v + r, we obtain

− ∑_{\begin{matrix}a_{1}+a_{2}=v \\ v>a_{1}>a_{2}⩾1 \end{matrix}} (a_{1}α_{a_{1}}(r)[Δ̄_{a_{1}+r,0}, Δ̄_{a_{2}0}] + a_{1}α_{a_{2}}(r)[Δ̄_{a_{1}0}, Δ̄_{a_{2}+r,0}]) ≡

2X ∑_{\begin{matrix}b_{1}+b_{2}=v+r \\ v>b_{1}>b_{2}⩾1 \end{matrix}} b_{1}[Δ̄_{b_{1}0}, Δ̄_{b_{2}0}] mod L̄_{triv}^{(v)} + p^{2}L̄_{W} .

The right-hand sum is not empty, because r < v = pb by the choice of b. Therefore, X = 0. This proves the case r ≡ 0 mod p.

The proposition is completely proved.

3.8. Now we can finish the proof of the Main Lemma.

As we have already noted, the part a) follows from Proposition of n.3.7.

Take v = 2v_{0}/p, then v ∈ ℕ, v ≡ 0 mod p and

F̄_{v}^{(v)} ≡ ∑ a_{1}[Δ̄_{a_{1}0}, Δ̄_{a_{2}0}] mod L̄_{triv}^{(v)} + pL̄_{W} ,

where the sum is taken for all a_{1}, a_{2} ∈ ℤ^{+}(p) such that a_{1} + a_{2} = v, 1 ⩽ a_{2} < v_{0}/p and v_{0}/p < a_{1} < v.

The terms of the form [Δ̄_{b_{1}0}, Δ̄_{b_{2},m}], where b_{2} < v_{0}/p and m ≠ 0 mod N_{0}, from the image F̄_{W} (F̄_{v}^{(v)}) modulo L̄_{triv}^{(v)} + pL̄_{W} can be written in the following way

∑ a_{1}α_{a_{2}b_{2}m}[Δ̄_{a_{1}0}, Δ̄_{b_{2}m}],

where the sum is taken for all a_{1}, a_{2}, b_{2} ∈ ℤ^{+}(p) such that a_{1} + a_{2} = v, 1 ⩽ a_{2} < b_{2} < v_{0}/p and v_{0}/p < a_{1} < v.

Because m ≠ 0 mod N_{0}, we have

γ(b_{2}, m, a_{1}, 0) ⩽ b_{2} + \frac{a_{1}}{p} < v.

Therefore, if α_{a_{2},b_{2},−m} ≠ 0, then γ(a_{1}, 0, b_{2}, −m) = a_{1}+b_{2}/pm̂ > v, i.e. b_{2}/pm̂ > a_{2}. This gives the part b) of the main lemma.

The part c) follows now from Proposition 3 of n.2.4.

The Main Lemma is proved.

4. Fundamental exact sequence.

4.1. Let [K : ℚ_{p}] < ∞, choose uniformiser π_{K} ∈ K and a sequence {π_{s}}_{s⩾0} ⊂ K̄ such that π_{0} = π_{K} and π_{s}^{p} = π_{s−1} for s ∈ ℕ. Set K_{s} = K(π_{s}) and denote by K̃ the union of all K_{s}.

We note that K̃ is an APF-extension of K (we use the terminology and constructions from [Wi]), so one can apply the Fontaine-Wintenberger field-of-norms functor X to obtain the field K = X (K̃). This is a complete discrete valuation field, its residue field canonically identified with the residue field k of K, and it has specific uniformiser t obtained from the above sequence {π_{s}}_{s⩾0}. The functor X induces equivalence of the category of extensions of K̃ in K̄ and the category of extensions of K in K_{sep} = X (K̄). This gives identification

G_{K} = Gal(K_{sep}/K) = Gal(K̄/K̃),

so we have the embedding ι : G_{K} ⟶ Γ_{K}. This embedding is compatible with ramification filtrations, i.e.

ι(G_{K}^{(x)}) = ι(G_{K}) ∩ Γ_{K}^{(φ_{_{K}/K}(x))}

for all x ⩾ 0, where φ_{K̃/K}(x) is the Herbrandt function of the infinite extension

K̃/K (by definition φ_{K̃/K}(x) = lim_{s→∞} φ_{K_{s}/K}(x), we note also that φ_{K̃/K}(x) = φ_{K_{s}/K}(x) for x ⩽ ep^{s+1}/(p − 1)).

4.2. Assume that a primitive root of degree p^{M} of unity ζ_{M} belongs to K. Then

Gal(K_{M}/K) = ℤ/p^{M}ℤτ_{0},

where τ_{0} ∈ Gal(K_{M}/K) is uniquely characterised by the property τ_{0}(π_{M}) = ζ_{M}π_{M}.

The ramification subgroup Γ_{K}(p)^{(a_{K})} (where a_{K} = e_{K}(p/(p − 1) + M), cf. Introduction) belongs to the kernel of the natural projection

Γ_{K}(p) ⟶ Gal(K_{M}/K),

because a_{K} = v(K_{M+1}/K) > v(K_{M}/K).

On the other hand, the embedding ι from n.4.1 induces the homomorphism ι(p) : G_{K}(p) ⟶ Γ_{K}(p) and for v_{K} = e_{K}p^{M+1}/(p − 1), we have

ι(p) (G_{K}(p)^{(v_{K})}) ⊂ Γ_{K}(p)^{(a_{K})}

(because φ_{K̃/K}(v_{K}) = a_{K}).

As a result of the above considerations we obtain the following sequence of morphisms of profinite groups

(1)  1 ⟶ G_{K}(p)/G_{K}(p)^{(v_{K})} ⟶ Γ_{K}(p)/Γ_{K}(p)^{(a_{K})} ⟶ ℤ/p^{M}ℤτ_{0} ⟶ 0.

Proposition. The sequence (1) is exact.

Proof.

Lemma 1. Γ_{K}^{(a_{K})} = Γ_{K_{M}}^{(v_{K})}.

Proof of lemma. We have

K̄^{Γ_{K}^{(a_{K})}} = ⋃_{α∈A} L_{α},

where {L_{α}}_{α∈A} is the family of finite extensions of K in K̄ such that v(L_{α}/K) < a_{K}. Because

v(K_{M}/K) = e_{K} (\frac{p}{p − 1} + M − 1) < a_{K}

we have

⋃_{α∈A} L_{α} = ⋃_{α∈A_{1}} L_{α},

where A_{1} = {α ∈ A | L_{α} ⊃ K_{M}}. From the composition property for Herbrandt’s functions φ_{L_{α}/K}(x) = φ_{K_{M} /K} (φ_{L_{α}/K_{M}} (x)) and interpretation of upper ramification number v(L_{α}/K) as the second coordinate of the maximal edge point of the graph of the function φ_{L_{α}/K}, it follows

v(L_{α}/K) = max {φ_{K_{M} /K}(v(L_{α}/K_{M})), v(K_{M}/K)} .

Therefore, for any α ∈ A_{1}, it holds

v(L_{α}/K) < a_{K} ⟺ v(L_{α}/K_{M}) < v_{K}

(because v(K_{M}/K) < a_{K} and φ_{K_{M} /K}(v_{K}) = a_{K}). Therefore,

⋃_{α∈A_{1}} L_{α} = K̄^{Γ_{K_{M}}^{(v_{K})}}

and Γ_{K}^{(a_{K})} = Γ_{K_{M}}^{(v_{K})}. The lemma is proved.

The above lemma gives a natural exact sequence

1 ⟶ Γ_{K_{M}} /Γ_{K_{M}}^{(v_{K})} ⟶ Γ_{K}/Γ_{K}^{(a_{K})} ⟶ Gal(K_{M}/K) ⟶ 1

and, therefore, for Galois groups of corresponding maximal p-extensions Γ_{K_{M}} (p) and Γ_{K}(p), we obtain the exact sequence

1 ⟶ Γ_{K_{M}} (p)/Γ_{K_{M}} (p)^{(v_{K})} ⟶ Γ_{K}(p)/Γ_{K}(p)^{(a_{K})} ⟶ Gal(K_{M}/K) ⟶ 1.

Clearly, our proposition is implied now by the following lemma.

Lemma 2. The embedding ι : G_{K} ⟶ Γ_{K} induces isomorphism

ῑ: G_{K}(p)/G_{K}(p)^{(v_{K})} ⟶ Γ_{K_{M}} (p)/Γ_{K_{M}} (p)^{(v_{K})}.

Proof. Consider the category E(K_{M}, v_{K}) of finite extensions L of K_{M} such that v(L/K_{M}) < v_{K}. Clearly, our lemma is implied by the following property

the correspondence L ⟶ X (LK̃) gives rise to equivalence of the category E(K_{M}, v_{K}) and the category E(K, v_{K}) of finite extensions L of K in K_{sep} = X (K̄) such that v(L/K) < v_{K}.

We explain briefly the construction of inverse correspondence. Let L ∈ E(K, v_{K}), then L = X (L̃), where L̃ = LK̃ for some finite extension L of K. For any s ∈ ℤ_{⩾0}, set L_{s} = LK_{s}, then there exists an index s_{0} such that v(L_{s}/K_{s}) = v(L/K) < v_{K} for all s ⩾ s_{0}, cf. [Wi].

Suppose that s_{0} is the minimal integer with such properties. Prove that s_{0} ⩽ M. Assume that s_{0} > M, then

v(L_{s_{0}}/K_{s_{0}−1}) = max {φ_{K_{s_{0}} /K_{s_{0}−1}}(v(L/K), v(K_{s_{0}}/K_{s_{0}−1})} =
v(K_{s_{0}}/K_{s_{0}−1}) = e_{K}p^{s_{0}}/(p − 1) ⩾ v_{K}.

Therefore, the group Γ_{K_{s_{0}−1}}^{(v(K_{s_{0}} /K_{s_{0}−1}))} acts nontrivially on L_{s_{0}}. From inequality v(L_{s_{0}}/K_{s_{0}}) < v(K_{s_{0}}/K_{s_{0}−1}) it follows easily that this group is of order p. Let L_{s_{0}−1} be the corresponding fixed subfield of L_{s_{0}}. Then v(L_{s_{0}−1}/K_{s_{0}−1}) < v(K_{s_{0}}/K_{s_{0}−1}), therefore, L_{s_{0}−1} and K_{s_{0}} are linearly disjoint over K_{s_{0}−1}, and it can be easily checked that v(L_{s_{0}−1}/K_{s_{0}−1}) = v(L_{s_{0}}/K_{s_{0}}). This gives contradiction, therefore, s_{0} ⩽ M.

The inverse map now is defined by the correspondence L ↦ L_{M}.

The lemma and the proposition are completely proved.

4.3. Let w_{K} = e_{K}p^{M}. If φ_{K_{M} /K}(b_{K}) = w_{K}, where φ_{K_{M} /K} is the Herbrandt function of the extension K_{M}/K, then the fundamental exact sequence (1) from n.4.2 induces the exact sequence

1 ⟶ G_{K}(p)/G_{K}(p)^{(w_{K})} ⟶ Γ_{K}(p)/Γ_{K}(p)^{(b_{K})} ⟶ ℤ/p^{M}ℤτ_{0} ⟶ 0.

If τ̂_{0} ∈ Γ_{K}(p)/Γ_{K}(p)^{(b_{K})} is a lifting of τ_{0}, then the conjugation by τ̂_{0} determines the class cl(τ_{0}) of automorphisms of the group G_{K}(p)/G_{K}(p)^{(w_{K})} modulo the subgroup of its inner automorphisms.

Consider a power series P (X) ∈ W [[X]] such that P (π_{K}) ≡ ζ_{M} mod p. Its essential part

1 + ∑_{1⩽i<e_{K}} α_{i}X^{i} = P (X) mod(p, X^{e_{K}}),

where all α_{i} ∈ k, comes from uniquely defined decomposition of ζ_{M} mod p in powers of π_{K} mod p with coefficients from k. We note that

P (X)^{p^{M}} ≡ 1 mod(p, X^{e_{K}}).

Set P_{M}(X) = P (X)^{p^{M}} and consider τ_{P} ∈ Aut^{1} K such that τ_{P} (t) = tP_{M}(t). We note that τ_{P} induces an automorphism of order p^{M} on O_{K} mod t^{w_{K}}.

If τ̂_{P} is a lifting of τ_{P} to the group of all contonuous isomorphisms of the field K_{sep}, then the conjugation by τ̂_{P} induces an automorphism of G_{K}(p) and this automorphism respects the ramification filtration of G_{K}(p). So, we obtain an automorphism of the quotient group G_{K}(p)/G_{K}(p)^{(w_{K})}. We denote by cl(τ_{P}) its class modulo the subgroup of inner automorphisms of this group.

Proposition. For the above P ∈ W [[X]], we have cl(τ_{0}) = cl(τ_{P}).

Proof.

4.3.1. Set for the simplicity w_{K} = w and consider any finite Galois extension L of K such that L ⊃ K_{M} and v(L/K_{M}) < w. In notation of n.4.2 consider L = X (LK̃). Then L is a Galois extension of K and the functor X provides us with the identification Gal(L/K) = Gal(L/K_{M}). With respect to this identification we must prove that

cl(τ_{P})|_{Gal(L/K)} = cl(τ_{0})|_{Gal(L/K_{M})}.

4.3.2. Deligne’s equivalence, cf. [De]. Consider the triple

T_{L} = (O_{L}/t^{w}O_{L}, m_{L}/t^{w}m_{L}, j) ,

where m_{L} is the maximal ideal of the valuation ring O_{L} of L and j : m_{L}/t^{w}m_{L} ⟶ O_{L}/t^{w}O_{L} is induced by the embedding m_{L} ⟶ O_{L}.

The family of triples coming from all finite extensions L of K such that v(L/K) < w gives the family of objects of the category (ext Tr_{w} K)^{w} introduced in [De]. This category can be characterised purely in intrinsic terms and the main result of [De] states that the functor L ↦ T_{L} establishes equivalence of the category (ext K)^{w} of extensions L/K such that v(L/K) < w and the category (ext Tr_{w} K)^{w}.

We note that the triple T_{L} has the group of automorphisms consisting of compatible pairs of O_{K}/t^{w}O_{K}-algebra automorphisms of O_{L}/t^{w}O_{L} and m_{L}/t^{w}m_{L}. Then the group Aut^{w}(T_{L}) of automorphisms of T_{L} in the category (ext Tr_{w} K)^{w} consists of equivalence classes modulo m_{L}^{w(L)} and modulo m_{L}^{w(L)+1}, respectively, of those automorphisms, where φ_{L/K}(w(L)) = w (the Herbrandt function φ_{L/K} can be recovered completely from the triple T_{L}). Then the group Gal(L/K) coincides with the group

Aut^{w}(T_{L}/T_{K}) = {a ∈ Aut^{w}(T_{L}) | a|_{T_{K}} = id}.

In particular, if morphism τ_{P} (T_{K}) ∈ Aut^{w}(T_{K}) corresponds to τ_{P}, then

cl τ_{P} |_{Gal(L/K)} = cl(τ_{P} (T_{K}))|_{Aut^{w}(T_{L}/T_{K})}.

4.3.3. Let Ō be the valuation ring of K̄. Consider Fontaine’s ring R = \underleftarrow{lim}_{s}(Ō/pŌ)_{s}, where s ∈ ℤ_{⩾0} and transition morphisms from (Ō/pŌ)_{s+1} to

(Ō/pŌ)_{s} are induced by taking p^{th} powers of elements of Ō. The maximal ideal m_{R} of R can be also presented in the form \underleftarrow{lim}_{s}(m̄/pm̄)_{s}, where m̄ is the maximal

ideal of Ō and transition morphisms are also induced by taking p^{th} powers in m̄.

If R_{0} is the quotient field of R, then there exists a universal procedure identifying any value of the field-of-norms functor X with a suitable subfield of R_{0}, cf. [Wi]. In particular, we can assume that K ⊂ L ⊂ R_{0} and these embeddings are compatible with action of the group G_{K} (recall that R has the natural Γ_{K}-module structure).

4.3.4. Consider two natural projections pr_{M} : R ⟶ (Ō/pŌ)_{M} and pr_{M} : m_{R} ⟶ (m̄/pm̄)_{M}. Clearly, pr_{M}(t) = π_{M} and we have induced isomorphisms

p̄r_{M} : R/t^{w}R ≃ Ō/pŌ, p̄r_{M} : O_{K}/t^{w}O_{K} ≃ O_{K_{M}} /pO_{K_{M}} ,
p̃r_{M} : m_{R}/t^{w}m_{R} ≃ m̄/pm̄, p̃r_{M} : m_{K}/t^{w}m_{K} ≃ m_{K_{M}} /pm_{K_{M}} .

Note that all these isomorphisms are σ^{−M}-linear with respect to natural k-module structures of involved objects. We note also that p̄r_{M}(P_{M}(t)) = ζ_{M} mod p.

So, we obtain isomorphism of triples

pr_{M} : T_{K} = (O_{K}/t^{w}O_{K}, m_{K}/t^{w}m_{K}, j) ≃ T_{K_{M}} = (O_{K_{M}} /pO_{K_{M}} , m_{K_{M}} /pm_{K_{M}} , j).

Consider the triple

pr_{M}(T_{L}) = (p̄r(O_{L}/pO_{L}), p̃r_{M}(m_{L}/pm_{L}), j).

This is an object of the category (ext Tr_{w} K_{M})^{w}. By Deligne’s equivalence (it works in any characteristics), this triple is of the form T_{L′} = (O_{L′}/pO_{L′}, m_{L′}/pm_{L′}, j), where L′ is a finite extension of K_{M} such that v(L′/K_{M}) = v(L/K) and [L′ : K_{M}] = [L : K]. Because pr_{M} commutes with Galois action, we have L′ ⊂ LK̃. Therefore, cf. n.4.1, L′ = L. Of course, pr_{M}(T_{L}) = T_{L} and

pr_{M} : Aut^{w}(T_{L}/T_{K}) ≃ Aut^{w}(T_{L}/T_{K})

corresponds to the identification of the Galois groups of extensions L/K and L/K given by the functor X.

4.3.5. Denote by τ_{0}(T_{K_{M}}) the automorphism of T_{K_{M}} coming from τ_{0}. then

pr_{M}(τ_{P} (T_{K})) = τ_{0}(T_{K_{M}}).

Therefore, pr_{M} maps cl(τ_{P} (T_{K}))|_{Aut^{w}(T_{L}/T_{K})} to

cl(τ_{0}(T_{K_{M}}))|_{Aut^{w}(T_{L}/T_{K_{M}})} = cl(τ_{0})|_{Gal(L/K_{M})}.

The proposition is proved.

4.4. Remark. a) Proposition n.4.3 provides us with an opportunity to compute the action of τ_{0} by conjugation on any quotient of the form G_{K}(p)/G_{K}(p)^{(v)} with v ⩽ w_{K}. We shall apply it in Section 5 below for ṽ_{K} = v_{K}/p = e_{K}p^{M}/(p−1) < w_{K};

b) In fact, we need only to know the action of τ_{0} on the maximal abelian quotient of period p of the group G_{K}(p)/G_{K}(p)^{(ṽ_{K})}. In this case the above proposition can be proved “by hands” by the use of the Artin-Schreier equations T^{p} − T = αt^{−a} in the characteristics p case, and the analogues of those T^{p} − T = [α]π_{M}^{−a} in the characteristics 0 case (here α ∈ k, [α] ∈ K is its Teichmuller representative, 1 ⩽ a < e_{K}p^{M} and (a, p) = 1).

5. Proof of the Theorem.

5.1. Prove first that the simplest invariants of the fields K and K′ coincide.

Proposition. f_{K} = f_{K′}, e_{K} = e_{K′} and ζ_{M} ∈ K′.

Proof. The isomorphism F induces isomorphism of corresponding maximal abelian quotients and when restricted to the subgroup Γ_{K}(p)^{(1)} mod Γ_{K}(p)^{(a_{K})} induces isomorphism of multiplicative groups

F_{ab}^{(1)} : U_{K}^{(1)} mod U_{K}^{(a_{K})} ⟶ U_{K′}^{(1)} mod U_{K′}^{(a_{K})},

where we use the notation U_{K}^{(n)} (resp., U_{K′}^{(n)}) for the group of principal units in K (resp., K′) congruent to 1 modulo n^{th} power of maximal ideal. Clearly, for 1 ⩽ n ⩽ a_{K}, the morphism F_{ab}^{(1)} respects “ramification filtration”

F_{ab}^{(1)} (U_{K}^{(n)} mod U_{K}^{(a_{K})}) = U_{K′}^{(n)} mod U_{K′}^{(a_{K})}.

This gives f_{K} = f_{K′}, because a_{K} ⩾ 2.

The followinmg two statements are standard facts about the structure of the multiplicative group of principal units:

a) if a ∈ ℕ and a > e_{K}^{∗} := e_{K}p/(p − 1), then (U_{K}^{(a−e)})^{p} = U_{K}^{(a)};

b) the properties — b_{1}) there exists n ∈ ℕ such that n ≡ 0 mod p and

U_{K}^{(n)} mod U_{K}^{(n+1)} ⊄ U_{K}^{(1)p} mod U_{K}^{(n+1)},

— b_{2}) n = e_{K}^{∗} and ζ_{1} ∈ K, are equivalent.

Because ζ_{M} ∈ K, a_{K} > e_{K}^{∗} and F_{ab}^{(1)} respects “ramification filtration”, we obtain

U_{K′}^{(e_{K}^{∗})} mod U_{K′}^{(e_{K}^{∗}+1)} ⊄ U_{K′}^{(1)p} mod U_{K′}^{(e_{K}^{∗}+1)},

and therefore, e_{K}^{∗} = e_{K′}^{∗} or e_{K′} = e_{K}.

Finally, if F_{ab}^{(1)} (ζ_{M} mod U_{K}^{(a_{K})}) = ξ mod U_{K′}^{(a_{K})}, where ξ ∈ U_{K′}^{(1)}, then ξ^{p^{M−1}} ∉

U_{K′}^{(a_{K})} but ξ^{p^{M}} ∈ U_{K′}^{(a_{K})}. Because U_{K′}^{(a_{K})} = U_{K′}^{(e_{K}^{∗})p^{M}}, we conclude that ξ ≡ ζ′_{M} mod U_{K′}^{(e_{K}^{∗})}, where ζ′_{M} is a primitive root of unity of degree p^{M}. In particular, ζ_{M} ∈ K′.

The proposition is proved.

Remark. The above proposition implies that ℤ_{p}-algebras O_{K} mod p^{1/(p−1)} and O_{K′} mod p^{1/(p−1)} are isomorphic. If M = 1, this gives already the assertion of our theorem. So, we can assume below that M ⩾ 2. In particular, we shall use later that v_{K} ≡ 0 mod p^{2}.

5.2. By the above n.5.1, we can set a_{0} = a_{K} = a_{K′}, v_{0} = v_{K} = v_{K′}, e = e_{K} = e_{K′} and e^{∗} = e_{K}^{∗} = e_{K′}^{∗}.

In the notation of Section 4 fix uniformising element π_{K} ∈ K, the sequence {π_{s}}_{s⩾0} ⊂ K̄ and consider corresponding fundamental sequence

1 ⟶ G(p)/G(p)^{(v_{0})} ⟶^{i} Γ(p)/Γ(p)^{(a_{0})} ⟶^{j} Gal(K^{(M)}/K) ⟶ 1

(where we set G_{K} = G and Γ_{K} = Γ).

Proposition. The isomorphism F transforms the fundamental exact sequence (3) to some fundamental exact sequence for the field K′.

Proof. First, we note that F transforms the above fundamental sequence to the following exact sequence

1 ⟶ Ker(F^{−1} ∘ j) ⟶ Γ_{K′}(p)/Γ_{K′}(p)^{(a_{0})} ⟶ Coker(i ∘ F) ⟶ 1.

Therefore, Coker(i ∘ F) = Gal(E/K′) for some cyclic extension E of K′ of degree p^{M}. So, E = K′(α^{1/p^{M}}) for some α ∈ K^{′∗}, and it is sufficient to prove that

(2)  α ∉ U_{K′}^{(1)} mod K^{′∗p^{M}} .

But the compatibility of F with ramification filtrations implies that all ramification jumps of the extension E/K′ are the same as of the extension K_{M}/K. In particular, if E_{1} is the unique subextension of E of degree p over K′, then v(E_{1}/K′) = e^{∗}. This implies already the property (2). The proposition is proved.

5.3. In this section we put together all necessary information about the fields K and K′.

5.3.1. The field K. Its algebraic closure K̄ is assumed to be chosen, we also fix its uniformising element π_{K}, a sequence {π_{s}}_{s⩾0} ⊂ K̄, cf. n.4.1, and a primitive root ζ_{M} ∈ K of unity of degree p^{M}.

To the above data we have assigned the following objects:

— the field K = k((t)), where k is the residue field of K and t is a fixed uniformiser (related to the sequence {π_{s}}_{s⩾0});

— the group Ḡ = G_{K}/G_{K}^{p^{3}}C_{3}(G_{K}), where G_{K} = Gal(K_{sep}/K) (and K_{sep} is canonically defined by K̄), together with induced ramification filtration {Ḡ^{(v)}}_{v⩾1};

— the k-module M_{k}^{(1)} = M^{(1)} ⊗ k, where M^{(1)} = Ḡ^{(1)} mod Ḡ^{p}C_{2}(Ḡ), together with its fixed k-basis

D^{(1)}(t) mod p = {D_{an} | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}};

— the k-linear automorphism m ↦ τ_{0} ∗ m of M_{k}^{(1)} commuting with Frobenius morphism σ (induced by conjugation of any lifting τ̂_{0} of τ_{0} ∈ Gal(K_{M}/K) on G_{K}(p) mod G_{K}(p)^{(v_{0})}, where τ_{0}(π_{M}) = ζ_{M}π_{M}).

5.3.2. The field K′. We assume that K′ ⊂ K̄, its residue field coincide with that of K, we fix a uniformiser π_{K′} of K′, the corresponding sequence {π′_{s}}_{s⩾0} and a primitive root ζ′_{M} ∈ K′ of degree p^{M} of unity.

Then we relate to the field K′ the objects similar to objects related to the field K in the above n.5.3.1. The notation for objects related to K′ is obtained from the corresponding notation for objects related to K by adding a stroke.

5.3.3. The above data introduced for the fields K and K′ are related in the following way:

— there exists isomorphism Ḡ : Ḡ ⟶ Ḡ′ such that Ḡ(Ḡ^{(v)}) = Ḡ′^{(v)} for all 1 ⩽ v ⩽ v_{0} (this property is equivalent to the existence of the isomorphism of the quotients Ḡ/Ḡ^{(v_{0})} and Ḡ′/Ḡ^{′(v_{0})} compatible with ramification filtrations because Ḡ and Ḡ′ are characteristical quotients of free pro-p-groups);

— if g = Ḡ mod Ḡ^{p}C_{2}(Ḡ) and g_{k} = g ⊗ k : M_{k}^{(1)} ⟶ M′_{k}^{(1)}, then for any a ∈ ℤ^{+}(p) and m ∈ M_{k}^{(1)}, we have

g_{k}(τ_{0} ∗ m) ≡ τ′_{0} ∗ g_{k}(m) mod M′_{k}^{(v_{0})}.

5.4. For a ∈ ℤ^{+}(p) and n ∈ ℤ mod N_{0}, consider C_{an} = g_{k}^{−1}(D′_{an}). Then the set

{C_{an} | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}}

creates a k-basis of M_{k}^{(1)} satisfying the conditions a) and b) from n.1.6. Therefore, there exist n_{0} ∈ ℤ mod N_{0} and uniformising element t̃ of the field K such that in terms of the special basis D^{(1)}(t̃) = {D̃_{an} | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}} we have for all a ∈ ℤ^{+}(p),

σ^{n_{0}}C_{a0} = ∑_{\begin{matrix}b∈ℤ^{+}(p) \\ m∈ℤ mod N_{0} \end{matrix}} α_{abm}D̃_{bm},

where all coefficients α_{abm} ∈ k, α_{110} = 1, α_{1b0} = 0 for all b ∈ ℤ^{+}(p), b > 1, and α_{2b0} = 0 for all b ≡ 1 mod p.

Consider the fields isomorphism h : K ⟶ K′ such that h(t̃) = t′ and h|_{k} = σ^{−n_{0}} (recall that the residue fields of K and K′ are identified with that of K). Clearly, h gives a compatible with ramification fitrations isomorphism H from G to G′. Therefore, we obtain the induced isomorphism H̄ : Ḡ ⟶ Ḡ′ and the k-linear isomorphism h_{k} : M_{k}^{(1)} ⟶ M′_{k}^{(1)} such that h_{k}(D̃_{an}) = σ^{n_{0}}D′_{an} for all a ∈ ℤ^{+}(p) and n ∈ ℤ mod N_{0}.

Consider the group automorphism

F̄ = H̄ ∘ Ḡ^{−1} : Ḡ ⟶ Ḡ,

then for the corresponding k-linear map f_{k} : M_{k}^{(1)} ⟶ M_{k}^{(1)}, we have f_{k}(D̃_{an}) = σ^{n_{0}}C_{an} := C̃_{an}.

Because F̄ respects ramification filtration and by the above special choice of the special basis D^{(1)}(t̃), we can apply the Main Lemma from n.3, cf. Remark n.5.1. This gives the existence of β_{c,−m} ∈ k, where c ∈ ℤ^{+}(p) and 1 ⩽ m < N_{0}, such that for a ∈ ℤ^{+}(p), a < ṽ_{0} = v_{0}/p = ep^{M}/(p − 1), we have

C̃_{a0} = D̃_{a0} + ∑_{c,m} β_{c,−m}D̃_{ap^{m}+c,−m} mod M_{k}^{(ṽ_{0})}.

Remark. We agree to use the same notation for residue m ∈ ℤ mod N_{0} and for its lifting from the interval [0, N_{0}).

Lemma. For c ∈ ℤ^{+}(p) and 1 ⩽ m < N_{0}, there exist γ_{c,−m} ∈ k such that for a ∈ ℤ^{+}(p) ∩ [1, ṽ_{0}), it holds

D̃_{a0} = C̃_{a0} + ∑_{c,m} γ_{c,−m}C̃_{ap^{m}+c,−m} mod M_{k}^{(ṽ_{0})}.

Proof. It is easy to see that these elements γ_{c,−m} can be found uniquely from the following equations

0 = β_{c,−m} + γ_{c,−m} + ∑ γ_{c_{1},−m_{1}}σ^{−m_{1}}(β_{c_{2},−m_{2}}),

where the sum is taken for c_{1}, c_{2} ∈ ℤ^{+}(p) and m_{1}, m_{2} ∈ ℤ mod N_{0} such that c_{1}p^{m_{2}} + c_{2} = c, m_{1} + m_{2} = m and m_{2} ≠ 0.

5.5. For a ∈ ℤ^{+}(p) ∩ [1, ṽ_{0}), set

ε_{a} = 1 + ∑_{c,m} t̃^{c}σ^{m}(γ_{c,−m}),

where the sum is taken for natural numbers m and c < a such that v_{p}(a − c) = m (v_{p} is the p-adic valuation such that v_{p}(p) = 1).

Choose a power series P = P (X) ∈ W [[X]] such that P (π_{K}) ≡ ζ_{M} mod p. As in n.4.3 consider τ_{P} ∈ Aut^{1} K such that τ_{P} (t) = tP_{M}(t), where P_{M}(X) = P (X)^{p^{M}}.

Proposition. In the above notation we have the following congruence

∑ t̃^{−a}ε_{a}(τ_{0} ∗ C̃_{a0}) ≡ ∑ τ_{P}^{−1}(t̃^{−a}ε_{a})C̃_{a0} mod(σ − id)M_{K} + M ⊗ O_{K},

where the sum is taken for all a ∈ ℤ^{+}(p) ∩ [1, ṽ_{0}).

Proof. By n.1.5, we have

∑ t̃^{−a}(τ_{P} ∗ D̃_{a0}) ≡ ∑(τ_{P}^{−1}t̃)^{−a}D̃_{a0} mod(σ − id)M_{K} + M ⊗ O_{K},

where the sum is taken for a ∈ ℤ^{+}(p).

By n.4.3, we have τ_{P} ∗ D̃_{a0} = τ_{0} ∗ D̃_{a0} for all a ∈ ℤ^{+}(p) ∩ [1, ep^{M}).

By the use of the formula from Lemma n.5.4, we obtain the congruence

∑_{a∈ℤ^{+}(p)} t̃^{−a}(τ_{0} ∗ D̃_{a0}) ≡ ∑_{a∈ℤ^{+}(p)} t̃^{−a}(τ_{0} ∗ C̃_{a0})+

+ ∑_{\begin{matrix}a,c∈ℤ^{+}(p) \\ m∈ℕ \end{matrix}} t̃^{−ap^{m}}σ^{m}(γ_{c,−m})(τ_{0} ∗ C̃_{ap^{m}+c,0})

≡ ∑_{a∈ℤ^{+}(p)} t̃^{−a}ε_{a}(τ_{0} ∗ C̃_{a0}) mod(σ − id)M_{K} + M ⊗ O_{K} + M^{(ṽ_{0})} ⊗ K.

Similarly,

∑_{a∈ℤ^{+}(p)}(τ_{P}^{−1}t̃)^{−a}D̃_{a0} ≡ ∑_{a∈ℤ^{+}(p)} τ_{P}^{−1}(t̃^{−a}ε_{a})C̃_{a0} mod(σ−id)M_{K}+M⊗O_{K}+M^{(ṽ_{0})}⊗K.

The proposition is proved.

5.6. Consider P′(X) ∈ W [[X]] such that P′(π_{K′}) ≡ ζ′_{M} mod p. Set P′_{M}(X) = P′(X)^{p^{M}}, and consider τ_{P′} ∈ Aut^{1} K′ such that τ_{P′}(t′) = t′P′_{M}(t′). For a ∈ ℤ_{⩾0}, set

P′_{M}(X)^{a} mod p = ∑_{i⩾0} A′_{i}^{(a)}X^{ip^{M}} ,

where all coefficients A′_{i}^{(a)} ∈ k. We note, that

P′_{M}(X)^{p^{M}} ≡ 1 mod(p, X^{ep^{M}})

and, therefore, for all a ∈ [p^{M}, ep^{M}), we have A′_{i}^{(a−p^{M})} = A′_{i}^{(a)} (we note also that A′_{0}^{(a)} = 1, A′_{i}^{(a)} = 0 if 0 < i < e/(p − 1), and A′_{e/(p−1)}^{(a)} ≠ 0).

Proposition. For all a ∈ ℤ^{+}(p) ∩ [1, ep^{M}), we have

τ_{0} ∗ C̃_{a0} = ∑_{i⩾0} σ^{n_{0}}(A′_{i}^{(a)})C̃_{a+ip^{M} ,0} mod M_{k}^{(ep^{M})}.

Proof. By nn.4.3 and 1.5, we have

∑_{a∈ℤ^{+}(p)} τ_{P′}^{−1}(t^{′−a})D′_{a0} ≡ ∑_{a∈ℤ^{+}(p)}t^{′−a}(τ′_{0}∗D′_{a0}) mod(σ−id)M′_{K′}+M′⊗O_{K′}+M^{′(ep^{M})}⊗K.

This gives

τ′_{0} ∗ D′_{a0} ≡ ∑_{i⩾0} A_{i}^{′(a)}D′_{a+ip^{M} ,0} mod M_{k}^{′(ep^{M})}.

It remains only to note that

g_{k}(τ_{0} ∗ C_{a0}) ≡ τ′_{0} ∗ D′_{a0} mod M_{k}^{(v_{0})},

cf. n.5.3.3, and C̃_{a0} = σ^{n_{0}}C_{a0}.

5.7. Propositions of nn.5.5 and 5.6 give the following congruence

∑_{a} τ_{P}^{−1}(t̃^{−a}ε_{a})C̃_{a0} ≡ ∑_{a,i} t̃^{−a+ip^{M}} σ^{n_{0}}(A_{i}^{′(a)})ε_{a−ip^{M}} C̃_{a0} mod(σ − id)M_{K} + M ⊗ O_{K},

where a runs over the set ℤ^{+}(p) ∩ [1, ṽ_{0}) and i runs over ℤ_{⩾0}.

Because p^{N_{0}} > ṽ_{0}, this gives for all a ∈ ℤ^{+}(p) ∩ [1, ṽ_{0}), that

(3)  τ_{P}^{−1}(t̃^{−a}ε_{a}) ≡ ∑_{i⩾0} t̃^{−a+ip^{M}} σ^{n_{0}}(A_{i}^{′(a)})ε_{a−ip^{M}} mod O_{K}.

Lemma. For all a ∈ ℤ^{+}(p) ∩ (p^{M}, ṽ_{0}), it holds

t̃^{−a}ε_{a−p^{M}} ≡ t̃^{−a}ε_{a} mod σ^{M}K.

Proof follows immediately from the definition of ε_{a}, cf. n.5.5.

Now the congruence (3) gives for all a ∈ ℤ^{+}(p) ∩ [1, ṽ_{0})

(4)  τ_{P}^{−1}(t̃^{−a}ε_{a}) ≡ t̃^{−a}ε_{a}(σ_{∗}^{n_{0}}P′_{M})(t̃)^{a} mod(σ^{M}K + O_{K}),

where the power series σ_{∗}^{n_{0}}P′_{M} is obtained from the power series P′_{M} by applying σ^{n_{0}} to all its coefficients.

5.8. For m ∈ ℤ mod p^{M}, denote by N_{m} the k-submodules of t^{−(ṽ_{0}−1)}O_{K} generated by elements of the set

{t^{−a} | a ∈ ℕ ∩ [1, ṽ_{0}), a mod p^{M} = m}.

Clearly, these modules are τ_{P} -invariant and

t^{−(ṽ_{0}−1)}O_{K} mod O_{K} = ⊕_{m∈ℤ mod p^{M}}N_{m}.

For m ∈ ℤ mod p^{M}, denote by pr_{m} the projection from t^{−(ṽ_{0}−1)}O_{K} mod O_{K} to N_{m}. Clearly, if a ∈ ℤ^{+}(p) ∩ [1, ṽ_{0}) and m = a mod p^{M}, then

pr_{m}(t̃^{−a}ε_{a}) = t^{−a}σ^{M}b_{a} mod O_{K},

where b_{a} ∈ O_{K}^{∗}. In this notation we have

pr_{m}(τ_{P}^{−1}(t̃^{−a}ε_{a}) mod O_{K}) = τ_{P}^{−1}(pr_{m}(t̃^{−a}ε_{a}) mod O_{K}) = t^{−a}(σ^{M}b)P_{M}(t)^{a} mod O_{K}.

On the other hand,

pr_{m}(t̃^{−a}ε_{a}(σ_{∗}^{n_{0}}P′_{M})(t̃)^{a} mod O_{K}) = pr_{m}(t̃^{−a}ε_{a})(σ_{∗}^{n_{0}}P′_{M})(t̃)^{a} mod O_{K}

= t^{−a}(σ^{M}b_{a})(σ_{∗}^{n_{0}}P′_{M})(t̃)^{a} mod O_{K}.

Therefore, the congruence (4) gives

t^{−a}P_{M}(t)^{a} ≡ t^{−a}(σ_{∗}^{n_{0}}P′_{M})(t̃)^{a} mod O_{K}.

By taking a = ṽ_{0} − 1, we obtain from this congruence that

P_{M}(t) ≡ (σ_{∗}^{n_{0}}P′_{M})(t̃) mod t^{ṽ_{0}−1}O_{K},

and finally

P (t) ≡ (σ_{∗}^{n_{0}}P′)(t̃) mod t^{e/(p−1)}O_{K}.

Let θ(X) ∈ W [[X]] be such that t = θ(t̃). Consider P̃(X) = P (θ(X)) ∈ W [[X]]. Then P (t) = P̃(t̃) and we obtain the relation

P̃(t̃) ≡ (σ_{∗}^{n_{0}})P′(t̃) mod t^{e/(p−1)}O_{K}.

Introduce new uniformiser π̃_{K} ∈ K by the relation π_{K} = θ(π̃_{K}), then P̃(π̃_{K}) ≡ ζ_{M} mod p, and

(5)  (σ_{∗}^{n_{0}}P′)(π̃_{K}) ≡ ζ_{M} mod p^{1/(p−1)}.

On the other hand, by definition we have that

(6)  P′(π_{K′}) ≡ ζ′_{M} mod p.

Let ϕ ∈ Gal(ℚ_{p}(ζ_{M})/ℚ_{p}) be such that ϕ(ζ_{M}) = ζ′_{M}. Denote by ϕ̃ the isomorphism from O_{K} mod p^{1/(p−1)} to O_{K′} mod p^{1/(p−1)} uniquely determined by conditions ϕ̃(π̃_{K}) = π_{K′} and ϕ̃|_{k} = σ^{−n_{0}}. Then the above formulae (5) and (6) imply that ϕ̃ is a ϕ-linear isomorphism of ℤ_{p}[ζ_{M}]-algebras.

The theorem is proved.

References

[Ab1] V.A. Abrashkin, Ramification filtration of the Galois group of a local field. II, Proceeding of Steklov Math. Inst. 208 (1995).

[Ab2] V.A.Abrashkin, Ramification filtration of the Galois group of a local field. III, Preprint MPI 96-103 (to appear in Izvestiya RAN, ser. matem.).

[Ab3] V.A.Abrashkin, One group-theoretic property of the ramification filtration,, Preprint MPI 96-153 (to appear in Izvestiya RAN, ser. matem.).

[De] P.Deligne, Les corps locaux de caracterisque p, limites de corps locaux de caracterisque 0., Representations des groups reductifs sur un corps local. Travaux en Cours., Paris:Hermann. 1984, p. 120-157.

[Go] N.L.Gordeev, Infinity of the number of relations in the Galois group of the maximal pextension of a local field with bounded ramification, Izvestia AN SSSR, Ser. Matem. 45 (1981), 592-607.

[Ma] E.Maus, Relationen in Verzweigungsgruppen, J. reine ang. Math. 258 (1973), 23-50.

[Po] F. Pop, Glimpses of Grothendieck’s anabelian geometry, London Mathematical Society, Lecture Note Series 242, 113-126.

[Se] J.-P. Serre, Corps locaux, Hermann, Paris, 1968.

[Wi] J.-P. Wintenberger, Le corps des normes de certaines extensions infinies des corps locaux; application, Ann. Sci. Ec. Norm. Super., IV. Ser 16 (1983), 59-89.

Steklov Math. Institute, ul. Gubkina 8, Moscow 117 966, Russia

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