A STRONG PROPERTY OF RAMIFICATION FILTRATION. II

Victor A. Abrashkin

For two finite extensions of the p-adic field ℚ_{p} in the same algebraic closure ℚ̄_{p}, where p ⩾ 5, it is proved that any isomorphism of their Galois groups respecting ramification filtrations comes from conjugation by some element of Gal(ℚ̄_{p}/ℚ_{p}). In particular, such field E can be uniquely recovered from the abstract group theoretic structure of its Galois group Γ_{E} together with ramification filtration {Γ_{E}^{(v)}}_{v⩾0}.

0. Introduction.

Throughout all this paper p is a prime number, we assume that an algebraic closure ℚ̄_{p} of the field of p-adic numbers ℚ_{p} is chosen, and if E is a finite extension of ℚ_{p} in ℚ̄_{p}, then we use the notation Γ_{E} = Gal(ℚ̄_{p}/E). We denote usually by ℤ^{+}(p) the set of natural numbers relatively prime to p.

0.1. Consider the category F_{ℚ_{p}} of finite extensions E/ℚ_{p} in ℚ̄_{p} such that for any E_{1}, E_{2} ∈ F_{ℚ_{p}}, the set of morphisms Hom_{F_{ℚp}} (E_{1}, E_{2}) consists of τ ∈ Γ_{ℚ_{p}} such that τ(E_{1}) ⊂ E_{2}, what is equivalent to the inclusion Γ_{E_{2}} ⊂ τ^{−1}Γ_{E_{1}}τ.

Let P be the category of profinite finitely generated groups, such that for any two its objects Γ_{1} and Γ_{2}, Hom_{P}(Γ_{1}, Γ_{2}) consists of continuous monomorphisms f : Γ_{1} ⟶ Γ_{2} such that [Γ_{2} : f(Γ_{1})] < ∞.

Consider the functor P G : F_{ℚ_{p}} ⟶ P such that for any E ∈ F_{ℚ_{p}}, P G(E) = Γ_{E}. This functor is very far from to be fully faithful (but in the case of the category of algebraic number fields F_{ℚ} its analogue is fully faithful by the Neukirch-Uchida-Iwasawa-Ikeda theorem).

0.2. The Galois group Γ_{E} of E ∈ F_{ℚ_{p}} has the additional structure given by its normal closed ramification subgroups Γ_{E}^{(v)}, v ⩾ −1. We note that Γ_{E}^{(−1)} = Γ_{E}, Γ_{E}^{(v)} = Γ_{E}^{(0)} for −1 < v ⩽ 0, this filtration is left-continuous and has very rich structure (e.g. any positive rational number appears as a jump of this filtration).

If τ ∈ Hom_{F_{ℚp}} (E_{1}, E_{2}), then τ induces the continuous group monomorphism τ^{∗} : Γ_{E_{2}} ⟶ Γ_{E_{1}} (given by the correspondence g ↦ τgτ^{−1} for g ∈ Γ_{E_{2}}), [Γ_{E_{1}} : τ^{∗}(Γ_{E_{2}})] < ∞, and τ^{∗} is compatible with ramification filtrations in the following sense, for v ⩾ 0 one has τ^{∗}(Γ_{E_{2}}) ∩ Γ_{E_{1}}^{(v)} = τ^{∗}(Γ_{E_{2}}^{(x)}), where v = φ_{E_{2}/τ(E_{1})}(x) is Herbrand’s function of the extension E_{2}/τ(E_{1}).

One can imitate the above properties of ramification filtration to define the category ΦP of profinite groups with “ramification” filtration. Then for any E ∈

Typeset by AMS-TEX

The author had a nice opportunity to work on this subject during his stay at the MPI, Bonn (the spring of 1998) and expresses his gratitude to this organisation for hospitality. He is also very grateful to Prof. F.Pop for helpful discussions resulted in the essential clarification of the final statement.

F_{ℚ_{p}}, the correspondence E ↦ {Γ_{E}^{(v)}}_{v⩾−1} gives the functor ΦG : F_{ℚ_{p}} ⟶ ΦP. The main result of this paper is the fully faithfulness of this functor if p ⩾ 5. We prove this result in the following equivalent form.

Theorem. Let p ⩾ 5, E, E′ ∈ F_{ℚ_{p}} and let F : Γ_{E} ⟶ Γ_{E′} be a continuous group isomorphism such that for all v ⩾ 0, F (Γ_{E}^{(v)}) = Γ_{E′}^{(v)}. Then there exists τ ∈ Γ_{ℚ_{p}} such that for all g ∈ Γ_{E}, one has F (g) = τ^{−1}gτ.

The proof of this theorem uses a slight development of arguments from [Ab], where it was shown that a field E ∈ F_{ℚ_{p}} can be uniquely recovered from the group Γ_{E} together with some segment of its ramification filtration under the assumption that the field E contains “sufficiently many” roots of unity (i.e. if its residue field is sufficiently big and it contains a primitive root of unity of degree p^{M} for a sufficiently large M ∈ ℕ).

1. An auxiliary statement.

1.1. Let K ∈ F_{ℚ_{p}}. Denote its residue field by k and absolute ramification index by e_{K}. In this section we choose an algebraic closure K̄ of K and shall use the notation Γ = Γ_{K}(p) for the Galois group of the maximal p-extension of K in K̄. We denote by {Γ^{(v)}}_{v⩾1} the ramification filtration of the group Γ in upper numbering. Suppose that p ⩾ 5, a primitive root ζ_{M} of degree p^{M} of unity, where M ∈ ℕ, belongs to K, and f_{K} = [k : 𝔽_{p}] ⩾ log_{p}(p^{2}v_{K}), where v_{K} = e_{K}p^{M+1}/(p − 1).

Let a_{K} = e_{K}(p/(p − 1) + M) and assume that for some finite extension K′ of ℚ_{p}, there exists a group isomorphism

G : Γ/Γ^{(a_{K})} ⟶ Γ′/Γ′^{(a_{K})},

which is compatible with induced ramification filtrations (here Γ′ = Γ_{K′}(p) is the Galois group of the maximal p-extension of K′ and Γ′^{(a_{K})} is the corresponding higher ramification subgroup). Under the above assumptions we proved in [Ab] that the simplest invariants of the fields K and K′ coincide (i.e. e_{K} = e_{K′}, f_{K} = f_{K′} and ζ_{M} ∈ K′) and constructed for some ϕ_{M} ∈ Gal(ℚ_{p}(ζ_{M})/ℚ_{p}), a ϕ_{M}-linear isomorphism of ℤ_{p}[ζ_{M}]-algebras

ϕ̃: O_{K} mod p^{1/(p−1)} ⟶ O_{K′} mod p^{1/(p−1)}.

Set f_{K} = N_{0}, v_{K} = v_{0}, e_{K} = e, a_{K} = a_{0} and denote by U_{K}^{(a_{0})} (resp., U_{K′}^{(a_{0})}) the subgroup of principal units of K (resp., K′) congruent to 1 modulo the a_{0}^{th} power of the maximal ideal of its valuation ring. Taking the group isomorphism G on the level of maximal abelian quotients and using the reciprocity map of local class field theory we obtain the group isomorphism

G^{ab} : U_{K}^{(1)} mod U_{K}^{(a_{0})} ⟶ U_{K′}^{(1)} mod U_{K′}^{(a_{0})}.

We note that e ⩾ e_{ℚ_{p}(ζ_{M})} = p^{M−1}(p − 1) and therefore, a_{0} > p^{M} + 1.

Proposition. If M ⩾ 2 then the isomorphism

G^{ab} mod U_{K}^{(1)}^{p}U_{K}^{(p^{M} +1)} : U_{K}^{(1)}/U_{K}^{(1)}^{p}U_{K}^{(p^{M} +1)} ⟶ U_{K′}^{(1)}/U_{K′}^{(1)}^{p}U_{K′}^{(p^{M} +1)},

is induced by ϕ̃.

Proof. The proof of this proposition is based completely on contructions from the proof of the main theorem from [Ab], which gives the existence of the above ϕ_{M}- linear isomorphism ϕ̃. We start in nn.1.2-1.6 by recalling all necessary concepts and notation from its construction.

1.2. Fix an algebraic closure K̄ of K, a primitive root ζ_{M} of degree p^{M} of unity, a uniformiser π of K and a sequence {π_{s}}_{s⩾0} of elements of K̄ such that π_{s+1} = π_{s} for all s and π_{0} = π. Then we have a fundamental exact sequence

(1)  1 ⟶ G/G^{(v_{0})} ⟶^{i} Γ/Γ^{(a_{0})} ⟶^{j} ℤ/p^{M}ℤτ_{0} ⟶ 0,

where:

a) ℤ/p^{M}ℤτ_{0} = Gal(K_{M}/K) with K_{M} = K(π_{M}) and τ_{0} is uniquely determined by the condition τ_{0}π_{M} = ζ_{M}π_{M};

b) the epimorphism j is induced by the natural projection of the Galois group Γ to Gal(K_{M}/K);

c) G is the Galois group of the maximal p-extension of the complete discrete valuation field K = X (K̃) of characteristics p in its separable closure K_{sep} = X (K̄), where X is the field-of-norms functor and K̃ = K({π_{s} | s ⩾ 0}). We note that the residue field of K is canonically identified with k and K has the “special” uniformiser t coming from the sequence {π_{s}}_{s⩾0};

d) the embedding i is induced by the identification Gal(K_{sep}/K) = Gal(K̄/K̃) coming from the construction of the field-of-norms functor. By standard properties of this functor the morphism i is compatible with ramification filtrations (corresponding upper ramification numbers are related via Herbrand’s function of the infinite extension K̃/K).

In [Ab] it was proved that the isomorphism G : Γ/Γ^{(a_{0})} ⟶ Γ′/Γ′^{(a_{0})} maps the above fundamental exact sequence (1) to some fundamental exact sequence for the field K′ (this field is assumed to be a subfield of K̄). In other words, there exist a primitive root ζ′_{M} of degree p^{M} of unity, a uniformiser π′ of K′, a sequence {π′_{s}}_{s⩾0} ⊂ K̄ such that π′_{s+1} = π′_{s} for all s and π′_{0} = π′, which give rise to the fundamental exact sequence for the field K′

1 ⟶ G′/G′^{(a_{0})} ⟶^{i′} Γ′/Γ′^{(a_{0})} ⟶^{j′} ℤ/p^{M}ℤτ′_{0} ⟶ 0,

where G′ is the Galois group of the maximal p-extension of K′ = X (K̃′) in X (K̄) and K̃′ is a composit of all K′(π′_{s}). The choice of the above data could be done to provide us with existence of two group isomorphisms G_{1} : G/G^{(v_{0})} ⟶ G′/G′^{(v_{0})} and G_{2} : ℤ/p^{M}ℤτ_{0} ⟶ ℤ/p^{M}ℤτ′_{0} such that G_{1} ∘ i′ = i ∘ G, j ∘ G_{2} = G ∘ j′ and G_{2}(τ_{0}) = τ′_{0} (where τ′_{0} is uniquely determined by the condition τ′_{0}(π′_{M}) = π′_{M}ζ′_{M}). We note that G_{1} is compatible with ramification filtrations of the Galois groups G and G′. We shall denote by t′ the uniformiser of the field K′ coming from the above sequence {π′_{s}}_{s⩾0}.

1.3. Consider the pro-p-group K^{∗}/K^{∗}^{p} as a profinite 𝔽_{p}-module. Introduce kmodule M_{k} = K^{∗}/K^{∗}^{p} ⊗ k together with its k-submodules M_{k}^{(a)}, where a ∈ ℕ, generated by the images of principal units of K congruent to 1 modulo t^{a}. All these modules have the natural semilinear action of Frobenius coming from the absolute Frobenius morphism of the field k. Consider the map

pr_{k}^{ab} : G/G^{(v_{0})} ⟶ (G/G^{(v_{0})})^{ab} ⊗ k = M_{k} mod M_{k}^{(v_{0})}

given by the composition of projection to maximal abelian quotient with extension of scalars. Consider the push down of the fundamental exact sequence (1) via this morphism pr_{k}^{ab}

0 ⟶ M_{k} mod M_{k}^{(v_{0})} ⟶ pr_{k∗}^{ab} (Γ/Γ^{(a_{0})}) ⟶ ℤ/p^{M}ℤτ_{0} ⟶ 0.

This exact sequence provides us with a k-linear automorphism m ↦ τ_{0} ∗ m of M_{k} mod M_{k}^{(v_{0})} given by conjugation by any lifting of τ_{0}. The following properties come from Artin-Schreier theory and provide us with an opportunity to compute this automorphism on the k-submodule M_{k}^{(1)} mod M_{k}^{(v_{0})}.

a) Existence of special bases. Any choice of uniformising element t̃of K gives rise to the special basis D^{(1)}(t̃) = {D̃_{an} | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}} of the k-module M_{k}^{(1)} (recall that we set f_{K} = N_{0}, i.e. k ≃ 𝔽_{p^{N_{0}}} ) . The definition of the special basis can be done as follows. Consider the identification U_{K}^{(1)} = ∏_{a∈ℤ^{+}(p)} W_{a}, where W_{a} = W is the additive group of Witt vectors with coefficients in k. This identification depends only on the choice of t̃ and is given by the correspondence

u ↦ {w_{a} |w_{a} ∈ W, a ∈ ℤ^{+}(p)},

where u = ∏_{(a,p)=1} E(w_{a}, t̃^{a}) and E(w, X) = exp(wX +⋯+σ^{n}(w)X^{p^{n}}/p^{n}+…) ∈ W [[X]]. Then

M_{k}^{(1)} = U_{K}^{(1)} ⊗ k = ∏_{a,n}(k)_{an},

where a runs over ℤ^{+}(p) and n runs over ℤ mod N_{0}. By definition, the element D̃_{an} ∈ M_{k}^{(1)} of the special basis D^{(1)}(t̃) comes from 1 ∈ k = (k)_{an}.

b) Relation between different special bases. Let

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

be the special basis of M_{k}^{(1)} corresponding to the uniformiser t from n.1.2 c). Then the relation between elements of the special bases D^{(1)}(t) and D^{(1)}(t̃) is given via the following formal congruence in M_{k}^{(1)} ⊗_{k} K

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

c) Action of analytical automorphisms. Consider the group Aut^{1} K of “analytical” automorphisms of the field K, i.e. the group of automorphisms ψ of K such that ψ|_{k} = id. Conjugation by any lifting of ψ to Aut K_{sep} induces the k-linear automorphism m ↦ ψ ∗ m of M_{k}^{(1)}. In terms of the special basis D^{(1)}(t̃) this automorphism can be computed via the following formal congruence in M_{k}^{(1)} ⊗_{k} K

∑_{a∈ℤ^{+}(p)} ψ^{−1}(t̃)^{−a}D̃_{a0} ≡ ∑_{a∈ℤ^{+}(p)} t̃^{−a}(ψ ∗ D̃_{a0}) mod(σ − id) (M_{k}^{(1)} ⊗ K) + M_{k}^{(1)} ⊗ O_{K}.

d) Computaion of elements τ_{0}∗m. Let P = P (X) ∈ W [[X]] be such that P (π) ≡ ζ_{M} mod p. Set P_{M} = P^{p^{M}} and consider τ_{P} ∈ Aut^{1} K given by the correspondence t ↦ tP_{M}(t). Then for any m ∈ M_{k}^{(1)}, we have τ_{P} ∗ m ≡ τ_{0} ∗ m mod M_{k}^{(w_{0})}, where w_{0} = ep^{M} (we note that v_{0} > w_{0} > v_{0}/p).

1.4. Consider the group isomorphism G_{1} from n.1.2. It can be given by a group isomorphism G′ : G ⟶ G′ such that for any v ∈ [1, v_{0}], we have G′(G^{(v)}) = G′^{(v)}. Denote by g′_{k} the k-linear morphism M_{k}^{(1)} ⟶ M′_{k}^{(1)} induced by G′. Then for any m ∈ M_{k}^{(1)}, we have

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

Take the uniformiser t′ of K′ from n.1.2 and consider the special basis D^{(1)}(t′) = {D′_{an} | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}} of M′_{k}^{(1)}. For all a ∈ ℤ^{+}(p) and n ∈ ℤ mod N_{0}, set C_{an} = g_{k}^{−1}(D′_{an}). These elements create a basis of M_{k}^{(1)} and this basis can be approximated by some uniquely determined special basis. More precisely, there exist a uniformiser t̃ of K and n_{0} ∈ ℤ mod N_{0} such that if D^{(1)}(t̃) = {D̃_{an} | a ∈ ℤ^{+}(p), n ∈ ℤ mod N_{0}}, then for any a ∈ ℤ^{+}(p),

σ^{n_{0}}C_{a0} = ∑_{b,m} α_{abm}D̃_{bm},

where the sum is taken for all b ∈ ℤ^{+}(p) such that b ⩾ min{a, v_{0}} and all m ∈ ℤ mod N_{0}, and all coefficients α_{abm} ∈ k satisfy the properties α_{110} = 1, α_{1b0} = 0 for all b > 1, and α_{2b0} = 0 for b ≡ −1 mod p (these properties of coefficients α_{abm} provide us with uniquiness of this special basis).

Consider the field isomorphism h : K ⟶ K′ uniquely given by conditions h(t̃) = t′ and h|_{k} = σ^{−n_{0}}. Then h induces the isomorphism H of the Galois groups G and G′. It induces also 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}.

Introduce the automorphism F = H ∘ G′^{−1} of the group G. Clearly, F respects the segment {G^{(v)}}_{1⩽v⩽v_{0}} of the ramification filtration. Denote by f_{k} the k-linear automorphism of M_{k}^{(1)} induced by F, and set f_{k}(D̃_{an}) = σ^{n_{0}}C_{an} := C̃_{an}. The Main Lemma from [Ab] shows that f_{k} must have very special form modulo M_{k}^{(ṽ_{0})}, where ṽ_{0} = v_{0}/p. More precisely, for c ∈ ℤ^{+}(p) and m ∈ ℕ, there exist γ_{c,−m} ∈ k such that for all a ∈ ℤ^{+}(p),

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

1.5. The above relation (3) together with the formalism of Artin-Schreier theory from n.1.3 allow us to relate k-linear automorphisms m ↦ τ_{o}∗m of M_{k}^{(1)} mod M_{k}^{(ṽ_{0})} and m ↦ τ′_{0} ∗ m of M′_{k}^{(1)} mod M′_{k}^{(ṽ_{0})}.

Namely, we can apply the properties 1.3 c) and d) to express the map m ↦ τ_{0}∗m in terms of the special basis D^{(1)}(t̃). In order to express the map m ↦ τ′_{0} ∗ m in terms of the special basis D′^{(1)}(t′), introduce a power series P′ = P′(X) ∈ W [[X]] such that P′(π′) = ζ′_{M} mod p, set P′_{M} = P′^{p^{M}} and consider τ_{P′} ∈ Aut^{1} K′ given by the correspondence t′ ↦ t′P′_{M}(t′). Then we use τ_{P′} to describe explicitly the k-linear morphism m ↦ τ′_{0} ∗ m, where m ∈ M′_{k}^{(1)} mod M′_{k}^{(ṽ_{0})}, in terms of the special basis D^{(1)}(t′). These formulae are transformed to the action of τ_{0} in terms of elements C̃_{an} by the relation (2) of n.1.4. Finally, the relation (3) provides us with the following clue property.

If a ∈ ℤ^{+}(p) ∩ [1, ṽ_{0}) and

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

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

(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}).

1.6. For m ∈ ℤ mod p^{M}, denote by N_{m} the k-submodules of t^{−(ṽ_{0}−1)}O_{K} mod 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 we have the following τ_{P} -invariant decomposition

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}.

For m ∈ ℤ mod p^{M} and a ∈ ℤ^{+}(p) ∩ [1, ṽ_{0}) denote by b(a, m) the minimal nonnegative integer such that

pr_{m}(t̃^{−a}ε_{a}) ∈ t^{−b(a,m)}O_{K} mod O_{K}.

We note that if b(a, m) ≠ 0, then b(a, m) mod p^{M} = m, and if m = a mod p^{M} then b(a, m) = a.

If m ≠ 0 mod p^{M}, then the formula (4) of n.1.5 implies that

τ_{P}^{−1}(pr_{m}(t̃^{−a}ε_{a}) mod O_{K}) = pr_{m}(t̃^{−a}ε_{a})(σ_{∗}^{n_{0}}P′_{M})(t̃)^{a} mod O_{K}.

But τ_{P}^{−1} acts by multiplication by P_{M}(t)^{m} on N_{m}, therefore,

(σ_{∗}^{n_{0}}P′_{M})(t̃)^{a} ≡ P_{M}(t)^{m} mod t^{b(a,m)}

for all m ∈ ℤ mod p^{M} such that m ≠ 0 mod p^{M} and all a ∈ ℤ^{+}(p) ∩ [1, ṽ_{0}).

If we take a = ṽ_{0} − 1 and m = a mod p^{M} = −1 mod p^{M} (we use that M ⩾ 2 and therefore, ṽ_{0} ≡ 0 mod p^{M}), then the above congruence gives

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

and as in [Ab, n.5] we obtain

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

Now we note that the ℤ_{p}[ζ_{M}]-algebra O_{K} mod p^{1/(p−1)} is isomorphic to the algebra O_{K} mod t^{e/(p−1)}, with action of ζ_{M} induced by analytical automorphism of K such that t ↦ tP (t). Analogously, the ℤ_{p}[ζ_{M}]-algebra O_{K′} mod p^{1/(p−1)} is isomorphic to the algebra O_{K′} mod t′^{e/(p−1)} with action of ζ′_{M} given by analytical automorphism of K′ such that t′ ↦ t′P′(t′). Now the relation (6) implies that the fields isomorphism h : K ⟶ K′ from n.1.4 induces ϕ_{M}-linear automorphism of ℤ_{p}[ζ_{M}]-algebras O_{K} mod t^{e/(p−1)} and O_{K′} mod t′^{e/(p−1)}, where ϕ_{M}(ζ_{M}) = ζ′_{M}.

This also implies that the statement of our proposition is equivalent to the congruence

D̃_{a0} ≡ C̃_{a0} mod M_{k}^{(p^{M} +1)},

for all a ∈ ℤ^{+}(p) ∩ [1, p^{M} + 1).

1.7. The congruence (5) of n.1.6 implies that for all a ∈ ℤ^{+}(p) ∩ [1, ṽ_{0}) and all m ∈ ℤ mod p^{M} such that m ≠ 0 mod p^{M}, we have

P_{M}(t)^{a} = P_{M}(t)^{m} mod t^{b(a,m)}.

Because P_{M}(t) = 1 + At^{e/(p−1)} + … with A ≠ 0 and m ≡ b(a, m) mod p^{M} if b(a, m) ≠ 0, this gives for any a ∈ ℤ^{+}(p) ∩ [1, ṽ_{0}) and any m ∈ ℤ mod p^{M} such that m ≠ 0 mod p^{M}, the inequality

b(a, m) ⩽ ep^{s+1}/(p − 1),

where s = v_{p}(a − b(a, m)). In particular,

(7)  pr_{m}(t̃^{−a}ε_{a}) ∈ t^{−ep/(p−1)}O_{K} mod O_{K},

if m ≢ a mod p, but m ≠ 0 mod p^{M}.

1.8. Consider a presentation of t̃ in the form

t̃= αt(1 + Q(t) + βt^{r} + …) mod tṽ_{0}O_{K},

where α, β ∈ k^{∗}, Q ∈ Xk[X] is such that deg Q < r and r ∈ ℤ^{+}(p).

Lemma. In the above notation we have

r ⩾ ṽ_{0} − 1 − ep/(p − 1).

Corollary. For all a ∈ ℤ^{+}(p)∩[1, ṽ_{0}) and m ∈ ℤ mod p^{M}ℤ such that m ≡ 0 mod p, we have

pr_{m}(t̃^{−a}) ∈ t^{−ep/(p−1)}O_{K} mod O_{K}.

Proof of lemma. Clearly,

t̃^{−ṽ_{0}+1} = α_{1}t^{−ṽ_{0}+1}(1 + Q_{1}(t)^{p} + β_{1}t^{r} + …) mod O_{K},

where α_{1}, β_{1} ∈ k^{∗}, Q_{1} ∈ Xk[X], deg Q_{1} < r.

If r + 1 ≠ 0 mod p, then for m = −(r + 1) mod p^{M}, we have

pr_{m}(t̃^{−ṽ_{0}+1}) = α_{1}β_{1}t^{−ṽ_{0}+1+r} + … mod O_{K}.

Because m ≢ (ṽ_{0} − 1) mod p, the formula (7) of n.1.7 gives

pr_{m}(t̃^{−ṽ_{0}+1}ε_{ṽ_{0}−1}) ∈ t^{−ep/(p−1)}O_{K} mod O_{K}.

Note that m ≢ 0 mod p implies that pr_{m}(t̃^{−ṽ_{0}+1}) = pr_{m}(t̃^{−ṽ_{0}+1}ε_{ṽ_{0}−1}). Therefore, −(ṽ_{0} − 1) + r ⩾ −ep/(p − 1), i.e. the lemma is proved in the case r + 1 ≠ 0 mod p.

If r + 1 ≡ 0 mod p, then we can proceed similarly, to prove for m = −(r + 2) mod p^{M} that pr_{m}(t̃^{−ṽ_{0}+2}) = α_{2}β_{2}t^{−ṽ_{0}+2+r} + … mod O_{K}. This gives r ⩾ ṽ_{0} − 2 − ep/(p − 1), but r ≡ −1 mod p and ṽ_{0} ≡ 0 mod p imply that r ⩾ ṽ_{0} − 1 − ep/(p − 1). The lemma is proved.

1.9. From nn.1.7 and 1.8 it follows that for all a ∈ ℤ^{+}(p) ∩ [1, ṽ_{0}) and m ∈ ℤ mod p^{M} such that m ≠ 0 mod p^{M} and m ≡ 0 mod p, we have

pr_{m}(t̃^{−a}ε_{a} − t̃^{−a}) ∈ t^{−ep/(p−1)}O_{K}.

For trivial reasons we have pr_{m}(t̃^{−a}ε_{a} − t̃^{−a}) = 0, if m ≢ 0 mod p.

Thus, we obtained that for all a ∈ ℤ^{+}(p) ∩ [1, ṽ_{0}),

t̃^{−a} ∑_{c,m} σ^{m}(γ_{c,−m})t̃^{c} ∈ t^{−ep/(p−1)}O_{K} + σ^{M}K,

where the sum is taken for 1 ⩽ m < M and c ∈ ℤ^{+}(p) such that v_{p}(a − c) = m. This gives the following property of elements γ_{c,−m}

if 1 ⩽ m < M and there exists a_{1} ∈ (c, ṽ_{0} − ep/(p − 1)) such that v_{p}(a_{1} − c) = m, then γ_{c,−m} = 0.

Assume that c ∈ ℤ^{+}(p) and 1 ⩽ m < M. If there exists a ∈ ℤ^{+}(p) such that ap^{m}+c ⩽ a^{∗}, where a^{∗} = min{p^{M} +1, ṽ_{0}−ep/(p−1)}, then we can use a_{1} = ap^{m}+c to deduce that γ_{c,−m} = 0 from the above property of these elements. Therefore, D̃_{a0} = C̃_{a0} for all a ∈ ℤ^{+}(p) ∩ [1, a^{∗}). It remains only to note that for M ⩾ 2,

ṽ_{0} − ep/(p − 1) = ep(p^{M−1} − 1)/(p − 1) ⩾ e_{ℚ_{p}(ζ_{M})}p(p^{M−1} − 1)/(p − 1) > p^{M} + 1,

i.e. a^{∗} = p^{M} + 1. The proposition is proved.

2. Proof of the theorem.

2.1. Let Ẽ ∈ F_{ℚ_{p}} be a Galois extension of the field E. Then ΓẼ ⊂ Γ_{E} and F (ΓẼ) = ΓẼ′ ⊂ Γ_{E′}, where Ẽ′ is a finite Galois extension of E′. Then F induces the group isomorphism FẼ : Γ_{Ẽ/E} ⟶ Γ_{Ẽ′/E′}.

Because the centraliser of the subgroup Γ_{E} in Γ_{ℚ_{p}} is trivial, our theorem is implied by the following proposition.

Proposition. There exists a field isomorphism fẼ : Ẽ ⟶ Ẽ′ such that fẼ(E) = E′ and for any g ∈ Γ_{Ẽ/E} one has FẼ(g) = f −1 ; ẼgfẼ.

Proof of proposition. Without loss of generality we can assume in the proof (cf. nn.2.2-2.6 below) that Ẽ is a finite Galois extension of ℚ_{p}.

2.2. For s ∈ ℤ_{⩾0} set Ẽ_{s} = Ẽ(ζ_{s}), where ζ_{s} is as usually a primitive root of unity of degree p^{s}. Denote by Ẽ_{∞} the union of all Ẽ_{s}. Then Ẽ_{∞} is an infinite strict arithmetical profinite extension of ℚ_{p}. The following two properties a) and b) of such extensions are well-known, cf. [Wi].

a) For x ⩾ 0, there exists a limit φ_{∞}(x) = lim_{s→∞}φ_{Ẽ_{s}/ℚ_{p}}(x), where for any s ∈ ℤ_{⩾0},

φ_{Ẽ_{s}/ℚ_{p}} is Herbrand’s function of the extension Ẽ_{s}/ℚ_{p}. For all indices s, there exist C_{s} ∈ ℝ such that lim_{s→∞}C_{s} = +∞ and for any x ∈ [0, C_{s}] and u ⩾ s, one has φ_{∞}(x) = φ_{Ẽ_{u}/ℚ_{p}}(x).

We apply this property to prove the following statement.

Lemma. There exists an index s_{0} and a number x̃_{0} > 0 such that for any s ⩾ s_{0} and any x > x̃_{0}, the ramification subgroup (in lower numbering) Γ_{Ẽ_{s}/ℚ_{p},x} acts

trivially on Ẽ.

Proof. Let ṽ_{0} = v(Ẽ/ℚ_{p}) be the maximal upper ramification number for the extension Ẽ/ℚ_{p} (recall that ṽ_{0} is the maximal number such that Γ_{ℚ_{p}}^{(ṽ_{0})} acts nontrivially

on Ẽ). Let x̃_{0} ⩾ 0 be such that ṽ_{0} = φ_{∞}(x̃_{0}) and let an index s_{0} be such that C_{s} > x̃_{0} for all s ⩾ s_{0}. If s ⩾ s_{0} and x > x̃_{0}, then x_{s} := min{x, C_{s}} > x̃_{0} and

Γ_{Ẽ_{s}/ℚ_{p},x} = Γ_{Ẽ_{s}/ℚ_{p}}^{(φ_{s}(x))} ⊂ Γ_{Ẽ_{s}/ℚ_{p}}^{(φ_{s}(x_{s}))} = Γ_{Ẽ_{s}/ℚ_{p}}^{(φ_{∞}(x_{s}))}

acts trivially on Ẽ because φ_{∞}(x_{s}) > φ_{∞}(x̃_{0}) = ṽ_{0}. The lemma is proved.

b) Let Ẽ = X (Ẽ_{∞}) and K_{0} = X ( ⋃_{s⩾0}ℚ_{p}(ζ_{s})), where X is the field-of-norms

functor. Then K_{0} is a complete discrete valuation field, E is its finite Galois extension, and there exists s_{1} such that for all s ⩾ s_{1}, Herbrand’s functions coincide φ_{Ẽ_{s}/ℚ_{p}(ζ_{s})} = φ_{Ẽ/K_{0}}.

In particular, if s ⩾ s_{1} then [Ẽ: K_{0}] = [Ẽ_{s} : ℚ_{p}(ζ_{s})], the ramification indices of all extensions Ẽ_{s}/ℚ_{p}(ζ_{s}) are equal to the ramification index ẽ of the extension Ẽ/K_{0}, and we have the equality of the largest upper ramification numbers v(Ẽ/K_{0}) = v(Ẽ_{s}/ℚ_{p}(ζ_{s})).

2.3. For any index s ⩾ s_{1}, choose an unramified extension L_{s} of E_{s} such that if k_{s} is the residue field of L_{s} then [k_{s} : 𝔽_{p}] ⩾ 2(s + 1) + log_{p} ẽ. We note that the field L_{s} satisfies the assumptions for the field K from n.1.1 with M = s. Indeed, log_{p}(p^{2}v_{L_{s}}) = 2(s+1)+log_{p} ẽ, because v_{L_{s}} = e_{L_{s}}p^{s+1}/(p−1) and e_{L_{s}} = ẽp^{s−1}(p−1).

If F (Γ_{L_{s}}) = Γ_{L′_{s}}, then L′_{s} is a finite Galois extension of E′, F induces the group isomorphisms

F_{L_{s}} : Γ_{L_{s}/E} ⟶ Γ_{L′_{s}/E′},
F_{s} : Γ_{L_{s}} ⟶ Γ_{L′_{s}},

and these isomorphisms are compatible with ramification filtrations.

2.4. By the above construction, the fields L_{s} are unramified over Ẽ_{s}, therefore, v(L_{s}/ℚ_{p}(ζ_{s})) = v(E_{s}/ℚ_{p}(ζ_{s})) = v(Ẽ/K_{0}) = ṽ do not depend on the index s ⩾ s_{1}. Let an index s_{2} ⩾ s_{1} be such that p^{s_{2}−1} − 1 > ṽ. By the main result of [Ab], cf. n.1.1, we have for all s ⩾ s_{2}, a ϕ_{s}-linear isomorphism of ℤ_{p}[ζ_{s}]-algebras

ϕ̃_{s} : O_{L_{s}} mod p^{1/(p−1)} ⟶ O_{L′_{s}} mod p^{1/(p−1)},

where ϕ_{s} is some automorphism of the field ℚ_{p}(ζ_{s}).

Because p^{s−1} − 1 > ṽ, this implies for all s ⩾ s_{2}, the existence of field isomorphisms f_{s} : L_{s} ⟶ L′_{s} such that f_{s} and ϕ̃_{s} induce the same map

O_{L_{s}} mod p^{1/(p−1)−ε_{s}} ⟶ O_{L′_{s}} mod p^{1/(p−1)−ε_{s}},

where all ε_{s} > 0 and lim_{s→∞}ε_{s} = 0.

Indeed, let m_{ℚ_{p}(ζ_{s})} and m_{L_{s}} be maximal ideals in ℤ_{p}[ζ_{s}] and O_{L_{s}} respectively.

Then (p^{1/(p−1)}) = m_{ℚ_{p}(ζ_{s})}^{p^{s−1}}. Now we obtain from [De, rem.A.6.3] the existence (and even the uniqiuness) of a fields isomorphism f_{s}, such that its restriction to O_{L_{s}}

and ϕ̃_{s} coincide when taken modulo m_{L_{s}}^{ψ_{Ls/ℚp(ζs)}(p^{s−1}−1)+1}, where ψ_{L_{s}/ℚ_{p}(ζ_{s})} is the inverse function for Herbrand’s function of the extension L_{s}/ℚ_{p}(ζ_{s}).

Our property follows now easily because for s ⩾ s_{2}, we have e_{L_{s}} = ẽe_{ℚ_{p}(ζ_{s})} = ẽp^{s−1}(p − 1), ψ_{L_{s}/ℚ_{p}(ζ_{s})} = ψ_{Ẽ/K_{0}} does not depend on s, and for some C > 0 one has for all x the following inequalities

ẽx − C < ψ_{Ẽ/K_{0}}(x) < ẽx.

2.5. Let s ⩾ s_{2} and choose τ_{s} ∈ Γ_{ℚ_{p}} such that τ_{s}|_{L_{s}} = f_{s}. Set Ẽ′′ = τ_{s}^{−1}Ẽ′ and E′′ = τ_{s}^{−1}E′, and consider the group isomorphism G_{s} = F ∘ τ_{s}^{∗} : Γ_{E} ⟶ Γ_{E′′}.

Take an arbitrary τ ∈ Γ_{E}, then G_{s}(τ) ∈ Γ_{E′′}, and set

γ_{s}(τ) = τ^{−1}G_{s}(τ) ∈ Γ_{ℚ_{p}}.

Because L_{s} is a Galois extension of ℚ_{p}, elements of the group Γ_{ℚ_{p}} act by conjugation on the maximal abelian quotient Γ_{L_{s}}^{ab} of Γ_{L_{s}}. By Proposition n.1, the action of γ_{s}(τ) on Γ_{L_{s}}^{ab} induces the identity map on its subquotient Γ_{L_{s}}^{ab(1)} mod (Γ_{L_{s}}^{ab})^{p} Γ_{L_{s}}^{ab(p^{s}+1)} (of course, we assume that s_{2} ⩾ 2). Indeed, by Proposition n.1 G_{s} induces the identity map on this subquotient. But G_{s} transforms the conjugation by τ to the conjugation by G_{s}(τ) therefore, these conjugations coincide on our subquotient.

Now we apply local class field theory to conclude that the automorphism γ_{s}(τ) of the field L_{s} acts trivially on U_{L_{s}}^{(1)} mod U_{L_{s}}^{p}U_{L_{s}}^{(p^{s}+1)}.

Let π be a uniformiser of L_{s}, then we have

γ_{s}(τ)(1 + π) = (1 + π)(1 + α_{1})^{p} mod m_{L_{s}}^{p^{s}+1},
γ_{s}(τ)(1 + π^{2}) = (1 + π^{2})(1 + α_{2})^{p} mod m_{L_{s}}^{p^{s}+1},

for some α_{1}, α_{2} ∈ m_{L_{s}}. Assume that β = γ_{s}(τ)(π) − π ∉ m_{L_{s}}^{p^{s}}. Then the above two congruences give the contradiction:

α_{1}^{p} ≡ β mod βm_{L_{s}}, α_{2}^{p} ≡ πβ mod βm_{L_{s}}^{2}.

Therefore, β ∈ m_{L_{s}}^{p^{s}}, i.e. the image γ̄_{s}(τ) of γ_{s}(τ) in the Galois group Γ_{L_{s}/ℚ_{p}} belongs to its ramification subgroup (in lower numbering) Γ_{L_{s}/ℚ_{p},p^{s}−1}.

So, we proved the following statement.

Lemma. There exists an index s_{2} ⩾ 2 such that for all s ⩾ s_{2} and τ ∈ Γ_{E}, one has γ̄_{s}(τ) ∈ Γ_{L_{s}/ℚ_{p},p^{s}−1}.

2.6. Choose an index s such that s ⩾ max{s_{2}, s_{0}} and p^{s} − 1 > x̃_{0}, cf. Lemma n.2.2. Then any element of the ramification subgroup Γ_{Ẽ_{s}/ℚ_{p},p^{s}−1} acts trivially on

Ẽ, i.e. Γ_{Ẽ_{s}/ℚ_{p},p^{s}−1} ⊂ Γ_{Ẽ_{s}/Ẽ}. But the extension L_{s}/Ẽ_{s} is unramified, therefore, the image of γ_{s}(τ) in Γ_{Ẽ_{s}/ℚ_{p}} belongs to the ramification subgroup Γ_{Ẽ_{s}/ℚ_{p},p^{s}−1}. This gives finally that γ_{s}(τ) ∈ ΓẼ or

(8)  G_{s}(τ) ≡ τ mod ΓẼ

for any τ ∈ Γ_{E}.

Therefore, in notation of n.2.5 we have E′′ = E and Ẽ′′ = Ẽ, because the above congruence (8) implies the following implications:

τ ∈ Γ_{E} ⇒ G_{s}(τ) ∈ Γ_{E} ⇒ τ ∈ Γ_{E′′},
τ ∈ ΓẼ ⇒ G_{s}(τ) ∈ ΓẼ ⇒ τ ∈ ΓẼ′′.

From the congruence (8) it follows also that the group morphism G_{s} = F ∘ τ_{s}^{∗} induces the identity automorphism of the Galois group Γ_{Ẽ/E}. Therefore, the group morphism

FẼ : Γ_{Ẽ/E} ⟶ Γ_{Ẽ′/E′}

comes from the conjugation by the τ_{s} ∈ Γ_{ℚ_{p}} chosen in n.2.5. So, the statement of our proposition holds for fẼ = τ_{s}|Ẽ.

The proposition and the theorem are completely proved.

References

[Ab] V.A. Abrashkin, A strong property of ramification filtration, Preprint MPI (Bonn) 38 (1998).

[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.

[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

   E-mail address: victor@abrvic.mian.su