L’INSTITUTFOURIER DE ANNALES

A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

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L’INSTITUT FOURIER

Victor ABRASHKIN

Groups of automorphisms of local fields of periodp^M and nilpotent class

< p

Tome 67, n^o2 (2017), p. 605-635.

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GROUPS OF AUTOMORPHISMS OF LOCAL FIELDS OF PERIOD p

AND NILPOTENT CLASS < p

by Victor ABRASHKIN

Abstract. — SupposeKis a finite field extension ofQpcontaining ap^M-th primitive root of unity. For 16s < pdenote byK[s, M] the maximalp-extension of Kwith the Galois group of periodp^M and nilpotent classs. We apply the nilpotent Artin–Schreier theory together with the theory of the field-of-norms functor to give an explicit description of the Galois groups ofK[s, M]/K. As application we prove that the ramification subgroup of the absolute Galois group ofKwith the upper indexvacts trivially onK[s, M] iffv > eK(M+s/(p−1))−(1−δ1s)/p, where e_K is the ramification index ofKandδ1sis the Kronecker symbol.

Résumé. — SoitKune extension finie deQpcontenant une racinep^M-ième primitive de l’unité. Pour 16s < pon noteK[s, M] lap-extension maximale de K dont le groupe de Galois est de période p^M et de classe de nilpotences. En utilisant la théorie d’Artin–Schreier nilpotente et la théorie du corps des normes on donne une description explicite du groupe de Galois de K[s, M]/K. Comme application de ce résultat on montre que le sous-groupe de ramification du groupe de Galois absolu de Kde ramification supérieurevagit trivialement surK[s, M]

si et seulement si v > eK(M+s/(p−1))−(1−δ1s)/p, oùeK est l’indice de ramification deKetδ1sest le symbole de Kronecker.

Introduction

Everywhere in the paper M ∈Nis fixed andp6= 2 is prime.

Let K be a complete discrete valuation field of characteristic 0 with finite residue field k ' Fq0, where q0 = p^N0, N0 ∈ N. Fix an algebraic closure ¯K of K and denote byK<p(M) the maximalp-extension ofK in K¯ with the Galois group of nilpotent class < p and exponent p^M. Then Γ<p(M) := Gal(K<p(M)/K) = Γ/Γ^pMCp(Γ), where Γ = Gal( ¯K/K) and C_p(Γ) is the closure of the subgroup of commutators of order>p.

Keywords:local fields, upper ramification numbers.

Math. classification:11S15, 11S20.

Let{Γ^(v)}v>0be the ramification filtration of Γ in upper numbering [14].

The importance of this additional structure on the Galois group Γ (which reflects arithmetic properties ofK) can be illustrated by the local analogue of the Grothendieck Conjecture [5, 6, 13]: the knowledge of Γ together with the filtration{Γ^(v)}v>0is sufficient to recover uniquely the isomorphic class ofKin the category of complete discrete valuation fields.

Let {Γ<p(M)^(v)}v>0 be the induced ramification filtration of Γ<p(M).

Then the problem of arithmetical description of Γ<p(M) is the problem of explicit description of the filtration{Γ<p(M)^(v)}v>0in terms of generators of Γ_<p(M).

An analogue of this problem was studied in [2, 3, 4] in the case of local fields K of characteristic p with residue field k. More precisely, let G = Gal(Ksep/K) and G<p(M) = G/G^pMCp(G). In [2, 3] we developed a nilpotent version of the Artin–Schreier theory which allows us to construct identification of profinite groupsG<p(M) =G(L). HereLis a profinite Lie Z/p^M-algebra of nilpotent class< pandG(L) is the pro-p-group, obtained fromLby the Campbell–Hausdorff composition law, cf. Subsection 1.2 be- low for more details and [7, Subsection 1.1] for non-formal comments about nilpotent Artin–Schreier theory.

On the one hand, the above identification ofG<p(M) withG(L) depends on a choice of uniformising element inKand, therefore, is not functorial (in particular, it can’t be used directly to develop a nilpotent analog of classical local class field theory). On the other hand, the ramification subgroups G_<p(M)^(v) can be now described in terms of appropriate idealsL^(v)of the Lie algebraL. The definition of these ideals essentially uses the extension of scalarsLk:=L⊗WM(k) ofL(such operation does not exist in the category ofp-groups) together with the appropriate explicit system of generators of Lk, cf. Subsection 1.4. This justifies the advantage of the language of Lie algebras in the theory ofp-extensions of local fields.

In this paper we apply the above characteristic presults to the study of similar properties in the mixed characteristic case, i.e. to the study of the group Γ<p(M) together with its ramification filtration. Our main tool is the Fontaine–Wintenberger theory of the field-of-norms functor [15]. Note also that we assume thatK contains a primitivep^M-th root of unity and our methods generalize the approach from [1] where we considered the case M = 1. In some sense our theory can be treated as nilpotent version of Kummer’s theory in the context of complete discrete valuation fields. As a result, we identify Γ_<p(M) with the group G(L), whereL is a Lie Z/p^M- algebra and for an appropriate idealJ of L, we have the following exact

sequence of Lie algebras

(0.1) 0−→ L/J −→L−→CM −→0.

Here C_M is a cyclic group of order p^M with the trivial structure of Lie algebra overZ/p^M.

As a first step in the study of L, we give an explicit description of the idealJ. More generally, ifCs(L) is the closure of the ideal of commutators of order > s in L, then for s > 2, we have Cs(L) ⊂ L/J and exact sequence (0.1) induces the exact sequences

0−→ L/L(s)−→L/Cs(L)−→CM −→0,

where allL(s) are ideals inL. The main result of Section 3, Theorem 3.3, describes these ideals L(s) with 2 6 s 6 p and gives in particular that J =L(p).

Extension (0.1) splits in the category ofZ/p^M-modules and its structure can be given by explicit construction of a lift τ<p of a generator of CM

toLand the appropriate differentiation adτ_<p∈End(L/J). The study of adτ<pwill be done in the next paper via methods used in the caseM = 1 in [1].

In Section 4 we apply our approach to find for 1 6 s < p, the maxi- mal upper ramification numbersv(K[s, M]/K) of the maximal extensions K[s, M] ofK with Galois groups of periodp^M and nilpotent classs. (The maximal upper ramification number for a finite extensionK⁰/Kin ¯Kis the maximalv₀ such that the ramification subgroups Γ^(v) act trivially onK⁰ ifv > v0.) This result can be stated in the following form, cf. Theorem 4.5 from Section 4:

If[K:Qp]<∞andζM ∈K then for16s < p, v(K[s, M]/K) =e_K

M + s p−1

−1−δ_s1 p .

wheree_K is the ramification index ofK/Qpandδis the Kronecker symbol.

Remark. — The cases= 1 is very well-known and can be established without the assumptionζ_M ∈K. Is it possible to remove this restriction whens >1?

Notation. — IfMis anR-module then its extension of scalarsM⊗_RS will be very often denoted byMS, cf. also another agreement in Subsec- tion 1.1. Very often we drop off the indication toM from our notation and use justK<p,Γ<p,G<petc. instead ofK<p(M),Γ<p(M),G<p(M), etc.

1. Preliminaries

Let K be a complete discrete valuation field of characteristic p with residue fieldk 'Fq₀, q0 =p^N0, and fixed uniformiser t0. In other words, K=k((t₀)).

As earlier, G = Gal(Ksep/K), K<p = K<p(M) is the subfield of Ksep

fixed byG^pMC_p(G) andG<p =G<p(M) = Gal(K<p/K). The ramification filtration ofG<pwas studied in details in [2, 3, 4]. We overview these results in the next subsections.

1.1. Compatible system of lifts modulop^M

The uniformizer t₀ of K gives a p-basis for any separable extension E ofK, i.e.{1, t0, . . . , t^p−1₀ } is a basis of theE^p-moduleE. We can uset0 to construct a functorial onE (and onM) system of lifts O(E)(=O_M(E)) of E modulop^M. Recall that these lifts appear in the form WM(σ^M−1E)[t], whereW_M is the functor of Witt vectors of lengthM, σis the Frobenius morphism of takingp-th power andt= (t0,0, . . . ,0)∈WM(K).

Note thatt∈O(K)⊂WM(K),tmodp=t0andσt=t^p. The liftO(K) is naturally identified with the algebra of formal Laurent seriesW_M(k)((t)) in the variabletwith coefficients inWM(k). A liftσof the absolute Frobenius endomorphism ofK toO(K) is uniquely determined by the conditionσt= t^p. For a separable extension E of K we then have an extension of the Frobenius σ from E to O(E)(= WM(σ^M−1E)[t]). As a result, we obtain a compatible system of lifts of the Frobenius endomorphism of Ksep to O(Ksep) = lim

−→E

O(E). For simplicity, we shall denote this lift also by σ.

Note thatσis induced by the standard Frobenius endomorphism WM(σ) ofW_M(Ksep)⊃O(Ksep).

Suppose η0 ∈ AutK and let WM(η0) be the induced automorphism of W_M(K). If W_M(η₀)(t) ∈ O(K) then η := W_M(η₀)|_O(K) is a lift of η₀ to O(K), i.e.η ∈ AutO(K) andηmodp=η0. With the above notation and assumption (in particular,η(t)∈O(K)) we have even more.

Proposition 1.1. — Suppose E is separable overK, η_E0 ∈AutE and η_E0|_K = η₀. Then η_E := W_M(η_E0)|_O(E) is a lift of η_E0 to O(E) such that ηE|_O(K)=η.

Proof. — Indeed, using thatO(E) =W_M(σ^M−1E)[t], we obtain ηE(WM(σ^M−1E)) =WM(ηE0)(WM(σ^M−1E))⊂WM(σ^M−1E))⊂O(E),

and η_E(t) = W_M(η_E0)(t) = W_M(η₀)(t) ∈ O(K) ⊂O(E). So, η_E(O(E))⊂

O(E). Obviously,η_Emodp=η_E0.

Remark. — The above lifts η_E commute with σ if and only if η com- mutes with σ, i.e. σ(η(t)) = η(t^p). In particular, if η(t) = tα^pM−1 with α∈O(K) thenσ(η(t)) =t^pα^pM =η(t^p) (use thatσ(α)≡α^pmodpO(K)).

A very special case of the above proposition appears as the following property:

If E/K is Galois then the elementsg of the group Gal(E/K)can be naturally lifted to (commuting withσ) automorphisms ofO(E) via settingg(t) =t. Therefore,O(Ksep)has a natural structure of a G-module, the action ofGcommutes withσ,O(Ksep)^G =O(K) andO(Ksep)|σ=id=W_M(Fp).

Everywhere below we shall use the following simplified notation.

Notation. — If M is a Z/p^M-module and E is a separable extension of K we set M_E := M_O(E)(= M⊗_Z/pM O(E)). Similarly, we agree that Mk :=M⊗_Z/pMW_M(k).

1.2. Categories ofp-groups and LieZ/p^M-algebras,[11, 12]

If L is a LieZ/p^M-algebra of nilpotent class < p, denote by G(L) the p-group obtained from L via the Campbell-Hausdorff composition law ◦ defined forl1, l2∈Lviagexp(l1◦l2) =gexpl1·explg2. Here

exp(x) = 1 +g x+· · ·+x^p−1/(p−1)!

is the truncated exponential from Lto the quotient of the enveloping al- gebra A of L modulo the p-th power of its augmentation ideal J. (This construction of the Campbell-Hausdorff operation was introduced in [2, Subsection 1.2].)

The correspondence L7→G(L) induces equivalence of the categories of finite LieZ/p^M-algebras and finite p-groups of exponent p^M of the same nilpotent class 16s0< p. This equivalence can be extended to the similar categories of profinite Lie algebras and groups.

1.3. Witt pairing and Hilbert symbol, [8, 9]

Let

E(α, X) = exp

αX+σ(α)X^p

p +· · ·+σⁿ(α)X^pn pⁿ . . .

∈W(k)[[X]],

where α ∈ W(k), be the Shafarevich version of the Artin–Hasse expo- nential. Set Z⁺(p) = {a ∈ N | gcd(a, p) = 1}. Then any element u ∈ K^∗modK^∗pM can be uniquely written as

u=t^a₀⁰ Y

a∈Z⁺(p)

E(α_a, t^a₀)^1/amodK^∗pM,

wherea₀=a₀(u)∈Zmodp^M and allα_a=α_a(u)∈W(k) modp^M. Let M be a profinite free WM(k)-module with the set of generators {D₀} ∪ {D_an |a∈Z⁺(p), n∈Z/N₀}. Use the correspondences

(1.1) t₀7→D₀, E(α, t^a₀)^1/a7→ X

nmodN₀

σⁿ(α)D_an,

to identify K^∗/K^∗pM with a closed Z/p^M-submodule in M. Under this identification we haveK^∗/K^∗pM⊗_Z/pMW_M(k) =M.

Define the continuous action of the group hσi = Gal(k/Fp) on M as an extension of the natural action on WM(k) by setting σD0 = D0 and σD_an=D_a,n+1. ThenK^∗/K^∗pM =M^Gal(k/Fp).

The Witt pairing

O(K)/(σ−id)O(K)× K^∗/K^∗pM −→Z/p^M,

is given explicitly by the symbol [f, g) = Tr (Res(f dlogColg)). Here Tr : WM(k) −→ Z/p^M is induced by the trace of the field extension k/Fp, f ∈O(K) and Colg is the image of g∈ K^∗/K^∗pM under the group homo- morphism Col :K^∗/K^∗pM −→O_M^∗ (K) uniquely defined on the above free generators ofK^∗/K^∗pM via the conditionst07→t andE(α, t^a₀)7→E(α, t^a).

The Witt pairing is non-degenerate and determines the identification K^∗/K^∗pM = Homcont(O(K)/(σ−id)O(K),Z/p^M).

It also coincides with the Hilbert symbol (in the case of local fields of characteristic p) and allows us to specify explicitly the reciprocity map κ:K^∗/K^∗pM −→ G_<p^ab of class field theory. Namely, in the above notation we haveκ(g)f =f+ [f, g).

1.4. Lie algebraL and identification ηM

Let Le be a free profinite Lie Z/p^M-algebra with the module of (free) generatorsK^∗/K^∗pM. Then theWM(k)-moduleLek has the set of free gen- erators

(1.2) {D0} ∪ {Dan|a∈Z⁺(p), n∈Z/N0}.

If C_p(L) is the closure of the ideal of commutators of ordere > p, then L=L/Ce p(L) is the maximal quotient ofe Leof nilpotent class< p.

Remark. — Lk is a free object in the category of profinite LieWM(k)- algebras of nilpotent class< pwith the set of free generators (1.2).

We shall use the same notationD0andDanfor the images of the elements of (1.2) inL. Chooseα₀∈W_M(k) such that Trα₀= 1.

Consider e = α0D0 +P

a∈Z⁺(p)t^−aDa0 ∈ G(LK). If we set D0n :=

(σⁿα₀)D₀ then e can be written as P

a∈Z⁰(p)t^−aD_a0, where Z⁰(p) = Z⁺(p)∪ {0}.

Fix f ∈G(L_Ksep) such thatσf =e◦f. Then forτ ∈ G, the correspon- dence

τ7→(−f)◦τ f∈G(LK_sep)|σ=id=G(L), induces the identification of profinite groupsηM :G<p'G(L).

Note thatf ∈ LK<p andG<pstrictly acts on theG-orbit off.

The above result is a covariant version of the nilpotent Artin–Schreier theory developed in [3], cf. also Subsection 1.1 in [7] for the relation between the covariant and contravariant versions of this theory and for appropriate non-formal comments.

We shall use below a fixed choice of f and use the notation fore andf without further references.

1.5. Relation to class field theory

The above identificationη_M taken moduloC₂(G_<p) gives an isomorphism of profinitep-groups

η_M^ab:G_<p^ab −→ L^ab=L/C2(L) =M^Gal(k/Fp)=K^∗/K^∗pM.

Proposition 1.2. — η_M^ab is induced by the inverse to the reciprocity map of local class field theoryκ.

Proof. — Indeed, let {βi}16i6N₀ be a Z/p^M-basis of WM(k) and let {γi}16i6N0 be its dual basis with respect to the bilinear form induced by the trace of the field extensionW(k)[1/p]/Qp.

If a ∈ Z⁺(p) and E(β_i, t^a₀)^1/a = D_ia, then D_ia = P

nσⁿ(β_i)D_an, and, therefore,Da0=P

iγiDia. This implies that e=X

i,a

t^−aγ_iD_ia+α₀D₀modC₂(L_K), f =X

i,a

f_iaD_ia+f₀D₀modC₂(LKsep),

where allf_ia, f₀ ∈O(K<p), σf_ia−f_ia=γ_it^−a and σf₀−f₀ =α₀. From the definition of ηM it follows formally that for τia = (η_M^ab)⁻¹Dia and τ₀= (η_M^ab)⁻¹D₀,τ_iaf_i1a1 =f_i1a1+δ(ii₁)δ(aa₁),τ₀f_i1a1 =f_i1a1,τ_iaf₀=f₀ andτ0f0=f0+ 1. (Hereδis the Kronecker symbol.)

Now the explicit formula for the Hilbert symbol from Subsection 1.3 shows thatκ(E(βi, t^a₀)^1/a) andκ(t0) act by the same formulae asτia and,

resp.,τ0.

1.6. Construction of lifts of analytic automorphisms Letη0∈AutK. Then there is a liftη<p,0∈AutK<pofη0. (Use that the subgroupG^pMC_p(G) ofGis characteristic.) For any another such liftη⁰_<p,0, we haveη_<p,0⁰ η⁻¹_<p,0∈ G<p.

The covariant version of the Witt–Artin–Schreier theory [3], Section 1 (cf. also [7, Subsection 1.1] and [1, Section 1]), gives explicit description of the automorphisms η<p,0 in terms of the identification ηM. Consider a special case of this construction when η₀ admits a lift η ∈ AutO(K) which commutes withσ, and therefore we have the appropriate liftsη<p∈ AutO(K_<p), cf. Subsection 1.1. Then in terms of our fixed elementseand f, we haveη<p(f) =c◦(A⊗id_O(K<p))f, wherec∈ LK andA∈AutLcan be found from the relation

(id_L⊗η)e=σc◦(A⊗id_O(K))e◦(−c),

cf. [3, Subsection 1.5], or [1, Proposition 1.1], and Subsection 3.2 below.

In other words, if (A⊗id_WM(k))(D_a0) =De_a0then X

a∈Z⁰(p)

η(t)^−aDa0=σc◦



 X

a∈Z⁰(p)

t^−aDea0



◦(−c).

Note that proceeding as in [3, Subsection 1.5.4], cf. also [1, Subsec- tion 1.2], we can verify (this fact will be used systematically below) that with respect to the identificationη_M, the automorphismAcoincides with the conjugation Adη<p:τ7→η⁻¹_<pτ η<p(hereτ ∈ G<p).

1.7. Ramification filtration in L

For v > 0, denote by G_<p^(v) the ramification subgroup of G<p with the upper indexv. Let L^(v) be the ideal of L such that ηM(G^(v)_<p) =G(L^(v)).

The idealsL^(v) have the following explicit description.

First, for any a ∈ Z⁰(p) and n ∈ Z, set D_an := D_a,nmodN0. In other words, we allow the second index in all Dan to take integral values and assume that D_an1 = D_an2 iff n₁ ≡ n₂modN₀. For s > 1, agree to use the notation (¯a,n)¯ s, where ¯a = (a1, . . . , as) has coordinates inZ⁰(p) and

¯

n = (n₁, . . . , ns) ∈ Z^s. Then we can attach to (¯a,n)¯ s the commutator [. . .[Da₁n₁, Da₂n₂], . . . , Dasns] and set γ(¯a,n)¯ s =a1pⁿ¹+· · ·+asp^ns. For anyγ>0, letF_γ,−N⁰ be the element fromLk given by

(1.3) F_γ,−N⁰ = X

γ(¯a,¯n)s=γ

pⁿ¹a1η(¯n)[. . .[Da₁n₁, Da₂n₂], . . . , Da_sn_s]

where η(¯n) equals (s₁!(s₂−s₁)!. . .(s−s_l)!)⁻¹ if 0 6n₁ = · · · = n_s1 >

ns₁+1 = · · · = ns₂ > · · · > ns_l = · · · = ns > −N, and equals to zero otherwise. Then the main result of [4] (translated into the covariant setting, cf. [5, Subsections 1.1.2 and 1.2.4]) states that:

There isNe(v)∈Nsuch that if we fix anyN >N(v), thene L^(v)is the minimal ideal ofLsuch that for allγ>v,F_γ,−N⁰ ∈ L^(v)_k .

2. Filtration {L(s)}_s>1

In this section we define a decreasing central filtration {L(s)}s>1in the Z/p^M-Lie algebraLfrom Subsection 1.4. Its definition depends on a choice of a special element S ∈ m(K) := tW_M(k)[[t]] ⊂ O(K). This element S (together with the appropriate elementsS0andS⁰ from its definition) will be specified in Section 4, where we apply our results to the mixed charac- teristic case.

2.1. Elements S₀, S⁰, S∈m(K)

Let [p] be the isogeny of multiplication by p in the formal group SpfZp[[X]] overZp with the logarithmX+X^p/p+· · ·+X^pn/pⁿ+. . ..

Choose S₀ ∈ m(K) and setS⁰ = [p]^M−1(S₀) and S = [p]^M(S₀). Then S, S⁰ ∈m(K), they both depend only on the residueS0modpandS =σS⁰. In particular, if e^∗ ∈ N is such that Smodpgenerates the ideal (t^e₀^∗) in k[[t0]] then e^∗≡0 modp^M.

Proposition 2.1.

(a) dS= 0in Ω¹_O(K);

(b) there isS⁰⁰∈m(K), such thatS=S⁰(p+S⁰⁰);

(c) there areη₀, η₁∈WM(k)[[t]]^× andη₂∈WM(k)[[t]]such that S=t^e∗η0+pt^e∗/pη1+p²η2.

Proof.

(a) The congruence [p]X ≡ X^pmodpZp[[X]] implies that d([p]X) ∈ pZp[[X]]. Therefore,dS= 0 in Ω¹_O(K).

(b) Note that [p](X) ≡pXmodX². Therefore, there arewi ∈Z^p such thatS= [p]S⁰=pS⁰+P

i>2wiS⁰ⁱ and we can takeS⁰⁰=P

i>1wi+1S⁰ⁱ. (c) The t0-adic valuation of S⁰modpequals e^∗/p. Then our property is implied by the following equivalence inZp[[X]]

[p](X)≡pX+X^pmod (pX^p2−p+1, p²X).

Remark. — We shall use below property (a) in the following form:

If s ∈ N and S^s = P

l>1γ_lst^l, where all γ_ls ∈ W_M(k), then lγls= 0.

2.2. Morphism ι

Let U = (1 + t₀k[[t₀]])^× be the Zp-module of principal units in K.

ThenU/U^pM is a closedZ/p^M-submodule inK^∗/K^∗pM. Note that m(K) = WM(mK)∩O(K), where mK is the maximal ideal in the valuation ring of K. Consider a (unique) continuous homomorphism

ι:U −→m(K)

such that for anyα∈WM(k) anda∈Z⁺(p),ι:E(α, t^a₀)7→αt^a (hereE is the Shafarevich function, cf. Subsection 1.3).

Then ι induces an identification of U/U^pM with the closed W_M(k)- submodule

Imι=





 X

a∈Z⁺(p)

α_at^a

α_a∈W_M(k)







inO(K). This submodule is topologically generated overW_M(k) by allt^a witha∈Z⁺(p).

2.3. Definition of {L(s)}s>1

Set (K^∗/K^∗pM)⁽¹⁾=K^∗/K^∗pM. Fors>1, let (K^∗/K^∗pM)^(s+1)= (Imι)S^s with respect to the identificationU/U^pM = Imιfrom Subsection 2.2. Note, thatS=σS⁰ implies that for anys∈N, (Imι)S^s⊂Imι.

Definition. — {L(s)}_s>1 is the minimal central filtration of ideals of the Lie algebraLsuch that for alls>1,L(s)⊃(K^∗/K^∗pM)^(s).

The ideals L(s) can be defined by induction on s as follows. Let L(1) = L; then for s > 1, the ideal L(s + 1) is generated by the elements of (K^∗/K^∗pM)^(s+1) and [L(s),L]. Note also that for any s, (K^∗/K^∗pM)∩ L(s) = (K^∗/K^∗pM)^(s). (Use that Z/p^M-module L(s) is iso- morphic to (K^∗/K^∗pM)^(s)⊕(L(s)∩C2(L)).

In addition, for any s>1, the quotients (K^∗/K^∗pM)^(s)/(K^∗/K^∗pM)^(s+1) are freeZ/p^M-modules. This easily implies that allL(s)/L(s+ 1) are also freeZ/p^M-modules.

2.4. Characterization of {L(s)}s>1 in terms of e∈ LK

Recall that e=P

a∈Z⁰(p)t^−aDa0, cf. Subsection 1.4.

Proposition 2.2. — The filtration {L(s)}s>1 is the minimal central filtration inL such thatL(1) =Land for alls>1,

S^se∈ L_m(K)+L(s+ 1)_K. Proof. — We need the following two lemmas.

Lemma 2.3. — For all s >1 and αa ∈ WM(k) where a ∈ Z⁺(p), we have

Y

a∈Z⁺(p)

E(αa, t^a₀)∈(K^∗/K^∗pM)^(s+1)

⇔ Y

a∈Z⁺(p)

E(α_a, t^a₀)^1/a∈(K^∗/K^∗pM)^(s+1). Proof of Lemma 2.3. — We must prove that

X

a∈Z⁺(p)

αat^a∈S^sm(K) ⇔ X

a∈Z⁺(p)

1

aαat^a∈S^sm(K).

Let S^s = P

l>1γ_lst^l with γ_ls ∈ W_M(k), then lγ_ls = 0, cf. Remark in Subsection 2.1.

Suppose

X

a∈Z⁺(p)

αat^a∈S^sm(K).

Then P

aαat^a = (P

bβbt^b)(P

lγlst^l), where P

bβbt^b ∈ m(K) and αa = P

a=b+lβ_bγ_ls. This implies 1

aα_a = X

a=b+l

1

aβ_bγ_ls= X

a=b+l

1 bβ_bγ_ls, because ifa=b+l anda∈Z⁺(p) thenb∈Z⁺(p) and

1 aγls−1

bγls= −lγls

ab = 0.

So,

X

a∈Z⁺(p)

1 aαat^a=



 X

b∈Z⁺(p)

1 bβbt^b



 X

l

γlst^l

!

and

X

a

1

aαat^a∈S^sm(K).

Proceeding in the opposite direction we obtain the inverse statement.

The lemma is proved.

Lemma 2.4. — Ifs>1 and allα_a∈W_M(k)then Y

a∈Z⁺(p)

E(αa, t^a₀)^1/a∈(K^∗/K^∗pM)^(s)⇔ X

a∈Z⁺(p)

αaDa0∈(K^∗/K^∗pM)^(s)_k Proof of Lemma 2.4. — Suppose

Y

a∈Z⁺(p)

E(α_a, t^a₀)^1/a∈(K^∗/K^∗pM)^(s).

Choose a WM(F^p)-basis {βi} of WM(k), and let {γi} be its dual with respect to the trace form. Then for anyi,

Y

a∈Z⁺(p)

E(βiαa, t^a₀)^1/a∈(K^∗/K^∗pM)^(s). In other words (use (1.1) from Subsection 1.3),

ci= X

a∈Z⁺(p) n∈Z/N0Z

σⁿ(βi)σⁿ(αa)Dan∈

K^∗/K^∗pM(s)

⊂ L(s),

and

X

i

γici= X

a∈Z⁺(p)

αaDa0∈ L(s)k.

Suppose now that P

a∈Z⁺(p)α_aD_a0∈ L(s)k. Then X

a∈Z⁺(p)

α_aD_a0∈(K^∗/K^∗pM)^(s)_k , and, therefore,

X

a∈Z⁺(p) n∈Z/N₀Z

σⁿ(α_a)D_an∈(K^∗/K^∗pM)^(s).

This means, that Y

a∈Z⁺(p)

E(αa, t^a₀)^1/a∈(K^∗/K^∗pM)^(s).

The lemma is proved.

Now we can finish the proof of our proposition. If, as earlier, S^s = P

l>1γlst^l with γls ∈ WM(k), then (Imι)S^s is the WM(k)-submodule in m(K) generated by the elements t^a1S^s =P

l>1γ_lst^l+a1, a₁ ∈Z⁺(p). The above lemmas imply then that{L(s)}s>1 is the minimal central filtration inLsuch thatL(1) =Land for alla₁∈Z⁺(p), s>1,

X

l>1

γlsDa₁+l,0∈ L(s+ 1)k. On the other hand,

S^se= X

a∈Z⁰(p) l>1

γlst^−(a−l)Da0≡ X

a1∈Z⁺(p)



 X

l>1

γlsDa₁+l,0



t^−a1

moduloL_m(K). Therefore, S^se∈ L_m(K)+L(s+ 1)_K

⇔X

l

γlsDa₁+l,0∈ L(s+ 1)k for alla1∈Z⁺(p).

The proposition is proved.

Definition. — N =P

s>1S^−sL(s)_m(K).

Note that N is a Lie W_M(Fp)-subalgebra in L_K. With this notation Proposition 2.2 implies the following characterization of the filtration {L(s)}s>1.

Corollary 2.5. — {L(s)}_s>1is the minimal central filtration inLsuch thatL(1) =L ande∈ N.

Proof. — It will be sufficient to verify that

e∈ N ⇔ ∀s>1, S^se∈ L_m(K)+L(s+ 1)K.

The “if” part is obvious. The “only if” part can be proved by induction on svia the following property:

If l⁰(s) ∈ L(s)_K and Sl⁰(s) ∈ L_m(K)+L(s+ 1)_K then l⁰(s) ∈ S⁻¹L(s)_m(K)+L(s+ 1)K (use thatL(s)/L(s+ 1)is free Z/p^M-

module).

2.5. Element e^†∈G(LK)

Recall that Smodp generates the ideal (t^e₀^∗) in k[[t0]]. Therefore, the projections of the elements of the set

S^−mt^b |16b < e^∗,gcd(b, p) = 1, m∈N ∪ {α0} form a basis ofO(K)/(σ−id)O(K) overWM(k).

Proposition 2.6. — There are V(0) ∈ L, x ∈ SN and V(b,m) ∈ Lk, wherem>1,16b < e^∗,gcd(b, p) = 1, such that

(a) e^†:=P

m,bS^−mt^bV_(b,m)+α0V₍₀₎ ∈ N; (b) e^†= (−σx)◦e◦x.

Proof. — Note thatS∈σm(K) implies that the sets{t^−a |a∈Z⁺(p)}

and{S^−mt^b |m∈N,gcd(b, p) = 1,16b < e^∗}generate the sameWM(k)- submodules in O(K)/m(K). This implies the existence of V₍₀₎⁽⁰⁾ ∈ L and V_(b,m)⁽⁰⁾ ∈ Lk such that

(2.1) e≡e^†₀modL_m(K)

wheree^†₀:=P

(b,m)S^−mt^bV_(b,m)⁽⁰⁾ +α0V₍₀₎⁽⁰⁾. Fori>1, let N⁽ⁱ⁾=P

s>iS^−sL(s)_m(K). Then

• N⁽ⁱ⁾=S⁻ⁱL(i)_m(K)+N⁽ⁱ⁺¹⁾;

•

N⁽ⁱ⁾,N

⊂ N⁽ⁱ⁺¹⁾.

In particular, relation (2.1) implies that e=e^†₀+σx0−x0, wherex0 ∈ L_m(K), and we obtain

(2.2) (−σx0)◦e◦x₀≡e^†₀modSN⁽²⁾

(use thatx0, σx0∈ L_m(K)⊂SN⁽¹⁾). Now we need the following lemma.

Lemma 2.7. — Suppose M is aZp-module and i₀ ∈ N. Then for any l∈S⁻ⁱ⁰M_m(K), there arel₍₀₎ ∈M,˜l∈S⁻ⁱ⁰M_m(K)andl_(b,m)∈Mk, where 16m6i₀,gcd(p, b) = 1and16b < e^∗, such that

l=X

b,m

S^−mt^bl_(b,m)+α0l₍₀₎+σ˜l−˜l.

Proof of Lemma 2.7. — It will be sufficient to consider the caseM=Zp. In other words, we must prove the following statement:

For anys∈S⁻ⁱ⁰m(K), there are β₍₀₎ ∈W_M(Fp), s˜∈S⁻ⁱ⁰m(K) and β_(b,m) ∈ WM(k), where 1 6m 6i0, gcd(b, p) = 1 and 1 6 b < e^∗, such that

s=X

b,m

β_(b,m)S^−mt^b+α₀β₍₀₎+σ˜s−s.˜

We can assume that s = t^a0/Sⁱ⁰, where 1 6 a₀ < e^∗, i₀ ∈ N and our lemma is proved for all elementssfrompS⁻ⁱ⁰m(K) +t^a0S⁻ⁱ⁰m(K).

If gcd (a₀, p) = 1 there is nothing to prove. Otherwise, a₀ = pa₁ and s = s⁰+σ(s⁰)−s⁰ with s⁰ = t^a1/S⁰ⁱ⁰ = t^a1(p+S⁰⁰)/Sⁱ⁰. It remains to note thats⁰∈pS⁻ⁱ⁰m(K) +t^a0S⁻ⁱ⁰m(K), becauseS⁰⁰modp∈(t^e₀⁰), where e⁰:=e^∗(1−1/p), anda1+e⁰=a0/p+e⁰> a0 (use thata0< e^∗).

Continue the proof of Proposition 2.6. Clearly, it is implied by the fol- lowing lemma.

Lemma 2.8. — For all i > 0, there are x_i ∈ SN, V_(b,m)⁽ⁱ⁾ ∈ Lk and V₍₀₎⁽ⁱ⁾∈ Lsuch that:

(a1) xi+1≡ximodSN⁽ⁱ⁺¹⁾; (a2) V_(b,m)⁽ⁱ⁺¹⁾≡V_(b,m)⁽ⁱ⁾ modL(i+ 2)k; (a₃) V₍₀₎⁽ⁱ⁺¹⁾≡V₍₀₎⁽ⁱ⁾modL(i+ 2)

(b) ife^†_i =P

b,mS^−mt^bV_(b,m)⁽ⁱ⁾ +α₀V₀⁽ⁱ⁾then (−σxi)◦e◦xi≡e^†_imodSN⁽ⁱ⁺²⁾.

Proof of Lemma 2.8. — Use the elements V_(b,m)⁽⁰⁾ , V₍₀₎⁽⁰⁾, e^†₀ and x₀ from the beginning of the proof of Proposition 2.6. Then part (b) holds fori= 0 by (2.2).

Let i₀ > 1 and assume that our Lemma is proved for all i < i₀. Let l∈S⁻ⁱ⁰L(i0+ 1)_m(K)be such that

e^†_i

0−1−(−σxi₀−1)◦e◦xi₀−1≡lmodSN⁽ⁱ⁰⁺²⁾.

Apply Lemma 2.7 to M = L(i₀ + 1) and l ∈ S⁻ⁱ⁰L(i₀ + 1)_m(K). This gives us the appropriate elements l(b,m) ∈ L(i0 + 1)k, l(0) ∈ L(i0 + 1)