# Ramification filtration via deformations, II

Victor Abrashkin  Department of Mathematical Sciences, Durham University,
Science Laboratories, South Rd, Durham DH1 3LE, United Kingdom & Steklov
Institute, Gubkina str. 8, 119991, Moscow, Russia
<EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{K}$ be a field of formal Laurent series with coefficients in a
finite field of characteristic $p$. For $M\in\mathbb{N}$, let
$\mathcal{G}_{<p,M}$ be the maximal quotient of the Galois group of
$\mathcal{K}$ of period $p^{M}$ and nilpotent class $<p$ and
$\\{\mathcal{G}_{<p,M}^{(v)}\\}_{v\geqslant 0}$ — the filtration by
ramification subgroups in upper numbering. Let
$\mathcal{G}_{<p,M}=G(\mathcal{L})$ be the identification of nilpotent Artin-
Schreier theory: here $G(\mathcal{L})$ is the group obtained from a suitable
profinite Lie $\mathbb{Z}/p^{M}$-algebra $\mathcal{L}$ via the Campbell-
Hausdorff composition law. We develop new techniques to obtain a “geometrical”
construction of the ideals $\mathcal{L}^{(v)}$ such that
$G(\mathcal{L}^{(v)})=\mathcal{G}_{<p,M}^{(v)}$. Given $v_{0}\geqslant 1$, we
construct a decreasing central filtration $\mathcal{L}(w)$, $1\leqslant
w\leqslant p$, on $\mathcal{L}$ and an epimorphism of Lie
$\mathbb{Z}/p^{M}$-algebras
$\bar{\mathcal{V}}:\bar{\mathcal{L}}^{{\dagger}}\to\bar{\mathcal{L}}:=\mathcal{L}/\mathcal{L}(p)$.
We construct also a unipotent action $\Omega$ of $\mathbb{Z}/p^{M}$, on
$\bar{\mathcal{L}}^{{\dagger}}$ which induces the identity action on
$\bar{\mathcal{L}}$. Suppose $d\Omega=B^{{\dagger}}$, where
$B^{{\dagger}}\in\operatorname{Diff}\bar{\mathcal{L}}^{{\dagger}}$, and
$\bar{\mathcal{L}}^{{\dagger}[v_{0}]}$ is the ideal of
$\bar{\mathcal{L}}^{{\dagger}}$ generated by the elements of
$B^{{\dagger}}(\bar{\mathcal{L}}^{{\dagger}})$. The main result of the paper
states that the ramification ideal $\mathcal{L}^{(v_{0})}$ appears as the
preimage of the ideal in $\bar{\mathcal{L}}$ generated by
$\bar{\mathcal{V}}B^{{\dagger}}(\bar{\mathcal{L}}^{{\dagger}[v_{0}]})$. In the
last section we relate this result to the explicit construction of generators
of $\bar{\mathcal{L}}^{(v_{0})}$ obtained earlier by the author. The results
of this paper generalise (and also considerably simplify) the author results
in the context of the maximal quotient $\mathcal{G}_{<p,1}$ of period $p$.

###### Key words and phrases:

local field, Galois group, ramification filtration

###### 2010 Mathematics Subject Classification:

11S15, 11S20

## Introduction

Everywhere in the paper $M\in\mathbb{N}$ and $v_{0}\in\mathbb{R}_{>0}$ are
fixed. For $s\in\mathbb{N}$ and a topological group $G$, we denote by
$C_{s}(G)$ the closure of the subgroup of $G$ generated by the commutators of
order $\geqslant s$. If $L$ is a toplogical Lie algebra then $C_{s}(L)$ is the
closure of the ideal generated by commutators of degree $\geqslant s$. For any
topological $A$-modules $M$ and $B$ we use the notation
$M_{B}:=M\hat{\otimes}_{A}B$.

Let $\mathcal{K}$ be a complete discrete valuation field of characteristic $p$
with finite residue field. Let $\mathcal{K}_{<p,M}$ be a maximal $p$-extension
of $\mathcal{K}$ with the Galois group
$\mathrm{Gal}(\mathcal{K}_{<p,M}/\mathcal{K}):=\mathcal{G}_{<p,M}$ of
nilpotence class $<p$ and exponent $p^{M}$. If $L$ is a Lie
$\mathbb{Z}/p^{M}$-algebra then $G(L)$ is the group obtained from the elements
of $L$ via Campbell-Hausdorff composition law, cf. Sect. 1.2.

Consider a decreasing filtration by ramification subgroups in the upper
numbering $\\{\mathcal{G}^{(v)}_{<p,M}\\}_{v\geqslant 0}$ of
$\mathcal{G}_{<p,M}$. First explicit results about the structure of these
ramification subgroups were obtained by the author in [1]-[3]. At that time it
became clear that there should be more general approach based on a suitable
deformation theory. This idea was realised recently by the author in [8] in
the case of Galois group $\mathcal{G}_{<p,1}$, i.e. in the case of Galois
groups of exponent $p$. Additional evidence that this should work for general
$M$ came recently from [9] where the behavior of ramification subgroups in
“small” $p$-adic representations was described in terms of differential forms
on the corresponding $\varphi$-modulles. In this paper, we obtain a
generalisation (and also a considerable simplification) of our approach from
[8] to the case of Galois groups $\mathcal{G}_{<p,M}$ with arbitrary exponemt
$p^{M}$. Its general overview could be presented as follows.

a) Let $\mathcal{K}=k((t_{0}))$, where $k\simeq\mathbb{F}_{p^{N_{0}}}$. Then
we can use the uniformiser $t_{0}$ to define the lifts modulo $p^{M}$,
$O_{M}(\mathcal{K})=W_{M}(k)((t))\subset W_{M}(\mathcal{K})$ and
$O_{M}(\mathcal{K}_{sep})\subset W_{M}(\mathcal{K}_{sep})$. Here $W_{M}$ is
the functor of Witt vectors of length $M$ and $t=[t_{0}]$ is the Teichmuller
representative of $t_{0}$. This allows us to apply the nilpotent version of
Artin-Shreier theory, cf. [2], to fix an identification
$\eta^{0}:\mathcal{G}_{<p,M}\simeq G(\mathcal{L})$, where $\mathcal{L}$ is
explicitly constructed pro-finite Lie $\mathbb{Z}/p^{M}$-algebra. It appears
via choosing a suitable $e\in\mathcal{L}_{O_{M}(\mathcal{K})}$, taking
$f\in\mathcal{L}_{O_{M}(\mathcal{K}_{sep})}$ such that $\sigma(f)=e\circ f$
and setting for any $\tau\in\mathcal{G}_{<p,M}$,
$\eta^{0}(\tau)=(-f)\circ\tau(f)\in G(\mathcal{L})$. This provides us also
with “ramification” ideals $\mathcal{L}^{(v)}$ in $\mathcal{L}$ such that
$\eta^{0}(\mathcal{G}_{<p,M}^{(v)})=\mathcal{L}^{(v)}$.

b) The extension of scalars $\mathcal{L}_{W_{M}(k)}$ appears with the set of
explicit generators $\\{D_{an}\ |\
\mathrm{gcd}(a,p)=1,n\in\mathbb{Z}/N_{0}\\}\cup\\{D_{0}\\}$. We use them to
define a decreasing filtration of ideals
$\mathcal{L}=\mathcal{L}(1)\supset\mathcal{L}(2)\supset\ldots\supset\mathcal{L}(p)$
and provide $\bar{\mathcal{L}}=\mathcal{L}/\mathcal{L}(p)$ with induced
filtration $\bar{\mathcal{L}}(w)$, $1\leqslant w\leqslant p$. The algebra
$\bar{\mathcal{L}}$ is considerably smaller than $\mathcal{L}$; it is
generated over $W_{M}(k)$ by $D_{an}$ with $0\leqslant a<(p-1)v_{0}$. However,
we prove that $\mathcal{L}(p)\subset\mathcal{L}^{(v_{0})}$ and, as a result,
it will be sufficient to describe the image $\bar{\mathcal{L}}^{(v_{0})}$ of
$\mathcal{L}^{(v_{0})}$ in $\bar{\mathcal{L}}$.

c) Our next step is a construction of a new Lie $\mathbb{Z}/p^{M}$-algebra
$\mathcal{L}^{{\dagger}}$ together with projection
$\bar{\mathcal{V}}:\bar{\mathcal{L}}^{{\dagger}}\to\bar{\mathcal{L}}$. As a
matter of fact, there is a splitting of Lie algebras
$\bar{\mathcal{L}}^{{\dagger}}=\mathrm{Ker}\bar{\mathcal{V}}\oplus\bar{\mathcal{L}}$.
The construction of $\bar{\mathcal{L}}^{{\dagger}}$ depends on a construction
of a special system of coefficients inside ${O_{M}(\mathcal{K})}$, cf. Sect.
2. We just mention that this system specifies explicitly the powers of $t$
which depend on a choice of a sufficiently large $N^{*}\in\mathbb{N}$ and a
sufficiently close to $v_{0}$ rational number $r^{*}<v_{0}$. More precisely,
if $q:=p^{N^{*}}$ then $r^{*}(q-1):=b^{*}$ is a natural number prime to $p$,
and there are no ramification breaks of the ramification filtration
$\mathcal{G}^{(v)}_{<p,M}$ inside the interval $(r^{*},v_{0})$. (The
ramification breaks of the filtration $\mathcal{G}^{(v)}_{<p,M}$ are discrete
because they are discrete for abelian extensions and $\mathcal{G}_{<p,M}$ has
a finite nilpotence class.) Then the admissible exponents $t^{-\iota}$,
$\iota\in\mathfrak{A}^{0}$, appear as
$\iota=qp^{M-1}A-b^{*}U\in\mathfrak{A}^{0}$, where $A$ is a finite sum of
various $p^{n}a$ with $n\geqslant 0$ and $a\in(0,(p-1)v_{0})$, and $U$ is a
finite sum of $u_{j}p^{m_{j}}$ with $u_{j},m_{j}\geqslant 0$ and
$1\leqslant\sum_{j}u_{j}<p$. One of properties of such exponents
$\iota\in\mathfrak{A}^{0}$ is that $\mathrm{ch}(\iota):=\sum_{i}u_{i}$ is an
invariant of $\iota$. In these terms, working with the Lie algebra
$\bar{\mathcal{L}}$ inside $\bar{\mathcal{L}}^{{\dagger}}$ is related to
$t$-exponents with $\mathrm{ch}(\iota)=0$, and working with
$\mathrm{Ker}\bar{\mathcal{V}}\subset\bar{\mathcal{L}}^{{\dagger}}$
corresponds to t-exponents with $\mathrm{ch}(\iota)\geqslant 1$.

d) Let us come back to the identification $\eta^{0}:\mathcal{G}_{<p,M}\simeq
G(\mathcal{L})$ from a) determined (up to conjugation) by a choice of a
suitable $e\in\mathcal{L}_{O_{M}(\mathcal{K})}$. Then the image $\bar{e}$ of
$e$ under the projection
$\mathcal{L}_{O_{M}(\mathcal{K})}\to\bar{\mathcal{L}}_{O_{M}(\mathcal{K})}$
determines the epimorphic map $\bar{\eta}^{0}:{\mathcal{G}}_{<p,M}\to
G(\bar{\mathcal{L}})$. On the other hand, choose an analog
$\bar{e}^{{\dagger}}$ of $\bar{e}$ for $\bar{\mathcal{L}}^{{\dagger}}$ which
is compatible with $\bar{e}$, i.e. such that
$(\bar{\mathcal{V}}\otimes\mathrm{id})\bar{e}^{{\dagger}}=\bar{e}$. This will
allow us to obtain a unique lift $\eta^{{\dagger}}:\mathcal{G}_{<p,M}\to
G(\bar{\mathcal{L}}^{{\dagger}})$ of $\bar{\eta}^{0}$ such that
$\bar{\mathcal{V}}\bar{\eta}^{{\dagger}}=\bar{\eta}^{0}$.

e) Note that $\bar{\mathcal{L}}^{{\dagger}}$ also could be provided with a
decreasing filtration of length $p$ such that the projection
$\bar{\mathcal{V}}:\bar{\mathcal{L}}^{{\dagger}}\to\bar{\mathcal{L}}$ is
epimorphism of filtered Lie algebras. As a matter of fact, the filtration on
$\bar{\mathcal{L}}^{{\dagger}}$ is defined in terms of invariant
$\mathrm{ch}(\iota)\in[0,p-1]$. This allows us to define the action $\Omega$
of the elements $\gamma\in\mathbb{Z}/p^{M}$ on the system of coefficients
$\mathfrak{A}^{0}$ by the correspondences $t^{-qp^{M-1}a}\mapsto
t^{-qp^{M-1}a}E(t^{b^{*}})^{a\gamma p^{M-1}}$, where $E$ is the Artin-Hasse
exponential.

f) We use the action $\Omega$ to obtain a deformation of the element
$\bar{e}^{{\dagger}}$ to
$\Omega({\bar{e}}^{{\dagger}})\in\bar{\mathcal{L}}^{{\dagger}}_{O_{M}(\mathcal{K})}$.
This element determines the deformation $\Omega(\bar{\eta}^{{\dagger}})$ of
$\bar{\eta}^{{\dagger}}$ which is given by a unipotent automorphism
$A^{{\dagger}}$ of the Lie algebra $\bar{\mathcal{L}}^{{\dagger}}$.
Equivalently, it appears from a differentiation $B^{{\dagger}}$ of
$\bar{\mathcal{L}}^{{\dagger}}$ such that
$A^{{\dagger}}=\widetilde{\exp}B^{{\dagger}}$ (this is the truncated
exponential). With this notation, our main result states that the ramification
ideal $\bar{\mathcal{L}}^{(v_{0})}$ appears as
$\bar{\mathcal{V}}B^{{\dagger}}(\mathrm{Ker}{\bar{\mathcal{V}}})$.

In the remaining part of the paper we are doing explicit calculations with
specially chosen element $\bar{e}$ to find explicit generators of
$\bar{\mathcal{L}}^{(v_{0})}$ in the style of paper [3].

As it has been mentioned above, this paper generalises and simplifies our
approach from [8], where we considered the case of Galois groups of period
$p$. The main difficulty to construct deformations in the modulo
$p^{M}$-situation was related to the construction of the filtration
$\mathcal{L}(w)$, cf. Sect. 1. This techniques could be applied now to develop
the modulo $p^{M}$-case of results obtained in the mixed characteristic case
in [5, 6] (in preparation). Note also that in [8] we gave a description of
segments of ramification filtration in the case of Galois groups of period
$p$. This approach also admits now a generalisation to the modulo
$p^{M}$-situation.

##  1\. Lie algebra $\mathcal{L}$ and its filtration

As earlier, $\mathcal{K}$ is a complete discrete valuation field of
characteristic $p$ with a finite residue field $k\simeq\mathbb{F}_{p^{N_{0}}}$
and a fixed uniformiser $t_{0}$, i.e. $\mathcal{K}=k((t_{0}))$. Denote by
$\mathcal{K}_{sep}$ a separable closure of $\mathcal{K}$. Let
$\mathcal{G}=\mathrm{Gal}(\mathcal{K}_{sep}/\mathcal{K})$. Then $\mathcal{G}$
acts on $\mathcal{K}_{sep}$ and for any $a\in\mathcal{K}_{sep}$ and
$g_{1},g_{2}\in\mathcal{G}$, $g_{1}(g_{2}a)=(g_{1}g_{2})a$. Denote by $\sigma$
the $p$-th power map on $\mathcal{K}$, $\mathcal{K}_{sep}$ and $k$.

###  1.1. Lie algebra $\mathcal{L}$

For any field extension $\mathcal{E}$ of $\mathcal{K}$ in $\mathcal{K}_{sep}$,
we have a lift modulo $p^{M}$,
$O_{M}(\mathcal{E}):=W_{M}(\sigma^{M-1}\mathcal{E})[t]\subset
W_{M}(\mathcal{E})$, where $t=[t_{0}]\in W_{M}(\mathcal{K})$ is the
Teichmuller representative of $t_{0}$, cf. e.g. [2]. The system of lifts
$O_{M}(\mathcal{E})$ is functorial in $\mathcal{E}$, the actions of
$\mathcal{G}$ and $\sigma$ can be naturally extended to
$O_{M}(\mathcal{K}_{sep})$. Note that
$O_{M}(\mathcal{K}_{sep})^{\mathcal{G}}=O_{M}(\mathcal{K})=W_{M}(k)((t))$ and
$O_{M}(\mathcal{K}_{sep})|_{\sigma=\mathrm{id}}=W_{M}(\mathbb{F}_{p})$.

Let $\mathbb{Z}^{+}(p)=\\{a\in\mathbb{N}\ |\ \mathrm{gcd}\,(a,p)=1\\}$,
$\mathbb{Z}^{0}(p)=\mathbb{Z}^{+}(p)\cup\\{0\\}$.

Let $\widehat{\mathcal{L}}_{W_{M}(k)}$ be a profinite free Lie
$W_{M}(k)$-algebra with the (topological) module of (free) generators

(1.1) $\\{D_{0}\\}\cup\\{D_{an}\ |\
a\in\mathbb{Z}^{+}(p),n\in\mathbb{Z}/N_{0}\\}\,.$

Define $\sigma$-action on $\widehat{\mathcal{L}}_{W_{M}(k)}$ by setting
$\sigma:D_{an}\mapsto D_{a,n+1}$ and $\sigma:D_{0}\mapsto D_{0}$. Then
$\widehat{\mathcal{L}}_{W_{M}(k)}|_{\sigma=\mathrm{id}}:=\widehat{\mathcal{L}}$
is a free profinite Lie $\mathbb{Z}_{p}$-algebra such that
$\widehat{\mathcal{L}}\otimes W_{M}(k)=\widehat{\mathcal{L}}_{W_{M}(k)}$. We
set $\mathcal{L}:=\widehat{\mathcal{L}}/C_{p}(\widehat{\mathcal{L}})$. This is
a maximal quotient of $\widehat{\mathcal{L}}$ of nilpotent class $<p$.

###  1.2. Ideals $\mathcal{L}(w)$, $w\geqslant 1$

Recall that we have a fixed $v_{0}\in\mathbb{R}_{>0}$.

######  Definition.

For $m\in\mathbb{Z}_{\geqslant 0}$ and $s\in\mathbb{N}$ , let $U(s,m)$ be the
minimal value of $1/p^{i_{1}}+\ldots+1/p^{i_{s}}$, where all $i_{j}$ run over
$\mathbb{Z}_{\geqslant 0}$ and satisfy the condition $i_{1}+\dots+i_{s}=m$. By
definition, set for any $m\geqslant 0$, $U(0,m)=0$.

######  Definition.

For $w\geqslant 1$, let $\mathcal{L}(w)$ be the minimal central filtration of
ideals in $\mathcal{L}$ such that if $a/v_{0}\geqslant U(w-1,m)$ then
$p^{m}D_{an}\in\mathcal{L}(w)_{W_{M}(k)}$.

Note that $\mathcal{L}(1)=\mathcal{L}$ and $\forall w_{1},w_{2}\in\mathbb{N}$,
$[\mathcal{L}(w_{1}),\mathcal{L}(w_{2})]\subset\mathcal{L}(w_{1}+w_{2})$.

######  Proposition 1.1.

Let $s,s^{\prime},m,m^{\prime}\in\mathbb{Z}_{\geqslant 0}$. Then:

a) $U(s,m)+U(s^{\prime},m^{\prime})\geqslant U(s+s^{\prime},m+m^{\prime})\,.$

b) For $s\geqslant 1$, let $m=su_{0}+s_{1}$, where $0\leqslant s_{1}<s$. Then

$U(s,m)=s_{0}/p^{u_{0}}+s_{1}/p^{u_{0}+1}\,\text{with }s_{0}=s-s_{1}\,.$

In particular, $u_{0},s_{0},s_{1}$ are uniquely determined by $m$ and $s$.

c) Suppose $s+s^{\prime}<p$. Then for any $M\geqslant 0$,

$\min\\{U(s,m)+U(s^{\prime},m^{\prime})\ |\
m+m^{\prime}=M\\}=U(s+s^{\prime},M)\,.$

###### Proof.

a) is obvious.

b) Suppose $m_{1},\dots,m_{s}\in\mathbb{Z}_{\geqslant 0}$,
$m_{1}+\ldots+m_{s}=m$ and

$U(s,m)=1/p^{m_{1}}+\cdots+1/p^{m_{s}}\,.$

If for some indices $1\leqslant i,j\leqslant s$ it holds that $m_{i}\geqslant
m_{j}+2$ then we obtain the contradiction by the use of the obvious inequality

$1/p^{m_{i}-1}+1/p^{m_{j}+1}<1/p^{m_{i}}+1/p^{m_{j}}\,.$

Therefore, the minimal value $U(s,m)$ can appear only if there is
$u_{0}\in\mathbb{N}$ such that among all numbers $m_{i}$, $1\leqslant
j\leqslant s$, there are $s_{0}>0$ of those which are equal to $u_{0}$ and all
remaining $s_{1}=s-s_{0}$ numbers are equal to $u_{0}+1$. It remains to note
that $m=m_{1}+\dots+m_{s}=u_{0}s_{0}+(u_{0}+1)s_{1}=su_{0}+s_{1}\,.$

c) Suppose $U(s,m)+U(s^{\prime},m^{\prime})$ is the minimal value of the LHS.
We can assume that $s,s^{\prime}\geqslant 1$. Let $m=si+s_{1}$ and
$m^{\prime}=s^{\prime}i^{\prime}+s_{1}^{\prime}$, where $0\leqslant s_{1}<s$
and $0\leqslant s_{1}^{\prime}<s^{\prime}$. Then for $s_{0}=s-s_{1}$ and
$s_{0}^{\prime}=s^{\prime}-s_{1}^{\prime}$,
$s_{0}+s_{1}+s_{0}^{\prime}+s_{1}^{\prime}<p$,
$U(s,m)=s_{0}/p^{i}+s_{1}/p^{i+1}$ and
$U(s^{\prime},m^{\prime})=s_{0}^{\prime}/p^{i^{\prime}}+s_{1}^{\prime}/p^{i^{\prime}+1}$.

$c_{1}$) Suppose $i=i^{\prime}$. Then
$U(s,m)+U(s^{\prime},m^{\prime})=U(s+s^{\prime},m+m^{\prime})$ and our
statement is proved.

$c_{2}$) Suppose $i>i^{\prime}$.

$c_{21}$) Let $s_{1}\neq 0$ and $s_{1}^{\prime}\neq s^{\prime}-1$. In this
case $m-1=si+(s_{1}-1)$ and
$m^{\prime}=s^{\prime}i^{\prime}+(s_{1}^{\prime}+1)$ imply that

(1.2) $U(s,m-1)+U(s^{\prime},m^{\prime}+1)<U(s,m)+U(s^{\prime},m^{\prime})$

because $U(s,m-1)=U(s,m)+(1/p^{i}-1/p^{i+1})$ and
$U(s^{\prime},m^{\prime}+1)=U(s^{\prime},m^{\prime})-(1/p^{i^{\prime}}-1/p^{i^{\prime}+1})$.

$c_{22}$) Let $s_{1}=0$ but $s_{1}^{\prime}\neq s^{\prime}-1$. In this case
$m-1=s(i-1)+(s-1)$ and $m+1=s^{\prime}i^{\prime}+(s^{\prime}_{1}+1)$ imply
that either we have again the contradiction (1.2) or $i=i^{\prime}+1$. But in
the later case, $m+m^{\prime}=(s+s^{\prime})i^{\prime}+s+s_{1}^{\prime}$ and

$U(s+s^{\prime},m+m^{\prime})=\frac{s^{\prime}}{p^{i^{\prime}}}+\frac{s+s_{1}^{\prime}}{p^{i^{\prime}+1}}=U(s,m)+U(s^{\prime},m^{\prime})\,.$

$c_{23}$) Let $s_{1}\neq 0$ but $s_{1}^{\prime}-1=s^{\prime}$. Here
$m-1=si+s_{1}-1$ and $m+1=(s^{\prime}+1)i^{\prime}$. Then
$U(s,m-1)=U(s,m)+(1/p^{i}-1/p^{i+1})$ and
$U(s^{\prime},m^{\prime}+1)=U(s^{\prime},m^{\prime})-(1/p^{i^{\prime}}-1/p^{i^{\prime}+1})$.
Then again we obtain contradiction (1.2) because $i>i^{\prime}$.

$c_{24}$) Let $s_{1}=0$ and $s_{1}^{\prime}=s^{\prime}-1$. Here
$m-1=s(i-1)+s-1$ and $m^{\prime}+1=s^{\prime}(i^{\prime}+1)$. We obtain either
contradiction (1.2) or $i-1=i^{\prime}$. In the later case
$m+m^{\prime}=s^{\prime}i^{\prime}+s+s^{\prime}-1$ and
$U(s+s^{\prime},m+m^{\prime})=U(s,m)+U(s^{\prime},m^{\prime})$. ∎

Example. Suppose $m=0$ and $s\geqslant 1$. Then $u_{0}=s_{1}=0$ and for all
$w\geqslant 1$, $U(w-1,0)=w-1$. If $(w-1)v_{0}\leqslant a<wv_{0}$ then
$D_{an}\in\mathcal{L}(w)_{W_{M}(k)}$. In particular, if $a\geqslant(p-1)v_{0}$
then $D_{an}\in\mathcal{L}(p)_{W_{M}(k)}$. Notice that in the modulo $p$
situation (i.e. when $M=1$) from [8] the ideals $\mathcal{L}(w)$ coincide with
the ideals introduced in that paper (and denoted by $\mathcal{L}(s)$).

Proposition 1.1b) means:

######  Proposition 1.2.

The filtration $\\{\mathcal{L}(w)\ |\ w\geqslant 1\\}$ is the minimal central
filtration of $\mathcal{L}$ such that for all $a\in\mathbb{Z}^{0}(p)$ and non-
negative integers $\alpha_{0}\neq 0,\alpha_{1},j$, hold the conditions

$\bullet$ $R_{a}(\alpha_{0},\alpha_{1},j)$: if
$a/v_{0}\geqslant\alpha_{0}/p^{j}+\alpha_{1}/p^{j+1}$ then

$p^{\alpha_{0}j+\alpha_{1}(j+1)}D_{an}\in\mathcal{L}(1+\alpha_{0}+\alpha_{1})_{W_{M}(k)}\,.$

###  1.3. Independent relations
$p^{M(a,s)}D_{an}\in\mathcal{L}(1+s)_{W_{M}(k)}$, $s<p$

######  Definition.

Given $a\in\mathbb{Z}^{0}(p)$ and $0\leqslant s<p$, let $M(a,s)$ be the
minimal non-negative integer such that
$p^{M(a,s)}D_{an}\in\mathcal{L}(1+s)_{W_{M}(k)}$.

######  Remark.

The restriction $s<p$ is introduced because we are going to work below with
the Lie algebra $\bar{\mathcal{L}}=\mathcal{L}/\mathcal{L}(p)$ provided with
the filtration by the ideals
$\bar{\mathcal{L}}(w)=\mathcal{L}(w)/\mathcal{L}(p)$.

Clearly, $M(a,0)=0$ and for all $n$,
$p^{m}D_{0}\in\mathcal{L}(1)_{W_{M}(k)}\setminus\mathcal{L}(2)_{W_{M}(k)}$. In
addition, if $a/v_{0}\geqslant U(p-1,0)=p-1$ then
$D_{an}\in\mathcal{L}(p)_{W_{M}(k)}$. Therefore, we can assume below that
$s\geqslant 1$ and $a<pv_{0}$.

Suppose $a/v_{0}=\sum_{i\geqslant 0}\theta_{i}(a)/p^{i}$ is the standard
decomposition with the $p$-digits $0\leqslant\theta_{i}(a)<p$. Let $u(a)$ be
the minimal index such that $\theta_{u(a)}(a)\neq 0$.

Set $u=u(a)$, $\kappa_{0}=\theta_{u}(a)$ and $\kappa_{1}=\theta_{u+1}(a)$.
Note that

$\kappa_{0}/p^{u}+\kappa_{1}/p^{u+1}\leqslant
a/v_{0}<\kappa_{0}/p^{u}+(\kappa_{1}+1)/p^{u+1}\,.$

Consider the relation $R_{a}(\alpha_{0},\alpha_{1},j)$ from Prop.1.2. Then

(1.3)
$\alpha_{0}/p^{j}+\alpha_{1}/p^{j+1}\leqslant\kappa_{0}/p^{u}+\kappa_{1}/p^{u+1}\,,$

and
$p^{\alpha_{0}j+\alpha_{1}(j+1)}D_{an}\in\mathcal{L}(1+\alpha_{0}+\alpha_{1})_{W_{M}(k)}\,.$

Note that $\alpha_{0}+\alpha_{1}\geqslant p-1$ implies
$p^{\alpha_{0}j+\alpha_{1}(j+1)}D_{an}\in\mathcal{L}(p)_{W_{M}(k)}$. If we
want to obtain the relations for the ideals $\mathcal{L}(s)$ modulo
$\mathcal{L}(p)$ then it makes sense to restrict ourselves with the cases
where $\alpha_{0}+\alpha_{1}<p$. In particular,
$\alpha_{0}/p^{j}+\alpha_{1}/p^{j+1}$ will appear as a standard decomposition
in base $p$ with $p$-digits $\alpha_{0},\alpha_{1}$. Now inequality (1.3)
implies one of the following options:

– $j\geqslant u+1$;

– $j=u$ and $\alpha_{0}<\kappa_{0}$;

– $j=u$, $\alpha_{0}=\kappa_{0}$ and $\alpha_{1}\leqslant\kappa_{1}$.

Proceeding along these lines we obtain the following list of relations of the
form $p^{M(a,s)}D_{an}\in\mathcal{L}(1+s)_{W_{M}(k)}$:

######  Proposition 1.3.

For $(\alpha_{0},\alpha_{1},u)$ such that:

a) $1\leqslant\alpha_{0}\leqslant\kappa_{0}$ and $\alpha_{1}=0$, we have
$s=\alpha_{0}$ and

$M(a,s)=\alpha_{0}u\,;$

b) $\alpha_{0}=\kappa_{0}$ and $1\leqslant\alpha_{1}\leqslant\kappa_{1}$, we
have $s=\kappa_{0}+\alpha_{1}$ and

$M(a,s)=M(a,\kappa_{0})+\alpha_{1}(u+1)\,;$

c) $\alpha_{0}=\kappa_{0}-1$, $\alpha_{1}=\kappa_{1}+2$, we have
$s=\kappa_{0}+\kappa_{1}+1$ and

$M(a,s)=M(a,\kappa_{0}+\kappa_{1})+u+2\,;$

d) $\alpha_{0}=\kappa_{0}-1$ and $\alpha_{1}=\kappa_{1}+2+t<p$ with
$t\geqslant 1$, we have
$s=\kappa_{0}+\alpha_{1}$ and

$M(a,s)=M(a,\kappa_{0}+\kappa_{1}+1)+t(u+1)\,.$

######  Remark.

Taking into account that we need relations modulo $\mathcal{L}(p)_{k}$:

– for $s\in[0,\kappa_{0}]$, $M(a,s)$ is increasing in $s$ with step $u$;

– for $s\in[\kappa_{0},\min\\{\kappa_{0}+\kappa_{1},p-1\\}]$ the step is
$u+1$;

– for $s\in[\kappa_{0}+\kappa_{1},\min\\{\kappa_{0}+\kappa_{1}+1,p-1\\}]$ the
step is $u+2$;

– if $s\in[\kappa_{0}+\kappa_{1}+1,p-1]$ the step is $u+1$.

######  Proposition 1.4.

Relations from Proposition 1.3 with $s<p$ are all independent relations of the
form $p^{A}D_{an}\in\mathcal{L}(1+s)_{W_{M}(k)}$, where $A\geqslant 0$ and
$0\leqslant s\leqslant p-1$.

###### Proof.

The independence is obvious because for each $a$, the numbers $M(a,s)$ are
strictly increasing in $s$.

Let $a/v_{0}\geqslant s_{0}/p^{j}+s_{1}/p^{j+1}$ with integers $j\geqslant 0$,
$s_{0}\geqslant 1$ and $s_{1}\geqslant 0$. Note that $s_{0}$ and $s_{1}$ are
not necessarily $p$-digits.

Prove that the corresponding condition
$p^{m}D_{an}\in\mathcal{L}(1+s)_{W_{M}(k)}$, where $m=s_{0}j+s_{1}(j+1)$ and
$s=\min\\{s_{0}+s_{1},p-1\\}$, is implied by one of relations listed in
Proposition 1.3.

Otherwise, there is at least one triple $(\alpha_{0},\alpha_{1},u)$ listed in
Prop. 1.3 such that the following two conditions hold

(1.4) $\alpha_{0}+\alpha_{1}\leqslant s_{0}+s_{1}$ (1.5)
$\alpha_{0}u+\alpha_{1}(u+1)>s_{0}j+s_{1}(j+1)\,.$

Then $s_{0}\geqslant
s_{0}-\alpha_{0}\geqslant(\alpha_{0}+\alpha_{1}-s_{0}-s_{1})(u+1)+(s_{0}-\alpha_{0})=\alpha_{0}u+\alpha_{1}(u+1)-(s_{0}u+s_{1}(u+1))>(s_{0}+s_{1})(j-u)$
implies that $j\leqslant u$.

If $j<u$ then $p/p^{u}>a/v_{0}\geqslant s_{0}/p^{j}$ implies
$p>s_{0}p^{u-j}\geqslant p$. The contradiction.

Therefore, $j=u$. Then (1.4) implies $\alpha_{1}>s_{1}$ because

$0\geqslant(\alpha_{0}+\alpha_{1}-s_{0}-s_{1})u>s_{1}-\alpha_{1}\,.$

In particular, $\alpha_{1}\neq 0$ and the couple $(\alpha_{0},\alpha_{1})$
does not come from condition a).

The condition $\alpha_{0}+\alpha_{1}\leqslant s_{0}+s_{1}$ also implies that
$\alpha_{0}<s_{0}$. This excludes the case b) because
$s_{0}\leqslant\kappa_{0}$ (use that $j=u$).

It remains to consider the cases c) and d), where $\alpha_{0}=\kappa_{0}-1$
and $s_{0}=\kappa_{0}$. Here $\alpha_{0}=\kappa_{0}-1$,
$\alpha_{1}\geqslant\kappa_{1}+2$, $s_{0}=\alpha_{0}+1$, and
$s_{1}\leqslant\kappa_{1}$. Then we obtain the contradiction
$\alpha_{0}+\alpha_{1}\geqslant\kappa_{0}+\kappa_{1}+1>s_{0}+s_{1}$. ∎

###  1.4. Explicit description of ideals $\mathcal{L}(w)_{W_{M}(k)}$

Recall that the ideal $\mathcal{L}(w)_{W_{M}(k)}$ is generated as
$W_{M}(k)$-module by all commutators

$p^{m}[\dots[D_{a_{1}n_{1}},D_{a_{2}n_{2}}],\dots,D_{a_{r}n_{r}}]\,:=p^{m}[D_{\bar{a}\bar{n}}]$

where $\bar{a}=(a_{1},\ldots,a_{r})$, for $1\leqslant i\leqslant
r:=r(\bar{a})$, there are $m_{i},s_{i}\geqslant 0$ such that
$m=\sum_{i}m_{i}$, $r+\sum_{i}s_{i}\geqslant w$ and all $a_{i}\geqslant
U(s_{i},m_{i})v_{0}$.

######  Definition.

Set $M(\bar{a},s)=\min\\{m\in\mathbb{Z}_{\geqslant 0}\ |\
p^{m}[D_{\bar{a}\bar{n}}]\in\mathcal{L}(r(\bar{a})+s)_{W_{M}(k)}\\}$.

We always assume that $r=r(\bar{a})<p$ because
$D_{\bar{a}\bar{n}}\in\mathcal{L}(r)_{W_{M}(k)}$.

For $1\leqslant i\leqslant r$, consider the following decompositions in the
“base $p$” with $p$-digits from $[0,p)$:

$a_{i}/v_{0}=\sum_{j\geqslant 0}\theta_{j}(a_{i})p^{-j}\,,\ 1\leqslant
i\leqslant r\,.$

Set $\bar{u}=u(\bar{a})=\min\\{\ j\ |\
(\theta_{j}(a_{1}),\ldots,\theta_{j}(a_{r}))\neq\bar{0}\\}$.

Set $\bar{\kappa}_{0}=\kappa_{0}(\bar{a})=\sum_{1\leqslant i\leqslant
r}\theta_{\bar{u}}(a_{i})$ and
$\bar{\kappa}_{1}=\kappa_{1}(\bar{a})=\sum_{1\leqslant i\leqslant
r}\theta_{\bar{u}+1}(a_{i})$.

######  Proposition 1.5.

a) $M(\bar{a},j)=j\bar{u}$ for $0\leqslant
j\leqslant\min\\{p-r,\bar{\kappa}_{0}\\}$;

b) $M(\bar{a},\bar{\kappa}_{0}+j)=\bar{u}\bar{\kappa}_{0}+(\bar{u}+1)j$, for
$1\leqslant j\leqslant\min\\{p-(r+\bar{\kappa}_{0}),\bar{\kappa}_{1}\\}$;

c)
$M(\bar{a},\bar{\kappa}_{0}+\bar{\kappa}_{1}+j)=\bar{\kappa}_{0}\bar{u}+\bar{\kappa}_{1}(\bar{u}+1)+1+j(\bar{u}+1)$,
where $j$ runs over $1\leqslant j\leqslant
p-(r+\bar{\kappa}_{0}+\bar{\kappa}_{1})$.

######  Remark.

If $\bar{u}=0$ then
$[D_{\bar{a}\bar{n}}]\in\mathcal{L}(r+\bar{\kappa}_{0})_{W_{M}(k)}$.

###  1.5. $\mathbb{Z}_{p}$-submodules
$\mathcal{L}_{\mathcal{N}}(w)\subset\mathcal{L}(w)$

######  Definition.

For $w\geqslant 1$, let $\mathcal{L}_{\mathcal{N}}(w)$ be the closed submodule
in $\mathcal{L}$ such that $\mathcal{L}_{\mathcal{N}}(w)_{W_{M}(k)}$ is
generated by all $p^{m}[D_{\bar{a}\bar{n}}]$ which satisfy the inequality
$|\bar{a}|\geqslant v_{0}U(w-1,m)$.

######  Proposition 1.6.

We have the following decomposition of $\mathbb{Z}_{p}$-modules

$\mathcal{L}_{\mathcal{N}}(w)\cap
C_{2}(\mathcal{L})\subset\sum_{w_{1}+w_{2}\geqslant
w}[\mathcal{L}_{\mathcal{N}}(w_{1}),\mathcal{L}_{\mathcal{N}}(w_{2})]$

It will be sufficient to prove the following lemma.

######  Lemma 1.7.

If
$p^{m}[D_{\bar{a}_{1}\bar{n}_{1}},D_{\bar{a}_{2}\bar{n}_{2}}]\in\mathcal{L}_{\mathcal{N}}(w)_{W_{M}(k)}$
then for $i=1,2$, there are $m_{i},w_{i}$ such that:

1) $m_{1}+m_{2}=m$, $w_{1}+w_{2}=w$;

2)
$p^{m_{i}}[D_{\bar{a}_{i}\bar{n}_{i}}]\in\mathcal{L}_{\mathcal{N}}(w_{i})_{W_{M}(k)}$.

###### Proof of Lemma.

$\bullet$ Choose $w_{1},w_{2}\geqslant 1$ with maximal $w_{1}+w_{2}$ s.t. for
some $m_{1},m_{2}$,  $m=m_{1}+m_{2}$ and for $i=1,2$, $|\bar{a}_{i}|\geqslant
v_{0}U(w_{i}-1,m_{i})\,.$

We must prove that $w_{1}+w_{2}\geqslant w$.

Suppose $w_{1}+w_{2}<w$.

We can assume that (otherwise, $w_{1}+w_{2}$ is not maximal) for $i=1,2$,

$v_{0}U(w_{i},m_{i})>|\bar{a}_{i}|\geqslant v_{0}U(w_{i}-1,m_{i})\,.$

Let $v_{0}U(w_{1}+w_{2},m)>|\bar{a}_{1}|+|\bar{a}_{2}|$. Then
$U(w_{1}+w_{2},m)>U(w-1,m)$ implies $w_{1}+w_{2}\geqslant w$, q.e.d.

Otherwise, $|\bar{a}_{1}|+|\bar{a}_{2}|\geqslant v_{0}U(w_{1}+w_{2},m)$. By
Prop. 1.1c), $\exists m_{1}^{\prime},m_{2}^{\prime}$ such that
$m_{1}^{\prime}+m_{2}^{\prime}=m$ $\&$
$U(w_{1},m_{1}^{\prime})+U(w_{2},m_{2}^{\prime})=U(w_{1}+w_{2},m)$.

Let $m_{1}^{\prime}\geqslant m_{1}$ and $m_{2}^{\prime}\leqslant m_{2}$. Then
$U(w_{1},m_{1}^{\prime})\geqslant U(w_{1},m_{1})>v_{0}|\bar{a}_{1}|$ and
$U(w_{2},m_{2}^{\prime})\leqslant U(w_{2},m_{2})$.

If $|\bar{a}_{2}|<v_{0}U(w_{2},m^{\prime}_{2})$ then
$v_{0}U(w_{1}+w_{2},m)>|\bar{a}_{1}|+|\bar{a}_{2}|$. The contradiction.

If $|\bar{a}_{2}|\geqslant v_{0}U(w_{2},m_{2}^{\prime})$ then $w_{2}$ can be
replaced with a bigger one and $w_{1}+w_{2}$ is not maximal. The lemma is
proved. ∎

######  Corollary 1.8.

$\mathcal{L}_{\mathcal{N}}(w)\subset\mathcal{L}(w)$.

###### Proof.

By induction on $w$ it will be enough to verify the statement for
$p^{m}[D_{\bar{a}\bar{n}}]$ with $r(\bar{a})=1$. But this follows from the
definition of $\mathcal{L}(w)$. ∎

######  Definition.

Let $\bar{\mathcal{L}}=\mathcal{L}/\mathcal{L}(p)$ with the induced filtration
by the ideals $\bar{\mathcal{L}}(w)=\mathcal{L}(w)/\mathcal{L}(p)$,
$w\geqslant 1$.

Clearly, $\bar{\mathcal{L}}(p)=0$ and $\bar{\mathcal{L}}_{W_{M}(k)}$ is
generated by $D_{0}$ and all $D_{an}$ with $a<(p-1)v_{0}$.

##  2\. Lie algebra $\bar{\mathcal{L}}^{\,{\dagger}}$ and ideal
$\bar{\mathcal{L}}^{[v_{0}]}\subset\bar{\mathcal{L}}$

###  2.1. Parameters $r^{*}$ and $N^{*}$

Fix $u^{*},w^{*}\in\mathbb{N}$. (Below we will specify $u^{*}=(p-1)(p-2)+1$
and $w^{*}=(p-1)v_{0}$.)

For $1\leqslant s<p$, $0\leqslant m<M$, denote by $\delta_{0}(s,m)$ the
minimum of positive values of

$v_{0}-\frac{1}{U(s,m)}(a_{1}p^{n_{1}}+a_{2}p^{n_{2}}+\dots+a_{u}p^{n_{u}})\,,$

where $u\leqslant u^{*}$, all integers $n_{i}\leqslant M-1$ and
$a_{i}\in[0,w^{*})\cap\mathbb{Z}$. The existence of such $\delta_{0}(s,m)$ can
be proved easily by induction on $u$ for any fixed $s$.

Set $\delta_{0}:=\min\\{\delta_{0}(s,m)\ |\ 1\leqslant s<p,\ 0\leqslant
m<M\\}$.

Let $r^{*}\in\mathbb{Q}$ be such that $r^{*}=b_{0}^{*}/(q^{*}_{0}-1)$, where
$q^{*}_{0}=p^{N^{*}_{0}}$ with $N^{*}_{0}\geqslant 2$,
$b^{*}_{0}\in\mathbb{N}$ and $\mathrm{gcd}(b_{0}^{*},\,p(q^{*}_{0}-1))=1$. The
set of such $r^{*}$ is dense in $\mathbb{R}_{>0}$ and we can assume that
$r^{*}\in(v_{0}-\delta_{0},v_{0})$.

For $1\leqslant u\leqslant u^{*}$, introduce the following subsets in
$\mathbb{Q}$ :

— $A[u]$ is the set of all

$a_{1}p^{n_{1}}+a_{2}p^{n_{2}}+\dots+a_{u}p^{n_{u}}\,,$

where $n_{1},\dots,n_{u}\leqslant 0$, $n_{1}=0$ and all
$a_{i}\in[0,w^{*})\cap\mathbb{Z}$. For $m\in\mathbb{Z}_{\geqslant 0}$, denote
by $A[u,m]$ the subset of $A[u]$ consisted of the elements such that all
$n_{u}\geqslant-m$.

— $B[u]$ is the set of all numbers

$r^{*}(b_{1}p^{m_{1}}+b_{2}p^{m_{2}}+\dots+b_{u}p^{m_{u}})\,,$

where all $m_{1},\ldots,m_{u}\leqslant M-1$, $m_{1}\geqslant 0$,
$b_{i}\geqslant 0$, $b_{1}\neq 0$, $b_{1}+\dots+b_{u}<p$. (In particular,
$0\notin B[u]$.) For $m\in\mathbb{Z}_{\geqslant 0}$, $B[u,m]$ is the subset of
$B[u]$ consisted of the elements such that all $m_{u}\geqslant-m$.

The sets $A[u]$ and an analog of the set $B[u]$ (where we required $m_{1}=0$
instead of $0\leqslant m_{1}\leqslant M-1$) were introduced in [8]. Proceeding
similarly to [8] we obtain that for any $u\leqslant u^{*}$, $A[u]\cap
B[u]=\emptyset$ and there is $\rho(u^{*})>0$ such that

$\rho(A[u],B[u]):=\inf\\{|\alpha-\beta|\ |\ \alpha\in A[u],\beta\in
B[u]\\}\geqslant\rho(u^{*})\,.$

Fix the values $u^{*}=(p-1)(p-2)+1$ and $w^{*}=(p-1)v_{0}$
(since $u^{*}\geqslant p-1$, $B[u^{*}]=B[p-1]$).

Choose $N^{*}\in\mathbb{N}$ satifying the following conditions:

C1) $N^{*}\equiv 0\,\mathrm{mod}\,N^{*}_{0}$;

C2) $p^{N^{*}}\rho(A[u^{*}],B[u^{*}])\geqslant 2r^{*}(p-1)$;

C3) $r^{*}(1-p^{-N^{*}})\in(v_{0}-\delta_{0},v_{0})$.

Introduce $q=p^{N^{*}}$ and $b^{*}=b^{*}_{0}(q-1)/(q_{0}-1)\in\mathbb{N}$.

Note that $r^{*}=b^{*}/(q-1)$ and $b^{*}\in\mathbb{Z}^{+}(p)$.

######  Proposition 2.1.

If $\alpha\in A[u^{*}]$ and $\beta\in B[u^{*}]$ then

$q\,|q\alpha-(q-1)\beta|>b^{*}(p-1)\,.$

###### Proof.

Indeed, the left-hand side of our inequality equals

$q\,|q\alpha-(q-1)\beta|=q^{2}|\alpha-\beta+\beta/q|\geqslant
q^{2}|\alpha-\beta|-\beta q\geqslant q^{2}\rho(A[u^{*}],\,B[u^{*}])$
$-r^{*}(p-1)q\geqslant 2r^{*}(p-1)q-r^{*}(p-1)q=r^{*}(p-1)q>b^{*}(p-1)\,.$

∎

###  2.2. The set $\mathfrak{A}^{0}$

Use the above parameters $r^{*}$, $N^{*}$, $q=p^{N^{*}}$.

######  Definition.

${\mathfrak{A}}^{0}$ is the set of all
$\iota=p^{m}(qp^{M-1}\alpha-(q-1)\beta)$, where $m\geqslant 0$, $\alpha\in
A[u^{*},m]$, $\beta\in B[u^{*},m]\cup\\{0\\}$ and $|\iota|\leqslant
p^{M-1}b^{*}(p-1)$. (Note that $p^{m}\alpha\in\mathbb{Z}_{\geqslant 0}$ and
$p^{m}\beta/r^{*}\in\mathbb{N}$.)

######  Remark.

a) Prop.2.1 implies that for all $\iota\in\mathfrak{A}^{0}$, we have
$m<N^{*}$.

b) Any $p^{m}\alpha$ appears in the form
$a_{1}p^{n_{1}}+\ldots+a_{u^{*}}p^{n_{u^{*}}}$, where all
$a_{i}\in[0,(p-1)v_{0})$ and $0\leqslant n_{i}<N^{*}$.

c) All $p^{m}\beta$ appear in the form
$b^{*}(b_{1}p^{m_{1}}+\ldots+b_{u^{*}}p^{m_{u^{*}}})$, where all
$m_{i}\geqslant 0$ and $1\leqslant b_{1}+\ldots+b_{u^{*}}\leqslant p-1$.

Let $\mathfrak{A}^{0}_{0}:=\\{\iota\in\mathfrak{A}^{0}\ |\ \beta=0\\}$.

######  Lemma 2.2.

Suppose $\iota=p^{m}(q\alpha-(q-1)\beta)\in\mathfrak{A}^{0}$. Then:

a) $\mathfrak{A}^{0}_{0}=\\{qp^{M-1}a\ |\
a\in[0,(p-1)v_{0})\cap\mathbb{Z}\,\\}$;

b) if $\beta\neq 0$ then $m<N^{*}$ (in particular, $\mathfrak{A}^{0}$ is
finite);

c) the integers $p^{m}\alpha$ and $p^{m}\beta/r^{*}$ do not depend on the
presentation of $\iota$ in the form $p^{m}(qp^{M-1}\alpha-(q-1)\beta)$ from
the definition of $\mathfrak{A}^{0}$.

###### Proof.

a) If $\iota\in\mathfrak{A}^{0}_{0}$ then
$\iota=qp^{M-1}(p^{m}\alpha)\in\mathfrak{A}^{0}_{0}\subset\mathfrak{A}^{0}$
means that

$p^{m}\alpha/(p-1)\leqslant
b^{*}/q=r^{*}(1-q^{-1})\in(v_{0}-\delta_{0},v_{0})\,.$

Note that $U(p-1,0)=p-1$ and $\delta_{0}\leqslant\delta(p-1,0)$, cf. Sect.2.1.
Threfore, the inequalities $p^{m}\alpha/(p-1)<v_{0}$ and
$p^{m}\alpha/(p-1)\leqslant v_{0}-\delta_{0}$ are equivalent. As a result,

$\mathfrak{A}^{0}_{0}\subset\\{qp^{M-1}a\ |\
a\in[0,(p-1)v_{0})\cap\mathbb{Z}\\}\,.$

The opposite embedding is obvious.

b) If $m\geqslant N^{*}$ then by Prop. 2.1, $|\iota|>p^{M-1}b^{*}(p-1)$ i.e.
$\iota\notin\mathfrak{A}^{0}$.

c) If $\iota=p^{m^{\prime}}(qp^{M-1}\alpha^{\prime}-(q-1)\beta^{\prime})$ is
another presentation of $\iota$ then $p^{m}\beta/r^{*}$ and
$p^{m^{\prime}}\beta^{\prime}/r^{*}$ are non-negative congruent modulo
$qp^{M-1}$ integers and the both are smaller than $qp^{M-1}$. Indeed, we have

$\beta/r^{*}=b_{1}p^{m_{1}}+b_{2}p^{m_{2}}+\dots+b_{u}p^{m_{u}}\,,$

where all $m_{i}\leqslant M-1$ and $u\leqslant u^{*}$, then

$p^{m}\beta/r^{*}\leqslant p^{m+M-1}(b_{1}+\dots+b_{u})\leqslant
p^{m+M-1}(p-1)<p^{m+M}\leqslant qp^{M-1}$

because $m<N^{*}$. Similarly, $p^{m^{\prime}}\beta^{\prime}/r^{*}<qp^{M-1}$.
Therefore, they coincide and this implies also that
$p^{m}\alpha=p^{m^{\prime}}\alpha^{\prime}$. ∎

######  Corollary 2.3.

Suppose that $\iota=p^{m}(qp^{M-1}\alpha-(q-1)\beta)\in\mathfrak{A}^{0}$. Then
the sum of the “ $p$-digits” $b_{1}+\dots+b_{u}$ of the appropriate
$\beta/r^{*}=b_{1}p^{m_{1}}+b_{2}p^{m_{2}}+\dots+b_{u}p^{m_{u}}$ depends only
on $\iota$.

######  Definition.

$\mathrm{ch}\,\iota:=b_{1}+\dots+b_{u}$ .

######  Remark.

If $\mathrm{ch}\,\iota\geqslant 1$ then $\iota\neq 0$. (Use that $A[u]\cap
B[u]=\emptyset$.)

In the notation from Sect. 2.2 let
$\iota=p^{m}(qp^{M-1}\alpha-(q-1)\beta)\in{\mathfrak{A}}^{0}$. By Lemma 2.2
$p^{m}\alpha$ depends only on $\iota$ and can be presented (non-uniquely) in
the form $a_{1}p^{n_{1}}+a_{2}p^{n_{2}}+\dots+a_{u}p^{n_{u}}$ where all
coefficients $a_{i}\in[0,(p-1)v_{0}))\cap\mathbb{Z}$, $0\leqslant
n_{i}\leqslant m$, $n_{1}=m$ and $u\leqslant u^{*}$.

######  Definition.

$\kappa(\iota)$ is the maximal natural number such that for any above
presentation of $p^{m}\alpha$, $\kappa(\iota)\leqslant u$.

######  Remark.

a) If $\iota\in\mathfrak{A}^{0}$ then $\kappa(\iota)\leqslant u^{*}$ and
$\mathrm{ch}(\iota)\leqslant p-1$;

b) if $\iota\in\mathfrak{A}^{0}_{0}$ then $\mathrm{ch}(\iota)=0$;

c) if $\iota\in\mathfrak{A}^{0}_{0}$ and $\iota\neq 0$ then $\kappa(\iota)=1$.

###  2.3. Lie algebras ${\mathcal{L}}^{\,{\dagger}}$ and
$\bar{\mathcal{L}}^{\,{\dagger}}$

Let $\iota=p^{m}(qp^{M-1}\alpha-(q-1)\beta)\in\mathfrak{A}^{0}$ be given in
the standard notation from Sect. 2.2.

######  Definition.

The subset $\mathfrak{A}^{+}(p)$ consists of $\iota\in\mathfrak{A}^{0}$ such
that

— $\iota>0$;

— $\mathrm{gcd}(p^{m}\alpha\,,p^{m}\beta/r^{*},\,p)=1$;

— $\kappa(\iota)\leqslant(p-2)\mathrm{ch}(\iota)+1$.

######  Remark.

For $\iota\in\mathfrak{A}^{+}(p)$,
$(p-2)\mathrm{ch}(\iota)+1\leqslant(p-2)(p-1)+1=u^{*}\,.$

The elements of $\\{t^{-\iota}\ |\ \iota\in\mathfrak{A}^{+}(p)\\}$ behave
“well” modulo $(\sigma-\mathrm{id})\mathcal{K}$, i.e. the natural map
$\sum\limits_{\iota\in\mathfrak{A}^{+}(p)}kt^{-\iota}\longrightarrow\mathcal{K}/(\sigma-\mathrm{id})\mathcal{K}$
is injective. This is implied by the following proposition.

######  Proposition 2.4.

Let $v_{p}$ be the $p$-adic valuation such that $v_{p}(p)=1$.

a) Then all $\iota p^{-v_{p}(\iota)}$, where $\iota\in\mathfrak{A}^{+}(p)$,
are pairwise different.

b) If $\iota\in\mathfrak{A}^{+}(p)$ and $\mathrm{ch}(\iota)=1$ then $\iota
p^{-v_{p}(\iota)}\geqslant qv_{0}-b^{*}$.

###### Proof.

a) Suppose $\iota=p^{m}(qp^{M-1}\alpha-(q-1)\beta)\in\mathfrak{A}^{+}(p)$.

If $\mathrm{ch}(\iota)=0$ then $\iota\mapsto\iota p^{-v_{p}(\iota)}$
identifies $\\{\iota\in\mathfrak{A}^{+}(p)\ |\ \mathrm{ch}(\iota)=0\\}$ with
$\mathbb{Z}^{+}(p)\cap[0,(p-1)v_{0})$, cf. Lemma 2.2a).

If $\mathrm{ch}(\iota)\geqslant 1$ then $\iota p^{-m}\notin p^{M}\mathbb{N}$,
i.e. $m+M>v_{p}(\iota)$.

Indeed, $\iota p^{-m}=qp^{M-1}\alpha-(q-1)\beta\in p^{M}\mathbb{N}$ implies
(use that $qp^{M-1}\alpha\in p^{M}\mathbb{N}$ because $m<N^{*}$) that
$\beta=p^{-m}(b_{1}p^{m_{1}}+b_{2}p^{m_{2}}+\dots+b_{u}p^{m_{u}})\in
p^{M}\mathbb{N}$ with all $m_{i}\leqslant m+M-1$. But this number is
$\leqslant p^{M-1}\mathrm{ch}\,\iota<p^{M}$. The contradiction. Then Prop. 2.1
implies that

$\iota p^{-v_{p}(\iota)}\geqslant\iota
p^{-m-M+1}=p^{-M+1}|qp^{M-1}\alpha-(q-1)\beta|>$
$(b^{*}/q)(p-1)=r^{*}(1-q^{-1})(p-1)>(v_{0}-\delta_{0})(p-1)$

(use property C3 from Sect. 2.1).

Finally, if $\iota\in\mathfrak{A}^{0}_{0}$ then $\iota
p^{-v_{p}(\iota)}=a<(p-1)v_{0}$ implies that $a<(v_{0}-\delta_{0})(p-1)$ by
the choice of $\delta_{0}$, cf. Sect. 2.1. On the other hand, for all
$\iota\in\mathfrak{A}^{+}(p)$ with $\mathrm{ch}(\iota)\geqslant 1$, the values
$\iota p^{-v_{p}(\iota)}$ are different (use that
$\mathrm{gcd}(p^{m}\alpha,p^{m}\beta/r^{*})\not\equiv 0\,\mathrm{mod}\,p$) and
are bigger than $(v_{0}-\delta_{0})(p-1)$.

b) Here there is $0\leqslant A\leqslant M-1$ such that
$\iota=p^{,}(qp^{M-1}\alpha-p^{-A}b^{*})$, i.e. $\iota
p^{-v_{p}(\iota)}=q(p^{A}\alpha)-b^{*}$ and b) holds when
$p^{A}\alpha\geqslant v_{0}$. But if $p^{A}\alpha=\alpha/U(1,A)<v_{0}$ then
$\iota p^{-v_{p}(\iota)}\leqslant q(v_{0}-\delta_{0}-r^{*}(q-1)/q))<0$, cf.
condition C3 from Sect. 2.1. The contradiction.

The proposition is completely proved. ∎

######  Definition.

$\mathfrak{A}^{0}(p)=\mathfrak{A}^{+}(p)\cup\\{0\\}$.

Let $\widehat{\mathcal{L}}_{W_{M}(k)}^{\,{\dagger}}$ be the Lie algebra over
$W_{M}(k)$ with the set of free generators

$\\{\ D^{\,{\dagger}}_{\iota n}\ |\
\iota\in\mathfrak{A}^{+}(p),n\in\mathbb{Z}/N_{0}\\}\cup\\{D^{\,{\dagger}}_{0}\\}\,.$

Use notation $\sigma$ for the $\sigma$-linear automorphism of
$\widehat{\mathcal{L}}^{\,{\dagger}}_{W_{M}(k)}$ such that
$\sigma:D^{\,{\dagger}}_{\iota n}\mapsto D^{\,{\dagger}}_{\iota,n+1}$ and
$\sigma:D^{{\dagger}}_{0}\mapsto D^{{\dagger}}_{0}$. Introduce the Lie
$\mathbb{Z}/p^{M}$-algebras
$\widehat{\mathcal{L}}^{\,{\dagger}}:=\widehat{\mathcal{L}}^{\,{\dagger}}_{W_{M}(k)}|_{\sigma=\mathrm{id}}$
and
$\mathcal{L}^{\,{\dagger}}=\widehat{\mathcal{L}}^{\,{\dagger}}/C_{p}(\widehat{\mathcal{L}}^{\,{\dagger}})$.
Note that $\widehat{\mathcal{L}}^{\,{\dagger}}\otimes
W_{M}(k)=\widehat{\mathcal{L}}^{\,{\dagger}}_{W_{M}(k)}$, and this matches the
agreement about extensions of scalars from the Introduction.

Denote by $\\{\mathcal{L}^{\,{\dagger}}(w)\\}_{w\geqslant 1}$ the minimal
central filtration of $\mathcal{L}^{\,{\dagger}}$ such that
$p^{m}D^{\,{\dagger}}_{\iota n}\in\mathcal{L}^{\,{\dagger}}(w)_{W_{M}(k)}$ if
$\iota\geqslant p^{M-1}b^{*}U(w-1,m)$.

This means that $\mathcal{L}^{\,{\dagger}}(w)_{W_{M}(k)}$ is an ideal in
$\mathcal{L}^{\,{\dagger}}_{W_{M}(k)}$ generated as $W_{M}(k)$-module by all
$p^{m}[\dots[D^{\,{\dagger}}_{\iota_{1}n_{1}},D^{\,{\dagger}}_{\iota_{2}n_{2}}],\dots,D^{\,{\dagger}}_{\iota_{r}n_{r}}]$
such that for some integers $m_{1},\dots,m_{r},s_{1},\dots,s_{r}\geqslant 0$
we have:

a) $\iota_{j}\geqslant p^{M-1}b^{*}U(s_{j},m_{j})$, where $j=1,\ldots,r$;

b) $r+\sum_{1\leqslant j\leqslant r}s_{j}\geqslant w$ and $\sum_{1\leqslant
j\leqslant r}m_{j}\leqslant m$.

Note that for any $w\geqslant 1$,
$C_{w}(\mathcal{L}^{\,{\dagger}})\subset\mathcal{L}^{\,{\dagger}}(w)$.

For $m\in\mathbb{Z}_{\geqslant 0}$, let
$\mathcal{L}^{\,{\dagger}}[m]_{W_{M}(k)}$ be the $W_{M}(k)$-submodule of
$\mathcal{L}^{{\dagger}}_{W_{M}(k)}$ generated by all monomials
$[D^{\,{\dagger}}_{\bar{\iota}\bar{n}}]:=[\ldots[D^{\,{\dagger}}_{\iota_{1}n_{1}},\ldots],D^{\,{\dagger}}_{\iota_{r}n_{r}}]$
such that $\mathrm{ch}(\iota_{1})+\dots+\mathrm{ch}(\iota_{r})=m$. Denote by
$\mathcal{L}^{{\dagger}}[m]$ the $\mathbb{Z}_{p}$-submodule of
$\mathcal{L}^{{\dagger}}$ such that $\mathcal{L}^{{\dagger}}[m]\otimes
W_{M}(k)=\mathcal{L}^{{\dagger}}[m]_{W_{M}(k)}$.

######  Definition.

For $\bar{\iota}=(\iota_{1},\dots,\iota_{r})$ set $r(\bar{\iota})=r$,
$\mathrm{ch}(\bar{\iota})=\mathrm{ch}(\iota_{1})+\dots+\mathrm{ch}(\iota_{r})$
and $|\bar{\iota}|=\iota_{1}+\dots+\iota_{r}$.

As a result, we have the grading $\mathcal{L}^{\,{\dagger}}=\oplus_{m\geqslant
0}\mathcal{L}^{\,{\dagger}}[m]$ in the category of Lie algebras. For
$w\geqslant 1$, set
$\mathcal{L}^{\,{\dagger}}(w)[m]=\mathcal{L}^{\,{\dagger}}(w)\cap\mathcal{L}^{\,{\dagger}}[m]$.
Then $\mathcal{L}^{\,{\dagger}}(w)=\oplus_{m\geqslant
0}\mathcal{L}^{\,{\dagger}}(w)[m]$.

######  Remark.

The algebra $\mathcal{L}^{{\dagger}}[0]$ can be naturally identified with the
Lie subalgebra of $\mathcal{L}$ from Section 1 via
$j^{0}:D^{{\dagger}}_{qp^{M-1}a,n}\mapsto D_{an}$, where $a$ runs over
$\mathbb{Z}^{0}(p)\cap[0,(p-1)v_{0})$. Then the special choice of $\delta_{0}$
and condition C3 from Section 2.1 imply that this identification is compatible
with filtrations, i.e. for any $w\geqslant 1$,
$j^{0}(\mathcal{L}^{{\dagger}}(w))=\mathcal{L}(w)$.

Let
$\bar{\mathcal{L}}^{\,{\dagger}}=\mathcal{L}^{\,{\dagger}}/\mathcal{L}^{{\dagger}}(p)$.
This algebra is provided with the induced central filtration
$\\{\bar{\mathcal{L}}^{\,{\dagger}}(w)\\}_{w\geqslant 1}$ such that
$\bar{\mathcal{L}}^{\,{\dagger}}(p)=0$. We also have the induced gradings
$\bar{\mathcal{L}}^{\,{\dagger}}=\oplus_{m\geqslant
0}\bar{\mathcal{L}}^{\,{\dagger}}[m]$ and
$\bar{\mathcal{L}}^{\,{\dagger}}(w)=\oplus_{m\geqslant
0}\bar{\mathcal{L}}^{\,{\dagger}}(w)[m]$, where
$\bar{\mathcal{L}}^{\,{\dagger}}(w)[m]:=\bar{\mathcal{L}}^{\,{\dagger}}(w)\cap\bar{\mathcal{L}}^{\,{\dagger}}[m]$.

######  Definition.

If $l\in\bar{\mathcal{L}}^{\,{\dagger}}[m]_{W_{M}(k)}$, $l\neq 0$, we set
$\mathrm{ch}(l)=m$.

Clearly, for any $m_{1},m_{2}$,
$[\bar{\mathcal{L}}^{\,{\dagger}}[m_{1}],\bar{\mathcal{L}}^{\,{\dagger}}[m_{2}]]\subset\bar{\mathcal{L}}^{\,{\dagger}}[m_{1}+m_{2}]$
and we have the induced identification
$\bar{\mathcal{L}}^{{\dagger}}[0]=\bar{\mathcal{L}}$ which is compatible with
filtrations.

###  2.4. Lie algebras $\mathcal{N}^{{\dagger}}$ and
$\bar{\mathcal{N}}^{{\dagger}}$

Consider $N^{*},r^{*},q$ from Sec. 2.1.

Our approach to the study of the ramification filtration in
$\mathcal{G}_{<p,M}$ substantially uses the construction of a totally ramified
separable extension $\mathcal{K}^{\prime}$ of $\mathcal{K}$ such that
$[\mathcal{K}^{\prime}:\mathcal{K}]=q$ and the Herbrand function
$\varphi_{\mathcal{K}^{\prime}/\mathcal{K}}$ has only one edge point
$(r^{*},r^{*})$. For a detailed construction of $\mathcal{K}^{\prime}$ cf.
e.g. [3], Sect. 1.5. We just need that $\mathcal{K}^{\prime}=k((t_{01}))$,
where our fixed local parameter $t_{0}$ of $\mathcal{K}$ equals
$t_{01}^{q}E(t_{01}^{b^{*}})^{-1}$ and $E=E(X)$ is the Hasse-Artin
exponential.

We use the uniformiser $t_{01}$ to construct (similarly to Sec. 1.1) the
corresponding lifts $O^{\prime}_{M}(\mathcal{E}^{\prime})$ of separable
extensions $\mathcal{E}^{\prime}/\mathcal{K}^{\prime}$. Notice that for any
separable extension $\mathcal{E}^{\prime}$ of $\mathcal{K}^{\prime}$,
$O^{\prime}_{M}(\mathcal{E}^{\prime})=W_{M}(\sigma^{M-1}\mathcal{E}^{\prime})[t_{1}]\subset
W_{M}(\mathcal{E}^{\prime})$, where $t_{1}=[t_{01}]$ is the Teichmuller
representative of $t_{01}$. In particular,
$O^{\prime}_{M}(\mathcal{K}^{\prime})=W_{M}(k)((t_{1}))$. There is no natural
embedding of $O_{M}(\mathcal{K})$ into $O^{\prime}_{M}(\mathcal{K}^{\prime})$.
However, we have the embedding

$\iota:\sigma^{M-1}O_{M}(\mathcal{K})\subset
O^{\prime}_{M}(\mathcal{K}^{\prime})$

determined by the “modulo $p^{M}$” lift
$t^{p^{M-1}}=t_{1}^{qp^{M-1}}E(t_{1}^{b^{*}})^{-p^{M-1}}\,$ to
$O^{\prime}_{M}(\mathcal{K}^{\prime})$ of the “modulo $p$” relation
$t_{0}=t_{01}^{q}E(t^{b^{*}}_{01})^{-1}$.

######  Definition.

Let ${\mathcal{N}}^{\,{\dagger}}$ be the $W_{M}(k)$-submodule in
${\mathcal{L}}^{\,{\dagger}}_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$ generated
by the elements $p^{A}t_{1}^{-b}[D^{\,{\dagger}}_{\bar{\iota}\bar{n}}]$, where

a) $b\in\mathfrak{A}^{0}$, $\mathrm{ch}\,b\geqslant\mathrm{ch}(\bar{\iota})$;

b) $-b\geqslant-|\bar{\iota}|+p^{M-1}b^{*}U(\mathrm{ch}(\bar{\iota}),A)\,\,.$

Clearly, $\mathcal{N}^{{\dagger}}$ is a Lie $W_{M}(k)$-subalgebra in
$\mathcal{L}^{{\dagger}}_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$. We can
define similarly $\bar{\mathcal{N}}^{{\dagger}}$ by replacing
$\mathcal{L}^{{\dagger}}$ with $\bar{\mathcal{L}}^{{\dagger}}$. The natural
projection $\mathcal{L}^{{\dagger}}\to\bar{\mathcal{L}}^{{\dagger}}$ induces
the projection $\mathcal{N}^{{\dagger}}\to\bar{\mathcal{N}}^{{\dagger}}$.

######  Definition.

For $j,m\geqslant 0$, let ${\mathcal{N}}^{\,{\dagger}}[j,m]$ be the
$W_{M}(k)$-submodule in ${\mathcal{N}}^{\,{\dagger}}$ generated by the
elements $p^{A}t_{1}^{-b}[D^{\,{\dagger}}_{\bar{\iota}\bar{n}}]$, where

a) $\mathrm{ch}(\bar{\iota})=m$, $\mathrm{ch}\,b=j+m$;

b) $-b\geqslant-|\bar{\iota}|+p^{M-1}b^{*}U(m+j,A)\,.$

Clearly, there is a decomposition
$\oplus_{j,m}{\mathcal{N}}^{\,{\dagger}}[j,m]={\mathcal{N}}^{\,{\dagger}}$ and

$[{\mathcal{N}}^{\,{\dagger}}[j_{1},m_{1}],{\mathcal{N}}^{\,{\dagger}}[j_{2},m_{2}]]\subset{\mathcal{N}}^{\,{\dagger}}[j_{1}+j_{2},m_{1}+m_{2}]\,.$

######  Definition.

For $s\geqslant 0$, ${\mathcal{N}}^{\,{\dagger}}(s):=\oplus_{j+m\geqslant
s}{\mathcal{N}}^{\,{\dagger}}[j,m]$ form a decreasing filtration in
${\mathcal{N}}^{\,{\dagger}}$ such that
$[{\mathcal{N}}^{{\dagger}}(s_{1}),{\mathcal{N}}^{{\dagger}}(s_{2})]\subset{\mathcal{N}}^{\,{\dagger}}(s_{1}+s_{2})$.

Similarly to Section 1.5 we can introduce for $w\geqslant 1$, a closed
submodule $\mathcal{L}^{{\dagger}}_{\mathcal{N}}(w)$ in
$\mathcal{L}^{{\dagger}}$ generated by all $p^{m}[D_{\bar{\iota}\bar{n}}]$
with $|\bar{\iota}|\geqslant p^{M-1}b^{*}U(w-1,m)$. Then there are analogs of
Proposition 1.6 and Corollary 1.8

$\mathcal{L}^{{\dagger}}_{\mathcal{N}}(w)\cap
C_{2}(\mathcal{L}^{{\dagger}}_{W_{M}(k)})\subset\sum_{w_{1}+w_{2}\geqslant
w}[\mathcal{L}_{\mathcal{N}}(w_{1}),\mathcal{L}_{\mathcal{N}}(w_{2})],\ \
\mathcal{L}^{{\dagger}}_{\mathcal{N}}(w)\subset\mathcal{L}^{{\dagger}}(w)\,.$

As a result, for any $w$ and $\mathrm{m}^{\prime}=W_{M}(k)[[t_{1}]]$, it holds

$\mathcal{N}^{{\dagger}}(s)\subset\mathcal{L}^{{\dagger}}_{\mathrm{m}^{\prime}}+\mathcal{L}^{{\dagger}}(1+s)_{O^{\prime}_{M}(\mathcal{K}^{\prime})}\,.$

######  Remark.

Similar definitions can be done in the context of the Lie algebra
$\bar{\mathcal{L}}^{{\dagger}}$. In particular, we obtain an analog
$\bar{\mathcal{N}}^{{\dagger}}$ of $\mathcal{N}^{{\dagger}}$ with
corresponding filtration $\bar{\mathcal{N}}^{{\dagger}}(s)$ and grading
$\bar{\mathcal{N}}[j,m]$. The projection
$\mathcal{N}^{{\dagger}}\to\bar{\mathcal{N}}$ allows us to transform above
properties to $\bar{\mathcal{N}}^{{\dagger}}$.

In particular, we obtain

######  Corollary 2.5.

$\bar{\mathcal{N}}^{\,{\dagger}}(p-1)\subset\mathcal{L}^{\,{\dagger}}_{\mathrm{m}^{\prime}}$.

######  Definition.

Set
$\widetilde{\mathcal{N}}^{\,{\dagger}}=\bar{\mathcal{N}}^{\,{\dagger}}/\bar{\mathcal{N}}^{\,{\dagger}}(p)$.

The above grading on $\bar{\mathcal{N}}^{\,{\dagger}}$ induces the grading
$\widetilde{\mathcal{N}}^{\,{\dagger}}=\oplus_{j,m}{\widetilde{\mathcal{N}}}^{\,{\dagger}}[j,m]$.
Note that $\widetilde{\mathcal{N}}^{\,{\dagger}}[j,m]=0$ if $j+m\geqslant p$.

###  2.5. The action $\Omega$

Suppose $\gamma\in\mathbb{Z}/p^{M}$.

For $b=p^{m}(qp^{M-1}\alpha-(q-1)\beta)\in\mathfrak{A}^{0}$ and
$l=p^{A}t_{1}^{-b}D^{\,{\dagger}}_{\bar{\iota}\bar{n}}\in\widetilde{\mathcal{N}}^{\,{\dagger}}\,$,
set

$\Omega(l):=E(t_{1}^{b^{*}})^{(p^{m}\alpha)p^{M-1}}l\,,$

where $E(X)$ is the Artin-Hasse exponential.

Note that
$E(t_{1}^{b^{*}})^{(p^{m}\alpha)p^{M-1}}\equiv\widetilde{\exp}\left((p^{m}\alpha)S_{b^{*}}\right)\mathrm{mod}\,S_{b^{*}}^{p}$
where
$S_{b^{*}}=t_{1}^{b^{*}p^{M-1}}+pt_{1}^{b^{*}p^{M-2}}+\ldots+p^{M-1}t_{1}^{b^{*}}$
and $\widetilde{\exp}$ is the truncated exponential. Therefore, in
$\widetilde{\mathcal{N}}^{\,{\dagger}}$ it holds
$\Omega(l)=\widetilde{\exp}\left((p^{m}\alpha)S_{b^{*}}\right)l$.

######  Remark.

Note that
$S_{b^{*}}^{p}\bar{\mathcal{N}}^{{\dagger}}\subset\bar{\mathcal{N}}^{{\dagger}}(p)$.
Therefore, it is essential that we could define the action $\Omega$ only in
the context of
$\widetilde{\mathcal{N}}^{{\dagger}}=\bar{\mathcal{N}}^{{\dagger}}/\bar{\mathcal{N}}^{{\dagger}}(p)$.
However, $\widetilde{\mathcal{N}}^{{\dagger}}$ is still related to the Lie
algebra $\bar{\mathcal{L}}^{{\dagger}}$, i.e. the natural embedding
$\bar{\mathcal{L}}^{{\dagger}}\to\bar{\mathcal{N}}^{{\dagger}}\subset\bar{\mathcal{L}}^{{\dagger}}_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$
still induces embedding
$\bar{\mathcal{L}}^{{\dagger}}\to\widetilde{\mathcal{N}}^{{\dagger}}$ after
taking the projection from $\bar{\mathcal{N}}^{{\dagger}}$ to
$\widetilde{\mathcal{N}}^{{\dagger}}$. (Use that
$\bar{\mathcal{N}}^{{\dagger}}(p)\subset\bar{\mathcal{L}}^{{\dagger}}_{\mathrm{m}^{\prime}}$.)

By linearity, $\Omega$ can be extended to the whole
$\widetilde{\mathcal{N}}^{\,{\dagger}}$.

The map $n\mapsto\Omega_{\gamma}(n):=\Omega^{\gamma}(n)$,
$\gamma\in\mathbb{Z}/p^{M}$, is a well-defined action of the additive group
$\mathbb{Z}/p^{M}$ on the Lie algebra $\widetilde{\mathcal{N}}^{\,{\dagger}}$.
This action is unipotent: for any
$n\in\widetilde{\mathcal{N}}^{\,{\dagger}}[j,m]$,
$\Omega_{\gamma}(n)-n\in\oplus_{i>j}\widetilde{\mathcal{N}}^{\,{\dagger}}[i,m]\,.$

Let $\bar{e}^{\,{\dagger}}\in\bar{\mathcal{N}}^{{\dagger}}$ be such that

(2.1) $\ \
\bar{e}^{\,{\dagger}}\equiv\sum\limits_{\iota\in\mathfrak{A}^{0}(p)}t_{1}^{-\iota}D^{\,{\dagger}}_{\iota
0}\,\mathrm{mod}\,C_{2}(\bar{\mathcal{L}}^{{\dagger}}_{O^{\prime}_{M}(\mathcal{K}^{\prime})})\,.$

######  Remark.

Such element is used usually in the nilpotent Artin-Shreier theory, cf.
Section 3. In the above expression the elemnt $D^{{\dagger}}_{00}$ is not
defined. In this and similar case for the element $e$ from Section 3 we set
$D^{{\dagger}}_{00}:=\alpha D^{{\dagger}}_{0}$, where
$\alpha=\hat{\alpha}\,\mathrm{mod}\,p^{M}\in W_{M}(k)$ and the trace of
$\hat{\alpha}$ over $\mathbb{Q}_{p}$ is equal to 1.

Denote by $\widetilde{e}^{\,{\dagger}}$ the image of $\bar{e}^{\,{\dagger}}$
in $\widetilde{\mathcal{N}}^{\,{\dagger}}$.

A choice of $\bar{e}^{\,{\dagger}}$ allows us to associate to the above
defined action $\Omega_{\gamma}$ the “conjugated” action of
$A^{\,{\dagger}}_{\gamma}$ on $\bar{\mathcal{L}}^{\,{\dagger}}$ as follows.

Let $\widetilde{\mathcal{N}}^{\,{\dagger}}\langle
j\rangle=\oplus_{m}\widetilde{\mathcal{N}}^{\,{\dagger}}[j,m]$ and
$\widetilde{\mathcal{N}}^{\,{\dagger}}\\{s\\}:=\oplus_{j\geqslant
s}\widetilde{\mathcal{N}}^{\,{\dagger}}\langle j\rangle$.

######  Proposition 2.6.

For $\gamma\in\mathbb{Z}/p^{M}$, there are unique
$\widetilde{c}_{\gamma}\in\widetilde{\mathcal{N}}^{\,{\dagger}}\\{1\\}$ and
$A^{\,{\dagger}}_{\gamma}\in\mathrm{Aut}_{\mathrm{Lie}}\bar{\mathcal{L}}^{\,{\dagger}}$
such that

a) $\sigma\tilde{c}_{\gamma}\in\widetilde{\mathcal{N}}^{\,{\dagger}}\\{1\\}$
and
$\Omega_{\gamma}(\widetilde{e}^{\,{\dagger}})=(\sigma\widetilde{c}_{\gamma})\circ(A^{\,{\dagger}}_{\gamma}\otimes\mathrm{id}_{\mathcal{K}})\widetilde{e}^{\,{\dagger}}\circ(-\widetilde{c}_{\gamma})$;

b) for any $\iota\in\mathfrak{A}^{0}(p)$,
$A^{\,{\dagger}}_{\gamma}(D^{\,{\dagger}}_{\iota 0})-D^{\,{\dagger}}_{\iota
0}\in\oplus_{m<\mathrm{ch}(\iota)}\bar{\mathcal{L}}^{\,{\dagger}}[m]_{W_{M}(k)}$.

###### Proof.

We need the following lemma.

######  Lemma 2.7.

Suppose $n\in\widetilde{\mathcal{N}}^{\,{\dagger}}[j,m]\cap
C_{s}(\bar{\mathcal{L}}^{\,{\dagger}})_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$
where $j+m\geqslant 1$. Then there are unique
$\mathcal{S}(n),\mathcal{R}(n)\in\widetilde{\mathcal{N}}^{\,{\dagger}}[j,m]\cap
C_{s}(\bar{\mathcal{L}}^{\,{\dagger}})_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$
such that:

a) $\mathcal{R}(n)=\sum\limits_{\iota}t^{-\iota}l_{\iota}$ where $\iota$ runs
over $\mathfrak{A}^{+}(p)$ with $\mathrm{ch}(\iota)=j+m$ and all
$l_{\iota}\in\left(\bar{\mathcal{L}}^{\,{\dagger}}[m]\cap
C_{s}(\bar{\mathcal{L}}^{\,{\dagger}})\right)_{W_{M}(k)}$;

b) $n=\mathcal{R}(n)+(\sigma-\mathrm{id})\mathcal{S}(n)$.

###### Proof of lemma.

It will be sufficient to specify the elements $\mathcal{R}(n)$ and
$\mathcal{S}(n)$ for
$n=p^{A}t_{1}^{-b}[D^{\,{\dagger}}_{\bar{\iota}\bar{n}}]$, where
$b\in\mathfrak{A}^{0}$, $\mathrm{ch}(b)=j+m$, $\mathrm{ch}(\bar{\iota})=m$.

It is essential that $\mathrm{ch}(b)\geqslant 1$ and, therefore, $b\neq 0$.

If $b>0$ we set $\mathcal{R}(n)=0$ and $\mathcal{S}(n)=-\sum_{i\geqslant
0}\sigma^{i}(n)$.

If $b=\iota_{0}p^{i_{0}}<0$ with $\iota_{0}\in\mathfrak{A}^{+}(p)$ we set
$\mathcal{R}(n)=\sigma^{-i_{0}}n$ and $\mathcal{S}(n)=\sum_{0\leqslant
j<i_{0}}\sigma^{-j}n$.

The uniqueness property holds because $\mathrm{ch}(b)\geqslant 1$ implies that
$b\neq 0$. Lemma is proved. ∎

Continue the proof of Prop. 2.6.

Use induction on $i\geqslant 1$ to prove the proposition modulo the ideals
$\widetilde{\mathcal{N}}^{\,{\dagger}}\\{1\\}\cap
C_{i}(\bar{\mathcal{L}}^{\,{\dagger}})_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$.

— If $i=1$ take $\widetilde{c}_{\gamma}=0$,
$A^{\,{\dagger}}_{\gamma}=\mathrm{id}$ and use that
$\Omega_{\gamma}(\widetilde{e}^{\,{\dagger}})-\widetilde{e}^{\,{\dagger}}\in\widetilde{\mathcal{N}}^{\,{\dagger}}\\{1\\}\,.$

— Suppose $i\geqslant 2$,
$\widetilde{c}_{\gamma}\in\widetilde{\mathcal{N}}^{{\dagger}}\\{1\\}$ and
$A_{\gamma}^{\,{\dagger}}\in\mathrm{Aut}_{\mathrm{Lie}}(\bar{\mathcal{L}}^{\,{\dagger}})$
are such that

$H=\Omega_{\gamma}\widetilde{e}^{\,{\dagger}}-(\sigma\widetilde{c}_{\gamma})\circ(A^{\,{\dagger}}_{\gamma}\otimes\mathrm{id}_{\mathcal{K}^{\prime}})\widetilde{\,e}^{\,{\dagger}}\circ(-\widetilde{c}_{\gamma})\in\widetilde{\mathcal{N}}^{\,{\dagger}}\\{1\\}\cap\,C_{i}(\widetilde{\mathcal{N}}^{\,{\dagger}})_{O^{\prime}_{M}(\mathcal{K}^{\prime})}\,.$

Then
$\mathcal{R}(H),\mathcal{S}(H)\in\widetilde{\mathcal{N}}^{\,{\dagger}}\\{1\\}\cap\,C_{i}(\widetilde{\mathcal{L}}^{\,{\dagger}})_{O^{\prime}_{M}(\mathcal{K}^{\prime}}$.

Set $\mathcal{R}(H)=\sum\limits_{\mathrm{ch}(\iota)\geqslant
1+m}t^{-\iota}H_{\iota m}$, where all $H_{\iota
m}\in\widetilde{\mathcal{L}}^{\,{\dagger}}[m]_{k}\cap
C_{i}(\bar{\mathcal{L}}^{\,{\dagger}})_{W_{M}(k)}$.

Introduce
$A^{\,{\dagger}\prime}_{\gamma}\in\mathrm{Aut}_{\mathrm{Lie}}(\bar{\mathcal{L}}^{\,{\dagger}})$
by setting for all involved $\iota$ and $m$,
$A_{\gamma}^{\,{\dagger}\prime}(D^{\,{\dagger}}_{\iota
0})=A^{\,{\dagger}}_{\gamma}(D^{\,{\dagger}}_{\iota 0})-\sum_{m}H_{\iota m}$.
Set also
$\widetilde{c}^{\,\prime}_{\gamma}=\widetilde{c}_{\,\gamma}-{\mathcal{S}}(H)$.
Then

$\Omega_{\gamma}\widetilde{e}^{\,{\dagger}}\equiv(\sigma\widetilde{c}^{\,\prime}_{\gamma
1})\circ(A^{\,{\dagger}\prime}_{\gamma}\
\otimes\mathrm{id}_{\mathcal{K}})\widetilde{e}^{\,{\dagger}}\circ(-\widetilde{c}^{\,\prime}_{\gamma})\,\mathrm{mod}\,\widetilde{\mathcal{N}}^{\,{\dagger}}\\{1\\}\cap
C_{i+1}(\widetilde{\mathcal{N}})_{O^{\prime}_{M}(\mathcal{K}^{\prime})}\,.$

The uniqueness follows similarly by induction on $i$ and the uniqueness part
of Lemma 2.7.

The proposition is proved. ∎

We have obviously the following properties.

######  Corollary 2.8.

For any $\gamma,\gamma_{1}\in\mathbb{Z}/p^{M}$,

a)
$A^{\,{\dagger}}_{\gamma+\gamma_{1}}=A^{\,{\dagger}}_{\gamma}A^{\,{\dagger}}_{\gamma_{1}}$;

b)
$\Omega_{\gamma}(\widetilde{c}_{\gamma_{1}})\circ(A^{\,{\dagger}}_{\gamma_{1}}\otimes\mathrm{id}_{\mathcal{K}})\widetilde{c}_{\gamma}=\widetilde{c}_{\gamma+\gamma_{1}}$;

c) if $l\in\bar{\mathcal{L}}^{\,{\dagger}}[m]$ then
$A^{\,{\dagger}}_{\gamma}(l)-l\in\oplus_{m^{\prime}<m}\bar{\mathcal{L}}^{\,{\dagger}}[m^{\prime}]$,
e.g.
$A^{\,{\dagger}}_{\gamma}|_{\bar{\mathcal{L}}^{\,{\dagger}}[0]}=\mathrm{id}\,.$

###  2.6. The action $\Omega_{U}$

Let $A^{\,{\dagger}}:=A^{\,{\dagger}}_{\gamma}|_{\gamma=1}$. Then for any
$\gamma=n\,\mathrm{mod}\,p^{M}$, $A^{\,{\dagger}}_{\gamma}=A^{\,{\dagger}n}$,
in particular,
$A^{\,{\dagger}p^{M}}=\mathrm{id}_{\bar{\mathcal{L}}^{\,{\dagger}}}$. By part
c) of the above corollary, for all $m\geqslant 0$, there is a differentiation
$B^{\,{\dagger}}\in\mathrm{End}_{\mathrm{Lie}}\bar{\mathcal{L}}^{\,{\dagger}}$
such that for all $m\geqslant 0$,
$B^{\,{\dagger}}(\bar{\mathcal{L}}^{\,{\dagger}}[m])\subset\oplus_{m^{\prime}<m}\bar{\mathcal{L}}^{\,{\dagger}}[m^{\prime}]$
and for all $\gamma\in\mathbb{Z}/p^{M}$,
$A^{\,{\dagger}}_{\gamma}=\widetilde{\exp}(\gamma B^{\,{\dagger}})$.

The derivation $B^{{\dagger}}$ can be recovered via methods from [5]-[6].

Let $U$ be an indeterminate. If
$b=p^{m}(qp^{M-1}\alpha-(q-1)\beta)\in\mathfrak{A}^{0}$ and
$l=p^{A}t_{1}^{-b}[D_{\bar{\iota}\bar{n}}]\in\widetilde{\mathcal{N}}^{\,{\dagger}}\,$
set

$\Omega_{U}l:=\widetilde{\exp}(U\otimes(p^{m}\alpha)S_{b^{*}})l\,\in\mathbb{Z}/p^{M}[U]\otimes\widetilde{\mathcal{N}}^{\,{\dagger}}\,.$

Proceeding similarly to the proof of Proposition 2.6 we obtain

(2.2)
$\Omega_{U}(\widetilde{e}^{\,{\dagger}})=\sigma(\widetilde{c}_{U})\circ(A^{\,{\dagger}}_{U}\otimes\mathrm{id})\widetilde{e}^{\,{\dagger}}\circ(-\widetilde{c}_{U})\,,$

where $A^{\,{\dagger}}_{U}=\widetilde{\exp}(UB^{\,{\dagger}})$. As a matter of
fact, when obtaining (2.2) we are working inside $\oplus_{m\geqslant
0}U^{j}\widetilde{\mathcal{N}}^{{\dagger}}[j,m]$.

It is easy to see that we have action of the additive group
$\mathbb{G}_{a}=\mathrm{Spec}\,\mathbb{Z}/p^{M}[U]$ on
$\widetilde{\mathcal{N}}^{\,{\dagger}}$. Similarly to [8] and by taking into
account the compatibility of this action with the double grading
$\widetilde{\mathcal{N}}^{\,{\dagger}}[j,m]$ we obtain the following:

######  Proposition 2.9.

a)
$\widetilde{c}_{U}=\widetilde{c}^{\,(1)}U+\dots+\widetilde{c}^{\,(p-1)}U^{p-1}$,
where all
$\widetilde{c}^{(j)},\sigma\widetilde{c}^{(j)}\in\widetilde{\mathcal{N}}^{\,{\dagger}}\langle
j\rangle$;

b) the cocycle $\widetilde{c}_{U}$ is determined uniquely by its linear part
$\widetilde{c}^{\,(1)}$;

c) the action $\Omega_{U}=\sum_{0\leqslant i<p}\Omega^{\,(i)}U^{i}$ (here
$\Omega^{(0)}=\mathrm{id}$) is recovered uniquely from its differential
$d\,\Omega_{U}:=\Omega^{(1)}U$.

d) $A^{{\dagger}}_{U}=\widetilde{\exp}B^{{\dagger}}_{U}$ and
$B^{{\dagger}}_{U}(\bar{\mathcal{L}}^{{\dagger}}[m])\subset\bar{\mathcal{L}}^{{\dagger}}[m-1]$.

######  Remark.

Prop. 2.9 is directly related to Corollary 2.8 as follows:

1) For any $\gamma\in\mathbb{Z}/p^{M}$,
$\widetilde{c}_{U}|_{U=\gamma}=\widetilde{c}_{\gamma}$ and
$\Omega_{U}|_{U=\gamma}=\Omega_{\gamma}$.

2) Vice versa, the coefficients $\widetilde{c}^{(j)}$ can be uniquely
recovered from the values $\widetilde{c}_{U}|_{U=[\alpha]}$, where $[\alpha]$
are the Teichmuller representatives of $\alpha\in\mathbb{F}_{p}$. (Use that
the matrix of Teichmuller representatives
$(([i\,\mathrm{mod}\,p]^{j}))_{1\leqslant i,j<p}$, is non-degenerate.)

######  Corollary 2.10.

For $m\in\mathbb{N}$,
$B^{\,{\dagger}}(\widetilde{\mathcal{L}}^{\,{\dagger}}[m])\subset\widetilde{\mathcal{L}}^{\,{\dagger}}[m-1]$.

###  2.7. Ideals $\bar{\mathcal{L}}^{\,{\dagger}[v_{0}]}$ and
$\bar{\mathcal{L}}^{[v_{0}]}$

Recall that $\bar{\mathcal{L}}^{\,{\dagger}}[0]$ is the minimal Lie subalgebra
of $\bar{\mathcal{L}}^{\,{\dagger}}$ containing all $D^{\,{\dagger}}_{\iota
n}$ with $\mathrm{ch}(\iota)=0$. By Remark from Sect. 2.3,
$\mathcal{L}^{{\dagger}}[0]_{W_{M}(k)}$ is generated by

(2.3) $\\{D^{{\dagger}}_{qp^{M-1}a,n}\ |\
a\in\mathbb{Z}^{0}(p)\cap[0,(p-1)v_{0}),n\in\mathbb{Z}/N_{0}\,\\}\,.$

Recall also that $\mathcal{L}^{{\dagger}}[0]$ is naturally identified with the
Lie algebra $\mathcal{L}$ via $j^{0}:D^{{\dagger}}_{qp^{M-1}a,n}\mapsto
D_{an}$. This identification agrees with filtrations and plays a crucial role
below, where we work with criterion from Prop. 3.2 in the algebra
$\mathcal{L}^{{\dagger}}[0]$.

Note also that ${\mathcal{L}}^{\,{\dagger}}[0]$ has the induced filtration
$\\{{\mathcal{L}}^{\,{\dagger}}(w)[0]\\}_{w\geqslant 1}$ and there is
epimorphism of filtered Lie algebras
$\mathcal{V}^{\,0}:{\mathcal{L}}^{\,{\dagger}}\longrightarrow{\mathcal{L}}[0]$
such that $D^{\,{\dagger}}_{\iota n}\mapsto D^{\,{\dagger}}_{\iota n}$ if
$\mathrm{ch}(\iota)=0$ and $D^{\,{\dagger}}_{\iota n}\mapsto 0$, otherwise.
Similarly, we have the epimorphism
$\bar{\mathcal{V}}^{0}:\bar{\mathcal{L}}^{{\dagger}}\to\bar{\mathcal{L}}^{{\dagger}}[0]$
and the identification
$\bar{\jmath}^{0}:\bar{\mathcal{L}}^{{\dagger}}[0]\simeq\bar{\mathcal{L}}$ of
filtered Lie algebras.

Let
$\bar{\mathcal{V}}:=\bar{j^{0}}\bar{\mathcal{V}}^{0}:\bar{\mathcal{L}}^{\,{\dagger}}\longrightarrow\bar{\mathcal{L}}$.

Define the ideal $\bar{\mathcal{L}}^{\,{\dagger}[v_{0}]}$ as the minimal ideal
in $\bar{\mathcal{L}}^{\,{\dagger}}$ containing all
$B^{\,{\dagger}}(\mathrm{Ker}\,\bar{\mathcal{V}})$. Set
$\bar{\mathcal{L}}^{[v_{0}]}=\bar{\mathcal{V}}(\bar{\mathcal{L}}^{\,{\dagger}[v_{0}]})$.

######  Definition.

Let $\mathfrak{A}^{0}_{1}=\\{\iota\in\mathfrak{A}^{0}\ |\
\mathrm{ch}(\iota)=1\\}$ and
$\mathfrak{A}_{1}^{+}(p)=\mathfrak{A}^{0}_{1}\cap\mathfrak{A}^{+}(p)$.

######  Proposition 2.11.

$\bar{\mathcal{L}}^{[v_{0}]}$ is the minimal ideal in $\bar{\mathcal{L}}$ such
that for all $\iota\in\mathfrak{A}_{1}^{+}(p)$,
$\bar{\mathcal{V}}(D^{{\dagger}}_{\iota
0})\in\bar{\mathcal{L}}^{[v_{0}]}_{W_{M}(k)}$.

###### Proof.

Clearly, $\mathrm{Ker}\bar{\mathcal{V}}$ is generated by the elements of
$\bar{\mathcal{L}}^{\,{\dagger}}[m]$ with $m\geqslant 1$. Therefore, for
$m\geqslant 2$,
$B^{\,{\dagger}}(\bar{\mathcal{L}}^{\,{\dagger}}[m])\subset\bar{\mathcal{L}}^{\,{\dagger}}[m-1]\subset\mathrm{Ker}\bar{\mathcal{V}}$.
As a result, the elements of
$\bar{\mathcal{V}}B^{{\dagger}}(\bar{\mathcal{L}}^{\,{\dagger}}[1])$ generate
the ideal $\bar{\mathcal{L}}^{[v_{0}]}$. If $r(\bar{\iota})>1$ and
$\mathrm{ch}(\bar{\iota})=1$ then $l=[D^{{\dagger}}_{\bar{\iota}\bar{n}}]$ can
be written in the form $[D^{{\dagger}}_{\iota n},l^{\prime}]$ where
$\iota\in\mathfrak{A}_{1}^{+}(p)$ and
$l^{\prime}\in\widetilde{\mathcal{L}}^{\,{\dagger}}[0]_{W_{M}(k)}$. This
implies that
$\bar{\mathcal{V}}B^{\,{\dagger}}(l)\in[\bar{\mathcal{L}}^{[v_{0}]},\bar{\mathcal{L}}]_{W_{M}(k)}$.

The proposition is proved. ∎

For each $j\geqslant 1$ set $\bar{\mathcal{N}}^{{\dagger}}\langle
j\rangle=\oplus_{m}\bar{\mathcal{N}}^{{\dagger}}[j,m]$, and let
$\pi:\bar{\mathcal{N}}\to\widetilde{\mathcal{N}}^{{\dagger}}$ be a natural
projection.

######  Proposition 2.12.

There are unique elements
$\bar{c}^{(j)}\in\bar{\mathcal{N}}^{{\dagger}}\langle j\rangle$ such that
$\pi(\bar{c}^{(j)})=\widetilde{c}^{(j)}$, cf. Prop. 2.9, and

$\Omega(\bar{e}^{\,{\dagger}})=\sigma(\bar{c})\circ(A^{{\dagger}}\otimes\mathrm{id}_{O^{\prime}_{M}(\mathcal{K}^{\prime})}\bar{e}^{\,{\dagger}}\circ(-\bar{c})\,$

where $\bar{c}=\bar{c}^{(1)}+\dots+\bar{c}^{(p-1)}$.

###### Proof.

It follows easily from the fact that
$\mathrm{Ker}\,\pi\subset\bar{\mathcal{L}}^{{\dagger}}_{\mathrm{m}^{\prime}}$
and for any $j$ and $m$, the map $\sigma-\mathrm{id}$ is a bijection on
$\mathrm{Ker}\pi\cap\bar{\mathcal{N}}^{{\dagger}}[j,m]$. ∎

##  3\. Relation to Galois theory

In [2] we developed a nilpotent analogue of the classical Artin-Schreier
theory of cyclic extensions of fields of characteristic $p$. Below we are
going to use the covariant analogue of this theory, cf. the discussion in [4],
for an explicit description of the ramification subgroup
$\mathcal{G}^{(v_{0})}_{<p,M}$ of
$\mathcal{G}_{<p,M}=\mathcal{G}/\mathcal{G}^{p^{M}}C_{p}(\mathcal{G})$.

###  3.1. Groups and Lie algebras of nilpotent class $<p$

Let $L$ be a finite Lie $\mathbb{Z}/p^{M}$-algebra of nilpotent class $<p$,
i.e. $C_{p}(L)=0$. Let $A$ be an enveloping algebra of $L$. There is a natural
embedding $L\subset A/J^{p}$ and in terms of this embedding the Campbell-
Hausdorff formula appears as

$(l_{1},l_{2})\mapsto l_{1}\circ
l_{2}=l_{1}+l_{2}+\frac{1}{2}[l_{1},l_{2}]+\dots,\ \ l_{1},l_{2}\in L\,,$

where
$\widetilde{\exp}(l_{1})\cdot\widetilde{\exp}(l_{2})\equiv\widetilde{\exp}(l_{1}\circ
l_{2})\,\mathrm{mod}\,J^{p}$. This composition law provides the set $L$ with a
group structure and we denote this group by $G(L)$. Clearly, $G(L)$ has period
$p^{M}$ and nilpotent class $<p$. Then the correspondence $L\mapsto G(L)$
induces equivalence of the category of finite $p$-groups of period $p^{M}$ and
nilpotent class $s<p$ and the category of finite Lie
$\mathbb{Z}/p^{M}$-algebras of the same nilpotent class $s$. This equivalence
can be naturally extended to similar categories of pro-finite ubjects.

###  3.2. Epimorphism $\eta^{0}$

Let $L$ be a finite $\mathbb{Z}/p^{M}$-Lie algebra of nilpotent class $<p$.
Set $L_{sep}:=L_{O_{M}(\mathcal{K}_{sep})}$. Then the elements of
$\mathcal{G}=\mathrm{Gal}(\mathcal{K}_{sep}/\mathcal{K})$ and $\sigma$ act on
$L_{sep}$ through the second factor, $L_{sep}|_{\sigma=\mathrm{id}}=L$ and
$(L_{sep})^{\mathcal{G}}=L_{\mathcal{K}}$. The covariant nilpotent Artin-
Schreier theory states that for any $e\in G(L_{\mathcal{K}})$, the set

$\mathcal{F}(e)=\\{f\in G(L_{sep})\ |\ \sigma(f)=e\circ f\\}$

is not empty and for any fixed $f\in\mathcal{F}(e)$, the map
$\tau\mapsto(-f)\circ\tau(f)$ is a continuous group homomorphism
$\pi_{f}(e):\mathcal{G}\longrightarrow G(L)$.

Suppose $\mathcal{L}$ is the Lie algebra from Section 1.1 and
$e=e(t)\in\mathcal{L}_{\mathcal{K}}$ satisfies
$e\equiv\sum_{a\in\mathbb{Z}^{0}(p)}t^{-a}D_{a0}\,\mathrm{mod}\,C_{2}(\mathcal{L})_{O_{M}(\mathcal{K})}$.
Recall that for $n\in\mathbb{Z}/N_{0}$, $D_{0n}$ is defined as
$\sigma^{n}(\alpha_{0})D_{0}$, where
$\alpha_{0}=\hat{\alpha}_{0}\,\mathrm{mod}p^{M}$ and $\alpha_{0}\in W(k)$ has
absolute trace 1. (We followed this agreement in [1]-[3] to relate the
identification $\eta^{0}\,\mathrm{mod}\,C_{2}(\mathcal{G})$, cf. below, to the
identification of class field theory.) Then a profinite version of the above
Artin-Schreier theory gives the existence of $f\in\mathcal{L}_{sep}$ such that
$\sigma f=e\circ f$ and the map
$\eta^{0}_{f}:=\pi_{f}(e)\,\mathrm{mod}\,\mathcal{G}^{p^{M}}C_{p}(\mathcal{G})$
induces a group isomorphism $\eta^{0}_{f}:\mathcal{G}_{<p}\simeq
G(\mathcal{L})$. We fix a choice of such $f$ and use the simpler notation
$\eta^{0}=\eta^{0}_{f}$.

Theorem 3.1 below requires the following assumption:

$\bf A)$ Let $\mathcal{N}$ be a $W_{M}(k)$-Lie subalgera of
$\mathcal{L}_{O_{M}(\mathcal{K})}$ generated by all $t^{-a}D_{an}$,
$a\in\mathbb{Z}^{0}(p)$. Then $e\in\mathcal{N}$ and
$e\equiv\sum_{a\in\mathbb{Z}^{0}(p)}t^{-a}D_{a0}\,\mathrm{mod}\,[\mathcal{N},\mathcal{N}]$.

######  Theorem 3.1.

$\eta^{0}(\mathcal{G}^{(v_{0})})=\mathcal{L}^{[v_{0}]}$.

######  Remark.

The above quite general result will be explicitly specified in Sec. 4, where
we present another proof of the explicit description of
$\mathcal{L}^{[v_{0}]}$ in terms of generators $\\{D_{an}\ |\
a\in\mathbb{Z}^{+}(p),n\in\mathbb{Z}/N_{0}\\}\cup\\{D_{0}\\}$ of
$\mathcal{L}_{W_{M}(k)}$ from [3].

The proof of Theorem 3.1 is given below in sections 3.3-3.5.

###  3.3. The criterion

Consider the parameters $N^{*},r^{*},q$ from Sec. 2.1.

Consider the field etension $\mathcal{K}^{\prime}/\mathcal{K}$ from Sec. 2.4.
Recall $\mathcal{K}^{\prime}=k((t_{01}))$ and the local parameter $t_{01}$ was
used to construct the lifts $\ O^{\prime}_{M}(\mathcal{K}^{\prime})$ and
$O^{\prime}_{M}(\mathcal{K}_{sep})$. There is a natural embedding
$\iota:\sigma^{M-1}O_{M}(\mathcal{K})\to O_{M}(\mathcal{K}^{\prime})$. This
allows us to identify the Lie $O^{\prime}_{M}(\mathcal{K}^{\prime})$-algebra
$\mathcal{N}^{{\dagger}}[0]$ with subalgebra of
$(\mathcal{L}_{\sigma^{M-1}O_{M}(\mathcal{K})})_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$.
In particular, $\sigma^{M-1}e(t)$ is identified with
$\Omega\left(\sigma^{M-1}e(t_{1}^{q})\right)$.

Clearly, we have $\sigma^{M}(f)=\sigma^{M-1}e(t)\circ\sigma^{M-1}f$ and for
any $\tau\in\mathcal{G}$,

$\eta^{0}(\tau)=(-\sigma^{M-1}f)\circ\tau(\sigma^{M-1}f)=(-f)\circ\tau(f)\,.$

Choose $f_{1}\in{\mathcal{L}}_{O^{\prime}_{M}(\mathcal{K}_{sep})}$ such that
$\sigma f_{1}=e(t_{1})\circ f_{1}$ and consider
$X\in{\mathcal{L}}_{O^{\prime}_{M}(\mathcal{K}_{sep})}$ such that

$\sigma^{M-1}f=X\circ(\sigma^{M-1+N^{*}}f_{1})\,.$

For $Y\in\mathcal{L}_{O^{\prime}_{M}(\mathcal{K}_{sep})}$ and an ideal
$\mathcal{I}$ in $\mathcal{L}$, define the field of definition of
$Y\,\mathrm{mod}\,\mathcal{I}_{O^{\prime}_{M}(\mathcal{K}_{sep})}$ over
$\mathcal{K}^{\prime}$ as
$\mathcal{K}^{\prime}(Y\,\mathrm{mod}\,\mathcal{I}_{O^{\prime}_{M}(\mathcal{K}_{sep})}):=\mathcal{K}_{sep}^{\mathcal{H}}$,
where
$\mathcal{H}=\\{g\in\mathcal{G}^{\prime}=\mathrm{Gal}(\mathcal{K}_{sep}/\mathcal{K}^{\prime})\
|\ (\mathrm{id}_{\mathcal{L}}\otimes g)Y\equiv
Y\,\mathrm{mod}\,\mathcal{I}_{sep}\\}$.

For a field extension $\mathcal{E}^{\prime}/\mathcal{K}^{\prime}$ in
$\mathcal{K}_{sep}$, define the biggest upper ramification number

$v(\mathcal{E}^{\prime}/\mathcal{K}^{\prime})=\mathrm{max}\\{v\ |\
\mathrm{Gal}(\mathcal{K}_{sep}/\mathcal{K}^{\prime})^{(v)}\text{acts non-
trivially on }\mathcal{E}^{\prime}\\}\,.$

The methods from [1, 2, 3] are based on the following criterion.

Set $e_{1}^{(q)}:=e(t_{1}^{q})$. Then
$\Omega\left(\sigma^{M-1}e_{1}^{(q)}\right)=e(t_{1}^{qp^{M-1}}E(t_{1}^{b^{*}})^{-p^{M-1}})$.

######  Proposition 3.2.

If $X\in\mathcal{L}_{sep}$ is such that

(3.1) $\Omega\left(\sigma^{M-1}e_{1}^{(q)}\right)=\sigma
X\circ\sigma^{M-1}e_{1}^{(q)}\circ(-X)\,$

then $\mathcal{L}^{(v_{0})}$ is the minimal ideal in the set of all ideals
$\mathcal{I}$ of $\mathcal{L}$ such that

$v(\mathcal{K}^{\prime}(X\,\mathrm{mod}\,\mathcal{I}_{O^{\prime}_{M}(\mathcal{K}_{sep})})/\mathcal{K}^{\prime})<qv_{0}-b^{*}\,.$

######  Remark.

The above inequality can be also rewritten in the following way
$v(\mathcal{K}^{\prime}(X\,\mathrm{mod}\,\mathcal{I}_{O^{\prime}_{M}(\mathcal{K}_{sep})})/\mathcal{K})<v_{0}\,.$

Suppose $\widetilde{\mathcal{J}}\subset\mathcal{L}$ is a closed ideal and
$\pi:\mathcal{L}\longrightarrow L:=\mathcal{L}/\widetilde{\mathcal{J}}$ is a
natural projection. Then we can use the appropriate images
$e_{L},e_{L}^{(q)}\in L_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$ and
$f_{L},X_{L}\in L_{O^{\prime}_{M}(\mathcal{K}_{sep})}$ to state the following
analog of Prop.3.2 for the ideal $L^{(v_{0})}:=\pi(\mathcal{L}^{(v_{0})})$ of
$L$.

######  Proposition 3.3.

$L^{(v_{0})}$ is the minimal ideal in the set of all ideals $\mathcal{I}$ of
$L$ such that
$v(\mathcal{K}^{\prime}(X_{L}\,\mathrm{mod}\,\mathcal{I}_{sep})/\mathcal{K}^{\prime})<qv_{0}-b^{*}$.

###  3.4. Relation between $\mathcal{L}^{(v_{0})}$ and $\mathcal{L}^{[v_{0}]}$

######  Definition.

For the projection $\mathrm{pr}:\mathcal{L}\to\bar{\mathcal{L}}$, set
$\mathcal{L}^{[v_{0}]}:=\mathrm{pr}^{-1}(\bar{\mathcal{L}}^{[v_{0}]})$.

Consider equality (3.1) from the above criterion. On the other hand, according
to Proposition 2.12 we have in $\bar{\mathcal{N}}^{\,{\dagger}}$ that

(3.2)
$\Omega\left(\sigma^{M-1}\bar{e}^{{\dagger}}\right)=\sigma(\bar{c}_{1})\circ(A^{{\dagger}}\otimes\mathrm{id})\sigma^{M-1}\bar{e}^{{\dagger}}\circ(-\bar{c}_{1})\,.$

Applying
$\bar{\mathcal{V}}^{0}:\bar{\mathcal{L}}^{{\dagger}}\to\bar{\mathcal{L}}^{{\dagger}}[0]$
and using the definition of the ideal $\bar{\mathcal{L}}^{[v_{0}]}$ we obtain
the following congruence in $\bar{\mathcal{N}}^{{\dagger}}[0]$ with
$\bar{x}=\bar{\mathcal{V}}^{0}(\bar{c}_{1})$:

(3.3)
$\Omega\left(\sigma^{M-1}\bar{e}_{1}^{(q)}\right)\equiv\sigma(\bar{x})\circ\sigma^{M-1}\bar{e}_{1}^{(q)}\circ(-\bar{x})\,\mathrm{mod}\,\bar{\mathcal{L}}^{[v_{0}]}_{O^{\prime}_{M}(\mathcal{K}^{\prime})}\,.$

Since
$\mathcal{L}^{{\dagger}}[0](p)=\mathrm{Ker}(\mathcal{L}^{{\dagger}}[0]\to\bar{\mathcal{L}}^{{\dagger}}[0])\subset\mathcal{L}^{[v_{0}]}$
we can take any $x\in\mathcal{N}^{{\dagger}}[0]$ such that
$\bar{x}=x\,\mathrm{mod}\,\mathcal{L}^{{\dagger}}[0](p)_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$
and consider congruence (3.3) in $\mathcal{N}^{{\dagger}}[0]$:

$\Omega\left(\sigma^{M-1}{e}_{1}^{(q)}\right)\equiv\sigma(x)\circ\sigma^{M-1}{e}_{1}^{(q)}\circ(-x)\,\mathrm{mod}\,{\mathcal{L}}^{[v_{0}]}_{O^{\prime}_{M}(\mathcal{K}^{\prime})}\,.$

In particular, the criterion from Prop. 3.2 implies that
${\mathcal{L}}^{(v_{0})}\subset{\mathcal{L}}^{[v_{0}]}$. (Use that $x$ is
defined just over a trivial field extension
$\mathcal{K}^{\prime}/\mathcal{K}^{\prime}$.)

This means also that we can assume $X\equiv
x\,\mathrm{mod}\,\mathcal{L}^{[v_{0}]}_{sep}$.

The following proposition implies the statement of Theorem 3.1 when $w=p$.

######  Proposition 3.4.

For any $1\leqslant w<p$, it holds

$a_{w})$
$\mathcal{L}^{(v_{0})}+C_{w}(\mathcal{L})=\mathcal{L}^{[v_{0}]}+C_{w}(\mathcal{L})\,.$

$b_{w})$ $\mathcal{L}(w)\subset\mathcal{L}^{(v_{0})}+C_{w}(\mathcal{L})$.

###  3.5. The proof of Proposition 3.4

Use induction on $w\geqslant 1$. If $w=1$ there is nothing to prove. So,
assume that proposition holds for some $1\leqslant w<p$.

For non-commuting variables $U$ and $V$, let

$\delta^{0}(U,V):=U\circ V-(U+V)\,.$

Note that if the values of $U$ and $V$ are defined modulo
$C_{w}(\mathcal{L})_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$ then the value of
$\delta^{0}(U,V)$ is well-defined modulo
$C_{w+1}(\mathcal{L})_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$.

######  Lemma 3.5.

There is a congruence

$\sigma X-X\equiv
y\,\mathrm{mod}\,\left([\mathcal{L}^{(v_{0})},\mathcal{L}]+C_{w+1}(\mathcal{L})\right)_{sep}\,,$

where the element $y\in\mathcal{N}^{{\dagger}}[0]$ is such that

$y=\Omega\left(\sigma^{M-1}e_{1}^{(q)}\right)-\sigma^{M-1}e_{1}^{(q)}+\delta^{0}\left(\Omega\left(\sigma^{M-1}e_{1}^{(q)}\right),x\right)-\delta^{0}\left(\sigma
x,\sigma^{M-1}e_{1}^{(q)}\right)\,.$

###### Proof of lemma.

Use that $\Omega\left(\sigma^{M-1}e_{1}^{(q)}\right)\circ X=\sigma
X\circ\sigma^{M-1}e_{1}^{(q)}$ and there are the following congruences modulo
$\left([\mathcal{L}^{(v_{0})},\mathcal{L}]+C_{w_{0}+1}(\mathcal{L})\right)_{sep}$:

$\delta^{0}\left(\Omega\left(\sigma^{M-1}e_{1}^{(q)}\right),X\right)\equiv\delta^{0}\left(\Omega\left(\sigma^{M-1}e_{1}^{(q)}\right),x\right)\,,$
$\delta^{0}(\sigma X,\sigma^{M-1}e_{1}^{(q)})\equiv\delta^{0}\left(\sigma
x,\sigma^{M-1}e_{1}^{(q)}\right)\,.$

∎

In order to prove part $b_{w+1})$ we need to verify the following condition:

$(\bullet_{1})$ if $qa\geqslant b^{*}U(w,m)$ then
$p^{m}D_{a0}\in(\mathcal{L}^{(v_{0})}+C_{w+1}(\mathcal{L}))_{W_{M}(k)}$.

This will be done by the use of the condition

(3.4)
$v(\mathcal{K}^{\prime}(X\,\mathrm{mod}\,\mathcal{L}^{(v_{0})}_{sep})/\mathcal{K}^{\prime})<qv_{0}-b^{*}\,.$

Suppose $y=\sum_{\iota}y_{\iota}+y_{+}$, where
$y_{+}\in\mathcal{L}_{\mathrm{m}^{\prime}}$, $\iota$ runs over
$\mathfrak{A}^{+}(p)$ and the corresponding $y_{\iota}$ appear in the form
$y_{\iota}=\sum_{r\geqslant 0}t_{1}^{-\iota p^{r}}l_{br}$ with all $l_{\iota
r}\in\mathcal{L}_{W_{M}(k)}$.

Then $X=\sum_{\iota}X_{\iota}+X_{+}$ where $\sigma
X_{\iota}-X_{\iota}=y_{\iota}$ and $\sigma X_{+}-X_{+}=y_{+}$. Here
$X_{+}\in\mathcal{L}_{\mathrm{m}^{\prime}}$ and does not contribute to (3.4).

The elements $X_{\iota}$ contribute to (3.4) as follows:

– if $\iota\in\mathfrak{A}^{+}(p)$ and $\iota/p^{v_{p}(\iota)}\geqslant
qv_{0}-b^{*}$ then

$\sum_{r}\sigma^{-r}l_{\iota
r}\in(\mathcal{L}^{(v_{0})}+C_{w+1}(\mathcal{L}))_{W_{M}(k)}\,.$

We can follow the contribution to upper ramification in (3.4) for each
$\iota\in\mathfrak{A}^{+}(p)$ separately because by Proposiion 2.4 these
contributions are independent.

The element $y\in\mathcal{N}^{{\dagger}}[0]\\{1\\}$ has a linear part

(3.5)
$L(y):=\Omega\left(\sigma^{M-1}e_{1}^{(q)}\right)-\sigma^{M-1}e_{1}^{(q)}=\sum_{a,s}t_{1}^{-ap^{M-1}q}aS_{b^{*}}^{s}D_{a0}$

and a non-linear part, which is a $W_{M}(k)$-linear combination of various

(3.6)
$Q(\iota,\bar{a}_{1},\bar{a}_{2})=p^{A}t_{1}^{-\iota}\left[[D_{\bar{a}_{1}\bar{n}_{1}}],[D_{\bar{a}_{2}\bar{n}_{2}}]\right]\,,$

where $\iota\in\mathfrak{A}^{0}$, $-\iota\geqslant-
qp^{M-1}(|\bar{a}_{1}|+|\bar{a}_{2}|)+p^{M-1-A}b^{*}$. (Recall that
$U(1,A)=p^{-A}$, cf. Section 1.)

Note that if the term $Q$ from (3.6) is such that

$|\bar{a}_{1}|+|\bar{a}_{2}|\geqslant(b^{*}/q)U(w,A)$

then $b^{*}/q$ could be replaced by $v_{0}$, cf. Section 2.1, and therefore

$p^{A}\left[[D_{\bar{a}_{1}\bar{n}_{1}}],[D_{\bar{a}_{2}\bar{n}_{2}}]\right]\in\mathcal{L}_{\mathcal{N}}(w+1)\subset\sum_{w_{1}+w_{2}=w+1}[\mathcal{L}_{\mathcal{N}}(w_{1}),\mathcal{L}_{\mathcal{N}}(w_{2})]\,.$

By inductive assumption $b_{w})$ this term belongs to the $W_{M}(k)$-extension
of scalars of

$\sum_{w_{1}+w_{2}=w+1}[\mathcal{L}^{(v_{0})}+C_{w_{1}}(\mathcal{L}),\mathcal{L}^{(v_{0})}+C_{w_{2}}(\mathcal{L})]\subset[\mathcal{L}^{(v_{0})},\mathcal{L}]+C_{w+1}(\mathcal{L})\,.$

This implies that the coefficients for $p^{A}t_{1}^{-\iota}$ in the non-linear
part of $y$ with $-\iota\leqslant-p^{M-1}b^{*}U(w,A)$ belong to
$[\mathcal{L}^{(v_{0})},\mathcal{L}]+C_{w+1}(\mathcal{L})$ and could be
ignored. Indeed, since $-\iota\geqslant-
qp^{M-1}(|\bar{a}_{1}|+|\bar{a}_{2}|+p^{M-1}b^{*}p^{M-1-A}$ it holds that
$|\bar{a}_{1}|+|\bar{a}_{2}|\geqslant(b^{*}/q)(U(w,m)+b^{*}p^{-m})$. All other
terms from the non-linear part will contribute just to smaller upper numbers
and also can be ignored. As a result, it remains to care just about the linear
part.

If $y_{\iota}=p^{m}t_{1}^{-\iota}l_{\iota}$ is a term from the linear part
$L(y)$ such that $\iota>0$ then $\iota\in\mathfrak{A}^{+}(p)$. Therefore, such
terms do not appear with $t_{1}^{-\iota p^{r}}$, $r\geqslant 1$. By Prposition
2.4 for different $\iota$, these terms give independent contribution to the
upper ramification number in (3.4).

Consider the term $p^{m}t^{-\iota}D_{a0}$ with
$-\iota=-aqp^{M-1}+b^{*}p^{M-1}U(w,m)$. If it satisfies the condition
$(\bullet_{1})$ then $\iota>0$ and it contributes to the upper number
$qp^{i}a-b^{*}(p^{i}U(w,m))$ where $p^{i}U(w,m)\in\mathbb{Z}^{+}(p)$ with
$i\geqslant 0$.

Therefore, $qp^{i}a-b^{*}p^{i}U(w,m)>0$ and by Sec.2.1, it is

$\geqslant
qp^{i}v_{0}U(w,m)-b^{*}p^{i}U(w,m)=(qv_{0}-b^{*})p^{i}U(w,m)\geqslant
qv_{0}-b^{*}\,.$

As a result, $p^{m}D_{a0}\in\mathcal{L}^{(v_{0})}+C_{w+1}(\mathcal{L})$. Part
$b_{w+1}$) is proved.

Prove part $a_{w+1}$).

Applying $\bar{\mathcal{V}}^{0}$ to (3.2) we obtain the following congruence
in $\bar{\mathcal{N}}^{{\dagger}}[0]$ modulo extension of scalars of the ideal
$[\bar{\mathcal{L}}^{{\dagger}[v_{0}]},\bar{\mathcal{L}}^{{\dagger}}]+C_{w+1}(\bar{\mathcal{L}}^{{\dagger}})$:

$\Omega\left(\sigma^{M-1}\bar{e}_{1}^{(q)}\right)\equiv\sigma(\bar{x})\circ(\sigma^{M-1}\bar{e}_{1}^{(q)})\circ(-\bar{x})+\sum_{\iota\in\mathfrak{A}^{+}(p)}t_{1}^{-\iota}\bar{\mathcal{V}}(A^{{\dagger}}-\mathrm{id})D_{\iota
0}\,.$

Comparing it with our criterion

$\Omega\left(\sigma^{M-1}\bar{e}_{1}^{(q)}\right)\equiv\sigma(\bar{X})\circ\sigma^{M-1}\bar{e}_{1}^{(q)}\circ(-\bar{X})\,\mathrm{mod}\,\left([\bar{\mathcal{L}}^{[v_{0}]},\bar{\mathcal{L}}]+C_{w+1}(\bar{\mathcal{L}})\right)_{sep}\,.$

we can assume that $\bar{X}=\bar{x}+\bar{Y}$, where
$\bar{Y}\in\bar{\mathcal{L}}^{[v_{0}]}_{sep}$ and satisfies the following
relation

$\sigma(\bar{Y})-\bar{Y}=\sum_{\iota\in\mathfrak{A}^{+}(p)}t_{1}^{-\iota}\bar{\mathcal{V}}(A^{{\dagger}}-\mathrm{id})(D^{{\dagger}}_{\iota
0})\,.$

Choose the $\iota$-components $\bar{Y}_{\iota}$ such that
$\sigma\bar{Y}_{\iota}-\bar{Y}_{\iota}=t_{1}^{-\iota}(A^{{\dagger}}-\mathrm{id})(D_{\iota
0})$ and $\bar{Y}=\sum_{\iota}\bar{Y}_{\iota}$. All $\iota$ give independent
contributions to upper ramification numbers. Take
$\iota\in\mathfrak{A}_{1}^{+}(p)$, then
$\sigma(\bar{Y}_{\iota})-\bar{Y}_{\iota}=t_{1}^{-\iota}B^{{\dagger}}(D_{\iota
0})$.

If $\iota=p^{m}(qp^{M-1}\alpha-p^{M-1-A}b^{*})$, cf. Prop. 2.4b), then it
gives the upper ramification $q(p^{A}\alpha)-b^{*}$. On the other hand,
$\alpha>p^{-A}b^{*}/q$ is equivalent to $\alpha\geqslant p^{-A}v_{0}$ or
$qp^{A}\alpha-b^{*}\geqslant qv_{0}-b^{*}$. So,
$\bar{\mathcal{V}}B^{{\dagger}}(D_{\iota 0})\in\bar{\mathcal{L}}^{(v_{0})}$.

Therefore,
$\bar{\mathcal{L}}^{(v_{0})}+C_{w+1}(\bar{\mathcal{L}})\supset\bar{\mathcal{L}}^{[v_{0}]}+C_{w+1}(\bar{\mathcal{L}}^{[v_{0}]})$.

Theorem 3.1 is completely proved.

###  3.6. Interpretation in terms of deformations

The above described construction of the ramification ideals
$\mathcal{L}^{(v_{0})}$ and $\bar{\mathcal{L}}^{(v_{0})}$ admits the following
interpretation if we replace everywhere $t_{1}$ by $t$.

We start, as earlier, with Lie algebra $\mathcal{L}$ and
$e=e(t)\in\mathcal{L}_{O_{M}(\mathcal{K})}$ such that (for simplicity)
$e=\sum_{a\in\mathbb{Z}^{0}(p)}t^{-a}D_{a0}$. Choose $f\in\mathcal{L}_{sep}$
such that $\sigma f=e\circ f$ and fix identification
$\eta^{0}:\mathcal{G}_{<p,M}\simeq G(\mathcal{L})$ via
$\tau\mapsto(-f)\circ\tau(f)$, $\tau\in\mathcal{G}_{<p,M}$. As earlier, define
the filtration $\mathcal{L}(p)$ and set
$\bar{\mathcal{L}}=\mathcal{L}/\mathcal{L}(p)$. Then
$\bar{\mathcal{L}}_{W_{M}(k)}$ is generated by $D_{an}$, where
$a\in\mathbb{Z}^{0}(p)\cap[0,(p-1)v_{0})$. If
$\bar{e}\in\bar{\mathcal{L}}_{O_{M}(\mathcal{K})}$ and
$\bar{f}\in\bar{\mathcal{L}}_{sep}$ are the images of $e$ and, resp., $f$
under extensions of scalars of the projection
$\mathcal{L}\to\bar{\mathcal{L}}$ then
$\tau\mapsto(-\bar{f})\circ\tau(\bar{f})$ determines projection
$\bar{\eta}^{0}:\mathcal{G}_{<p,M}\to G(\bar{\mathcal{L}})$. Clearly,
$\bar{\eta}^{0}$ is induced by $\eta^{0}$.

As earlier, choose $r^{*}$, $N^{*}$, set $q=p^{N^{*}}$, $b^{*}=r^{*}(q-1)$ and
introduce the system of coefficients $\\{t^{-\iota}\ |\
\iota\in\mathfrak{A}^{0}\\}$ from Section 2.2. Consider the quotient
${\mathcal{L}}^{{\dagger}}$ of $\mathcal{L}$ generated by

$\\{D_{an}\ |\ a\in[0,(p-1)v_{0}\\}\cup\\{D_{\iota n}\ |\
\iota\in\mathfrak{A}^{+}(p)\\}\subset\\{D_{an}\ |\
a\in\mathbb{Z}^{0}(p)\\}\,.$

As earlier, define a filtration on $\mathcal{L}^{{\dagger}}$ and set
$\bar{\mathcal{L}}^{{\dagger}}=\mathcal{L}^{{\dagger}}/\mathcal{L}^{{\dagger}}(p)$.
Note that by Prop. 2.4, all $D_{\iota n}\in\bar{\mathcal{L}}(p)_{W_{M}(k)}$,
and therefore, we have the induced projection
$\bar{\mathcal{V}}:\bar{\mathcal{L}}^{{\dagger}}\to\bar{\mathcal{L}}$.

Take

$e^{{\dagger}}=\sum_{a\in[0,(p-1)v_{0})}t^{-qp^{M-1}a}D_{a,M-1}+\sum_{\iota\in\mathfrak{A}^{+}(p)}t^{-\iota}D_{\iota
0}\,.$

Then the image of $\bar{e}^{{\dagger}}$ in
$\bar{\mathcal{L}}_{O_{M}(\mathcal{K})}$ coincides with
$\sigma^{M-1+N^{*}}(\bar{e})$. Therefore, we can choose
$f^{{\dagger}}\in\bar{\mathcal{L}}^{{\dagger}}_{sep}$ such that its image in
$\bar{\mathcal{L}}_{sep}$ coincides with $\sigma^{M-1+N^{*}}\bar{f}$. This
means that we have epimorphism $\mathcal{G}_{<p,M}\to
G(\bar{\mathcal{L}}^{{\dagger}})$ which gives $\bar{\eta}^{0}$ when composed
with projection $G(\bar{\mathcal{V}}):G(\bar{\mathcal{L}}^{{\dagger}})\to
G(\bar{\mathcal{L}})$.

Now Theorem 3.1 states that a unipotent $\mathbb{Z}/p^{M}$-action determined
by $t^{qp^{M-1}}\mapsto t^{qp^{M-1}}\widetilde{\exp}(-t^{b^{*}})^{p^{M-1}}$
induces via formalism of nilpotent Artin-Schreier theory a deformation
$B^{{\dagger}}:\mathrm{Ker}\,\bar{\mathcal{V}}\to\mathrm{Ker}\,\bar{\mathcal{V}}$.
This deformation allows us to recover a “substantial” part
$\bar{\mathcal{L}}^{(v_{0})}$ of the ramification ideal
$\mathcal{L}^{(v_{0})}$. It appears as
$\bar{\mathcal{V}}(\mathrm{Ker}\,\bar{\mathcal{V}})$, i.e. as a deformation of
the “trivial” part $\mathrm{Ker}\,\bar{\mathcal{V}}\subset\mathcal{L}(p)$ via
nilpotent operator $B^{{\dagger}}$.

##  4\. Explicit computations

Now we are going to specify our approach to obtain explicit description of
generators of the ramification ideal $\bar{\mathcal{L}}^{[v_{0}]}$ and to see
explicitly how the idea of deformation works in our case.

###  4.1. Agreements

We need to treat the Campbell-Hausdorff group law via exponentials in the
appropriate enveloping algebras $\bar{\mathcal{A}}$ and
$\bar{\mathcal{A}}^{{\dagger}}$ of $\bar{\mathcal{L}}$ and, resp.,
$\bar{\mathcal{L}}^{{\dagger}}$, by involving extension of scalars to
$O^{\prime}_{M}(\mathcal{K}^{\prime})$. As a matter of fact, the definition of
the action $\Omega$ requires working inside
$\widetilde{\mathcal{N}}^{{\dagger}}$ but can be formally treated in
$\bar{\mathcal{N}}^{{\dagger}}$, cf. Prop. 2.12. So, it makes sense to work
mainly with the Lie algebra
$\bar{\mathcal{L}}^{{\dagger}}[0]\simeq\bar{\mathcal{L}}$ and its enveloping
algebra $\bar{\mathcal{A}}^{{\dagger}}[0]$.

We are going also to use the advantage of special coefficients from
$\mathfrak{A}^{0}$. So, we need just a part of the quotient
$(\bar{\mathcal{A}}^{{\dagger}}/\bar{\mathcal{J}}^{{\dagger}p})_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$
generated by the elements of $\bar{\mathcal{N}}^{{\dagger}}[0]$. (Here
$\bar{\mathcal{J}}^{{\dagger}}$ is the augmentation ideal of
$\bar{\mathcal{A}}^{{\dagger}}$.) Even more, the ideal
$\bar{\mathcal{L}}^{[v_{0}]}$ is completely determined by the elements
$\bar{\mathcal{V}}^{0}B^{{\dagger}}(D_{\iota 0})$ with $\mathrm{ch}(\iota)=1$.
Therefore, it would be reasonable to ignore all $\iota$ with
$\mathrm{ch}(\iota)\geqslant 2$, i.e. to work with the part generated by
$\bar{\mathcal{N}}^{{\dagger}}_{\leqslant
1}[0]:=\bar{\mathcal{N}}^{{\dagger}}[0,0]\oplus\bar{\mathcal{N}}^{{\dagger}}[1,0]$.

As a result we are going to perform calculations inside the $W_{M}(k)$-module
$\bar{\mathcal{A}}^{{\dagger}}_{\leqslant 1}[0]$ generated by all
$p^{A}t_{1}^{-\iota}D_{\bar{a}\bar{n}}$, where for
$\bar{a}=(a_{1},\dots,a_{r})$ and $\bar{n}=(n_{1},\dots,n_{r})$,
$D_{\bar{a}\bar{n}}=D_{a_{1}n_{1}}\ldots D_{a_{r}n_{r}}$,
$\iota\in\mathfrak{A}^{0}$, $\mathrm{ch}(\iota)\in\\{0,1\\}$ and it holds
$-\iota\geqslant-qp^{M-1}|\bar{a}|+p^{M-1-A}b^{*}$ (recall that
$U(1,A)=p^{-A}$). Since we do not need to follow the effects related to
$\mathrm{ch}(\iota)\geqslant 2$, we can agree that $t_{1}^{-\iota}\cdot
t_{1}^{-\iota^{\prime}}=0$ if $\iota,\iota^{\prime}\in\mathfrak{A}^{0}_{1}$.
We do need to go to $\bar{\mathcal{A}}^{{\dagger}}_{\leqslant 1}[0]$ after
applying exponential map to $\bar{\mathcal{N}}^{{\dagger}}_{\leqslant 1}$ but
almost immediately we come back to $\bar{\mathcal{N}}^{{\dagger}}_{\leqslant
1}$ and even to $\bar{\mathcal{N}}^{{\dagger}}[1,0]$.

###  4.2. Choice of $e\in\mathcal{L}_{O_{M}(\mathcal{K})}$

We keep sufficiently general choice of the element
$e=e(t)\in{\mathcal{L}}_{O_{M}(\mathcal{K})}$ used for the construction of the
identification $\eta^{0}:\mathcal{G}_{<p,M}\simeq G(\mathcal{L})$. We fix its
choice by specifying

$E=\widetilde{\exp}(e)=1+\sum_{r(\bar{a})\geqslant
1}\eta(\bar{a})t^{-(a_{1}+\dots+a_{r})}D_{a_{1}0}\ldots D_{a_{r}0}\,.$

The structural constants $\eta(\bar{a})\in W_{M}(k)$ reflect the fact that $E$
is a “diagonal element” in ${\mathcal{A}}/J^{p}$ or, equivalently,
$E\in\widetilde{\exp}(\mathcal{L}_{O_{M}(\mathcal{K})})$ or, equivalently,
$\Delta E\equiv E\otimes E$ modulo $p$-th power of augmentation ideal in
$\mathcal{A}^{\otimes 2}_{O_{M}(\mathcal{K})}$. Explicitly, this can be stated
via the following condition:

$(*_{e})$ if $0\leqslant s_{1}\leqslant s<p$ then

$\eta(a_{1},\dots,a_{s_{1}})\eta(a_{s_{1}+1},\dots,a_{s})=\sum_{\pi\in
I_{s_{1}s}}\eta(a_{\pi(1)},\dots,a_{\pi(s)}),$

where $I_{s_{1}s}$ consists of all permutations $\pi$ of order $s$ such that
the sequences $\pi^{-1}(1),\dots,\pi^{-1}(s_{1})$ and
$\pi^{-1}(s_{1}+1),\dots,\pi^{-1}(s)$ are increasing (i.e. $I_{s_{1}s}$ is the
set of all “insertions” of the ordered set $\\{1,\dots,s_{1}\\}$ into the
ordered set $\\{s_{1}+1,\dots,s\\}$).

Note that in papers [1, 2, 3] we treated on the first place the simplest
choice of such constants $\eta(\bar{a})$ related to the case
$e=\sum_{a\in\mathbb{Z}^{0}(p)}t^{-a}D_{a0}$. In this case the constants are
quite simple $\eta(\bar{a})=1/r!$, where $\bar{a}=(a_{1},\dots,a_{r})$.
However, our approach works equally well also in the above general situation.

######  Remark.

There is very interesting possibility to take

$e(t)=D_{00}\circ(t^{-1}D_{10})\circ\ldots\circ(t^{-a}D_{a0})\circ\ldots\,.$

Then $E=\sum_{r(\bar{a})\geqslant
0}\eta(\bar{a})t^{-|\bar{a}|}D_{\bar{a}\bar{0}}$, where $\eta(\bar{a})=0$ if
the condition $0\leqslant a_{1}\leqslant a_{2}\leqslant\ldots\leqslant a_{r}$
for $D_{\bar{a}\bar{0}}$ is not satisfied. Note that under this condition
$\eta(\bar{a})=1/(r_{1}!r_{2}!\ldots r_{s}!)$ where
$\bar{a}=(\bar{a}^{(1)},\ldots,\bar{a}^{(s)})$ and each of the involved
vectors $\bar{a}^{(j)}$, $1\leqslant j\leqslant s$, has length $r_{j}$ and
equal components. This could be very helpful for studying the ramification
breaks of the filtration $\\{\mathcal{L}^{(v)}\\}_{v\geqslant 0}$.

###  4.3. Statement of the main result

######  Definition.

Let $\bar{n}=(n_{1},\dots,n_{s})$ with $s\geqslant 1$. Suppose there is a
partition $0=i_{0}<i_{1}<\dots<i_{r}=s$ such that if $i_{j}<u\leqslant
i_{j+1}$ then $n_{u}=m_{j+1}$ and $m_{1}>m_{2}>\dots>m_{r}$. Then set

$\eta(\bar{a},\bar{n})=\sigma^{m_{1}}\eta(\bar{a}^{(1)})\dots\sigma^{m_{r}}\eta(\bar{a}^{(r)})\,,$

where $\bar{a}^{(j)}=(a_{i_{j-1}+1},\dots,a_{i_{j}})$. If such a partition
does not exist we set $\eta(\bar{a},\bar{n})=0$.

If $s=0$ we set $\eta(\bar{a},\bar{n})=1$. As earlier, set
$D_{\bar{a}\bar{n}}=D_{a_{1}n_{1}}\dots D_{a_{s}n_{s}}$.

Note, if
$e_{(N^{*},\,0]}:=\sigma^{N^{*}-1}(e)\circ\sigma^{N^{*}-2}(e)\circ\dots\sigma(e)\circ
e$ then

$\widetilde{\exp}\,e_{(N^{*},\,0]}\equiv\sum_{\bar{a},\bar{n}}\eta(\bar{a},\bar{n})t^{-|\bar{a}|}D_{\bar{a}\bar{n}}\,.$

For $\gamma\geqslant 0$ and $N\in\mathbb{Z}_{\geqslant 0}$, introduce
$\mathcal{F}^{0}_{\gamma,-N}\in{\mathcal{L}}_{W_{M}(k)}$ such that

$\mathcal{F}^{0}_{\gamma,-N}=\sum_{\begin{subarray}{c}\bar{a},\bar{n}\\\
\gamma(\bar{a},\bar{n})=\gamma\end{subarray}}a_{1}p^{n_{1}}\eta(\bar{a},\bar{n})[\dots[D_{a_{1}n_{1}},D_{a_{2}n_{2}}],\dots,D_{a_{s}n_{s}}]\,.$

Here:

— $\bar{a}=(a_{1},\dots,a_{s})$, $\bar{n}\in\mathbb{Z}^{s}$, $n_{1}\geqslant
0$ and all $n_{i}\geqslant-N$;

—
$\gamma=\gamma(\bar{a},\bar{n})=a_{1}p^{n_{1}}+a_{2}p^{n_{2}}+\dots+a_{s}p^{n_{s}}$
.

######  Remark.

By definition everywhere $D_{a_{i}n_{i}}:=D_{a_{i},n_{i}\mathrm{mod}N_{0}}$,
but when computing $\eta(\bar{a},\bar{n})$ or $\gamma(\bar{a},\bar{n})$ it is
essential that $n_{i}\in\mathbb{Z}$.

Note that the non-zero terms in the above expression for
$\mathcal{F}_{\gamma,-N}^{0}$ can appear only if $n_{1}\geqslant
n_{2}\geqslant\ldots\geqslant n_{s}$ and $\gamma\in A[p-1,N]$, cf. Sect. 2.1.

The result about explicit generators of $\bar{\mathcal{L}}^{[v_{0}]}$ can be
stated in the following form.

Let $\bar{\mathcal{F}}^{0}_{\gamma,-N}$ be the image of
$\mathcal{F}^{0}_{\gamma,-N}$ in $\bar{\mathcal{L}}_{W_{M}(k)}$.

Let $\iota=p^{m}(qp^{M-1}\alpha-(q-1)\beta)\in\mathfrak{A}_{1}^{+}(p)$ be the
standard presentation from Section 2.2. Note that by definition
$\iota\leqslant p^{M-1}(p-1)b^{*}$, $p^{m}\alpha$ and $p^{m}(q-1)\beta/r^{*}$
are integers not both divisible by $p$. We have also that
$p^{m}(q-1)\beta_{\iota}/b^{*}=p^{m+M-1-A}$ where $0\leqslant A\leqslant M-1$,
compare to Prop. 2.4b). So, by setting $\gamma(\iota):=p^{A}\alpha$ and
$M(\iota):=m-A$ we obtain the following presentation of $\iota$:

(4.1) $\iota=p^{M-1+M(\iota)}(q\gamma(\iota)-b^{*})\,.$

######  Remark.

The condition $\iota\leqslant p^{M-1}(p-1)b^{*}$ holds automatically when we
work with elements of $\bar{\mathcal{N}}^{{\dagger}}[0]$. Indeed, if
$t^{-\iota}D_{\bar{a}\bar{n}}\in\bar{\mathcal{N}}^{{\dagger}}[0]$ then
$-\iota\geqslant-qp^{M-1}|\bar{a}|$. Therefore, if $\iota>p^{M-1}(p-1)b^{*}$
then $|\bar{a}|>(p-1)b^{*}/q$. By condition C3 from Section 2.1 it is
equivalent to $|\bar{a}|\geqslant(p-1)v_{0}$ and then by Corollary 1.8,
$D_{\bar{a}\bar{n}}\in\bar{\mathcal{L}}(p)_{W_{M}(k)}=0$.

Let $m(\iota)$ be the maximal integer such that $\iota p^{m(\iota)}\leqslant
p^{M-1}(p-1)b^{*}$. For any $\iota\in\mathfrak{A}^{+}_{1}(p)$, fix a choice of
$m_{\iota}\geqslant m(\iota)$.

######  Theorem 4.1.

$\bar{\mathcal{L}}^{[v_{0}]}$ is the minimal ideal in $\bar{\mathcal{L}}$ such
that for all $\iota\in\mathfrak{A}^{+}_{1}(p)$ given in notation (4.1),
$\bar{\mathcal{F}}^{0}_{\gamma(\iota),-(M(\iota)+m_{\iota})}\in\bar{\mathcal{L}}^{[v_{0}]}_{W_{M}(k)}$.

The proof is given in Sect. 4.4-4.7 below.

###  4.4. A recurrent relation

Consider the part of relation from Section 3.5 which is adjusted to
$\bar{\mathcal{N}}^{{\dagger}}_{\leqslant 1}$

(4.2)
$\Omega(\sigma^{M-1}\bar{e}_{1}^{(q)})=\sigma(\bar{x}_{1})\circ\sigma^{M-1}\bar{e}_{1}^{(q)}\circ(-\bar{x}_{1})+\qquad\qquad\qquad$
$\qquad\qquad\qquad\sum_{\iota\in\mathfrak{A}_{1}^{+}(p)}t_{1}^{-\iota}V_{\iota}\,\mathrm{mod}\,[\bar{\mathcal{L}}^{[v_{0}]},\bar{\mathcal{L}}]_{O^{\prime}_{M}(\mathcal{K}^{\prime})}\,.$

where $\bar{x}_{1}\in\bar{\mathcal{N}}^{{\dagger}}[1,0]$ and
$V_{\iota}=\bar{\mathcal{V}}^{0}B^{{\dagger}}(D_{\iota 0}^{{\dagger}})$.

Let $E_{0}=\exp(\sigma^{M-1}\bar{e}_{1}^{(q)})$ and
$\exp(\Omega(\sigma^{M-1}\bar{e}_{1}^{(q)})=E_{0}+\mathcal{E}_{0}$. In
particular,
$E_{0}=\sum_{a\in\mathbb{Z}^{0}(p)}\sigma^{M-1}\eta(\bar{a})t_{1}^{-qp^{M-1}|\bar{a}|}D_{\bar{a},M-1}$
and

$\mathcal{E}_{0}=\sum_{r(\bar{a})\geqslant
1}\sigma^{M-1}\eta(\bar{a})|\bar{a}|\sum_{A\geqslant
0}p^{A}t_{1}^{-qp^{M-1}|\bar{a}|+p^{M-1-A}b^{*}}D_{\bar{a},{M-1}}\,.$

Then (4.2) can be rewritten as follows

(4.3) $\sum_{\iota\in\mathfrak{A}_{1}^{+}(p)}t_{1}^{-\iota}V_{\iota}\equiv
E_{0}^{-1}\mathcal{E}_{0}+\bar{x}_{1}-E_{0}^{-1}\sigma(\bar{x}_{1})E_{0}\,\mathrm{mod}\,[\bar{\mathcal{L}}^{[v_{0}]},\bar{\mathcal{L}}]_{O^{\prime}_{M}(\mathcal{K}^{\prime})}\,.$

Note that this relation is still in
$\bar{\mathcal{L}}_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$ due to the
following easy lemma:

######  Lemma 4.2.

If $l\in\bar{\mathcal{N}}^{{\dagger}}_{\leqslant 1}[0]$,
$l^{\prime}\in\bar{\mathcal{N}}^{{\dagger}}[1,0]$ then

$\exp(l+l^{\prime})\cdot\exp(-l)-1\in\bar{\mathcal{N}}^{{\dagger}}[1,0]\,.$

As a result, the ideal $\bar{\mathcal{L}}^{[v_{0}]}$ appears now as the
minimal ideal in $\bar{\mathcal{L}}$ such that
$\bar{\mathcal{L}}^{[v_{0}]}_{W_{M}(k)}$ contains the elements
$V_{\iota}\in\bar{\mathcal{L}}_{W_{M}(k)}$ from (4.3).

###  4.5. Solving relation (4.2)

Let $\iota\in\mathfrak{A}^{0}_{1}$. Set

— $r(\iota):=\max\\{r\ |\ \iota p^{r}\in\mathfrak{A}_{1}^{0}\\}(=\max\\{r\ |\
|\iota p^{r}|\leqslant p^{M-1}(p-1)b^{*}\\})$.

— $\mathfrak{A}_{1}^{\mathrm{prim}}=\mathfrak{A}^{0}_{1}\setminus
p\mathfrak{A}^{0}_{1}$. Note that
$\mathfrak{A}^{+}_{1}(p)=\\{\iota\in\mathfrak{A}^{\mathrm{prim}}_{1}\ |\
\iota>0\\}$.

For any $j\geqslant 0$, let $E_{j}=\sigma^{j}E_{0}$ and
$\mathcal{E}_{j}=\sigma^{j}\mathcal{E}_{0}$.

Consider the following formal sum in
$\bar{\mathcal{A}}_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$

$\sum_{j\geqslant 0}E_{0}^{-1}E_{1}^{-1}\ldots
E_{j}^{-1}\mathcal{E}_{j}E_{j-1}\ldots E_{0}=\sum_{\iota,r}t_{1}^{-\iota
p^{r}}\bar{\mathcal{F}}[\iota p^{r}]\,$

where the last sum is taken over all
$\iota\in\mathfrak{A}^{\mathrm{prim}}_{1}$ and $r\geqslant 0$.

######  Proposition 4.3.

For all $\iota\in\mathfrak{A}^{\mathrm{prim}}_{1}$ and $r\geqslant 0$,
$\bar{\mathcal{F}}[ip^{r}]\in\bar{\mathcal{L}}_{W_{M}(k)}$.

The proof will be given in Sections 4.6-4.7 below.

For $r\geqslant 0$, let $\bar{\mathcal{F}}[\iota
p^{r}]_{0}:=\bar{\mathcal{F}}[\iota p^{r}]-\sigma\bar{\mathcal{F}}[\iota
p^{r-1}]\in\bar{\mathcal{L}}_{W_{M}(k)}$. Then

$\bar{\mathcal{F}}[\iota p^{r}]=\sum_{0\leqslant j\leqslant
r}\sigma^{j}\bar{\mathcal{F}}[\iota p^{r-j}]_{0}$

######  Proposition 4.4.

a) $\bar{x}_{1}\equiv-\sum_{\iota,r}\bar{\mathcal{F}}[\iota
p^{r}]t_{1}^{-\iota
p^{r}}\,\mathrm{mod}\,\bar{\mathcal{L}}^{[v_{0}]}_{O^{\prime}_{M}(\mathcal{K}^{\prime})}+\bar{\mathcal{N}}^{{\dagger}}(p)$,
where the sum is taken over all $\iota\in\mathfrak{A}^{\mathrm{prim}}_{1}$ and
$0\leqslant r\leqslant r(\iota)$;

b) if $\iota\in\mathfrak{A}_{1}^{+}(p)$, then

$V_{\iota}\equiv\sigma^{-r(\iota)}\bar{\mathcal{F}}[\iota
p^{r(\iota)}]\,\mathrm{mod}\,\left[\bar{\mathcal{L}}^{[v_{0}]},\bar{\mathcal{L}}\right]_{W_{M}(k)}$

###### Proof.

Apply induction on $1\leqslant w<p$ to prove the proposition modulo
$C_{w}(\bar{\mathcal{L}})_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$ for all
$1\leqslant w\leqslant p$. Clearly, a) holds with $w=1$.

Suppose $1\leqslant w<p$ and a) holds modulo
$C_{w}(\bar{\mathcal{L}})_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$.

Rewrite (4.2) in the following form

(4.4)
$\sum_{\mathrm{ch}\iota=1}t_{1}^{-\iota}V_{\iota}+\sigma{\bar{x}}_{1}^{1}-\bar{x}_{1}^{1}\equiv$
$E_{0}^{-1}\mathcal{E}_{0}-\left(E_{0}^{-1}\sigma\bar{x}_{1}^{1}E_{0}-\sigma\bar{x}^{1}_{1}\right)\,\mathrm{mod}\,[\bar{\mathcal{L}}^{[v_{0}]},\bar{\mathcal{L}}]_{O^{\prime}_{M}(\mathcal{K}^{\prime})}\,.$

Now the RHS belongs to $\bar{\mathcal{N}}^{{\dagger}}[1,0]$ modulo extension
of scalars of
$[\bar{\mathcal{L}}^{[v_{0}]},\bar{\mathcal{L}}]+C_{w+1}(\bar{\mathcal{L}})$.
It is easy to see that the RHS appears in the form

$E_{0}^{-1}\mathcal{E}_{0}+\sum_{\iota,r,a}\bar{\mathcal{F}}[qp^{M-1}a+\iota
p^{r+1}]_{0}t_{1}^{-(qp^{M-1}a+\iota p^{r+1})}=\sum_{r\leqslant
r(\iota)+1,\iota}\bar{\mathcal{F}}[\iota p^{r}]_{0}t_{1}^{-\iota p^{r}}\,.$

We can assume that in the last sum $r\leqslant r(\iota)$ due to the following
Lemma.

######  Lemma 4.5.

Let $r\geqslant r(\iota)+1$.

a) If $\iota<0$ then $\bar{\mathcal{F}}[\iota p^{r}]t_{1}^{-\iota
p^{r}}\in\bar{\mathcal{N}}^{{\dagger}}[0](p)$.

b) If $\iota>0$ then $\bar{\mathcal{F}}[\iota
p^{r}]\in\left([\bar{\mathcal{L}}^{{\dagger}[v_{0}]},\bar{\mathcal{L}}]+C_{w+1}(\bar{\mathcal{L}})\right)_{W_{M}(k)}$.

###### Proof of Lemma.

Part a) follows just from definition of $\bar{\mathcal{N}}^{{\dagger}}(p)$.
Part b) provided by the inductive assumption about the RHS. ∎

Continue the proof of proposition.

Apply the operators $\mathcal{S}$ and $\mathcal{R}$ from Lemma 2.7 to the RHS
of (4.3).

If $\iota<0$ then the expression for $\bar{x}_{1}$ will be provided with

$\mathcal{S}\left(\sum_{r}\bar{\mathcal{F}}[\iota p^{r}]_{0}t_{1}^{-\iota
p^{r}}\right)=-\sum_{j\geqslant
0}\sigma^{j}\left(\sum_{\iota,r}\bar{\mathcal{F}}[\iota
p^{r}]_{0}t_{1}^{-\iota p^{r}}\right)=-\sum_{\iota,r}\bar{\mathcal{F}}[\iota
p^{r}]t^{-\iota p^{r}}\,.$

Obviously, the elements from
$\bar{\mathcal{N}}^{{\dagger}}[0](p)\subset\bar{\mathcal{L}}_{\mathrm{m}^{\prime}}$
will not contribute to the part
$\sum_{\iota\in\mathfrak{A}^{+}(p)}t^{-\iota}V_{\iota}$.

For $\iota>0$, $V_{\iota}t_{1}^{-\iota}$ is congruent modulo
$\left([\bar{\mathcal{L}}^{[v_{0}]},\bar{\mathcal{L}}]+C_{w+_{1}}(\bar{\mathcal{L}})\right)_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$
to

$\mathcal{R}\left(\sum_{r\leqslant r(\iota)}\bar{\mathcal{F}}[\iota
p^{r}]_{0}t_{1}^{-\iota p^{r}}\right)\equiv\sum_{r\leqslant
r(\iota)}\sigma^{-r}\bar{\mathcal{F}}[\iota
p^{r}]_{0}t_{1}^{-\iota}\equiv-\sigma^{-r(\iota)}\bar{\mathcal{F}}[\iota
p^{r(\iota)}]\,.$

This proves b).

In this case $\bar{x}_{1}$ will be contributed modulo
$\bar{\mathcal{L}}^{[v_{0}]}_{O^{\prime}_{M}(\mathcal{K}^{\prime})}$ with

$\mathcal{S}\left(\sum_{r}\bar{\mathcal{F}}[\iota p^{r}]_{0}t_{1}^{-\iota
p^{r}}\right)=\sum_{r}\sum_{0\leqslant r_{1}<r}t_{1}^{-\iota
p^{r_{1}}}\sigma^{-r+r_{1}}\bar{\mathcal{F}}[\iota p^{r}]_{0}$
$=\sum_{r}t_{1}^{-\iota p^{r}}\sum_{r<r_{1}\leqslant
r(\iota)}\sigma^{-r_{1}+r}\bar{\mathcal{F}}[\iota
p^{r_{1}}]_{0}=\sum_{r}t_{1}^{-\iota
p^{r}}\sum_{\begin{subarray}{c}r_{1}+r_{2}=r\\\
r_{2}<0\end{subarray}}\sigma^{r_{2}}\bar{\mathcal{F}}[\iota p^{r_{1}}]_{0}$
$\equiv-\sum_{r}t_{1}^{-\iota p^{r}}\sum_{\begin{subarray}{c}r_{1}+r_{2}=r\\\
r_{1},r_{2}\geqslant 0\end{subarray}}\sigma^{r_{2}}\bar{\mathcal{F}}[\iota
p^{r_{1}}]_{0}\equiv-\sum_{r}\bar{\mathcal{F}}[\iota p^{r}]t_{1}^{-\iota
p^{r}}$

because

$\sum_{r_{2}+r_{1}=r}\sigma^{r_{2}}\bar{\mathcal{F}}[\iota
p^{r_{1}}]_{0}\equiv\sigma^{r}\left(\sigma^{-r(\iota)}V_{\iota}\right)\,\mathrm{mod}\,\left([\bar{\mathcal{L}}^{[v_{0}]},\bar{\mathcal{L}}]+C_{w+1}(\bar{\mathcal{L}}))\right)_{W_{M}(k)}\,.$

∎

###  4.6. A combinatorial statement

Recall that the values $\eta(\bar{a},\bar{n})$ were defined in Section 4.3. If
$e_{(N^{*},0]}:=\sigma^{N^{*}-1}e\circ\sigma^{N^{*}-2}e\circ\dots\circ\sigma
e\circ e$ then

$\widetilde{\exp}\,e_{(N^{*},0]}=E_{N^{*}-1}\ldots
E_{1}E_{0}=\sum_{\bar{a},\bar{n}}\eta(\bar{a},\bar{n})t_{1}^{-qp^{M-1}\gamma(\bar{a},\bar{n})}D_{\bar{a}\bar{n}}\in\bar{\mathcal{L}}_{O^{\prime}_{M}(\mathcal{K}^{\prime})}\,.$

As earlier, for all $i\geqslant 0$, $E_{i}=\sigma^{i}(E_{0})$ and
$\gamma(\bar{a},\bar{n})=a_{1}p^{n_{1}}+\ldots+a_{s}p^{n_{s}}$.

Let $-e_{[0,N^{*})}:=(-e)\circ(-\sigma e)\dots(-\sigma^{N^{*}-1}e)$. Introduce
similarly the constants $\eta^{o}(\bar{a},\bar{n})\in W_{M}(k)$ by the
following identity

$\widetilde{\exp}(-e_{[0,N^{*})})=\sum\eta^{o}(\bar{a},\bar{n})D_{\bar{a}\bar{n}}\,.$

Set $\eta^{o}(\bar{a}):=\eta^{o}(\bar{a},\bar{0})$. It can be easily seen that
if there is a partition from the definition of $\eta$-constants in Sect, 4.3
but such that $m_{1}<m_{2}<\dots<m_{r}$, then

$\eta^{o}(\bar{a},\bar{n})=\sigma^{m_{1}}\left(\eta^{o}(\bar{a}^{(1)})\right)\sigma^{m_{2}}\left(\eta^{o}(\bar{a}^{(2)})\right)\dots\sigma^{m_{r}}\left(\eta^{o}(\bar{a}^{(r)})\right)\,.$

Otherwise, $\eta^{o}(\bar{a},\bar{n})=0$.

If we don’t care what vectors $\bar{a}$ and $\bar{n}$ are we can agree just to
write $\eta(1,\dots,s)$ instead of $\eta(\bar{a},\bar{n})$ and use similar
agreement for $\eta^{o}$. E.g. the equality
$e_{(N^{*},0]}\circ(-e_{[0,N^{*})})=0$ means that

(4.5) $\sum_{0\leqslant s_{1}\leqslant
s}\eta(1,\dots,s_{1})\eta^{o}(s_{1}+1,\dots,s)=\delta_{0s}\,.$

For $1\leqslant s_{1}\leqslant s<p$, consider the subset $\Phi_{ss_{1}}$ of
permutations $\pi$ of order $s$ such that $\pi(1)=s_{1}$ and for any
$1\leqslant l\leqslant s$, the subset $\\{\pi(1),\dots,\pi(l)\\}\subset[1,s]$
is “connected”, i.e. for some $n(l)\in\mathbb{N}$,

$\\{\pi(1),\dots,\pi(l)\\}=\\{n(l),n(l)+1,\dots,n(l)+l-1\\}.$

By definition, $\Phi_{s0}=\Phi_{s,s+1}=\emptyset$.

Set
$B_{s_{1}}(1,\dots,s)=\sum_{\pi\in\Phi_{ss_{1}}}\eta(\pi(1),\dots,\pi(s))$.
Then, cf. [8], Sect.4,

######  Lemma 4.6.

Suppose $0\leqslant s_{1}\leqslant s<p$. Then

a)
$B_{s_{1}}(1,\dots,s)+B_{s_{1}+1}(1,\dots,s)=\eta(s_{1},\dots,1)\eta(s_{1}+1,\dots,s)$;

b) $\eta^{o}(1,\dots,s)=(-1)^{s}\eta(s,s-1,\dots,1)$;

c) for indeterminates $X_{1},\dots,X_{s}$,

$\sum_{\begin{subarray}{c}1\leqslant s_{1}\leqslant s\\\
\pi\in\Phi_{ss_{1}}\end{subarray}}(-1)^{s_{1}-1}X_{\pi^{-1}(1)}\dots
X_{\pi^{-1}(s)}=[\dots[X_{1},X_{2}],\dots,X_{s}].$

###  4.7. Lie elements $\bar{\mathcal{F}}[\iota]$ and
$\bar{\mathcal{F}}[\iota]_{0}$

Agree to write $\bar{n}=(n_{1},\dots,n_{s})\geqslant M$ if all $n_{i}\geqslant
M$, and treat similarly $\bar{n}>M$, $\bar{n}\leqslant M$ and $\bar{n}<M$.

For $1\leqslant s_{1}\leqslant s$, let
$\gamma^{*}_{[s_{1},s]}(\bar{a},\bar{n})=\sum\limits_{s_{1}\leqslant
u\leqslant s}a_{u}p^{n^{*}_{u}}$ where $n_{u}^{*}=0$ if
$n_{u}=M(\bar{n}):=\max\\{n_{1},\dots,n_{s}\\}$ and $n_{u}^{*}=-\infty$ (i.e.
$p^{n_{u}^{*}}=0$), otherwise.

Suppose $\iota\in\mathfrak{A}^{0}_{1}$ is given as in (4.1). Then

$\bar{\mathcal{F}}[\iota]=\sum_{\bar{a},\bar{n},A}\sum_{1\leqslant
s_{1}\leqslant
s}p^{A}\eta^{o}(1,\dots,s_{1}-1)\eta(s_{1},\dots,s)\gamma_{[s_{1},s]}^{*}(\bar{a},\bar{n})D_{\bar{a}\bar{n}}$

Here the first sum is taken over all $(\bar{a},\bar{n})$ of lengths
$1\leqslant s<p$ such that $\bar{n}\geqslant 0$,
$\gamma(\bar{a},\bar{n})=p^{M(\iota)}\gamma(\iota)$ and
$M(\bar{n})=M(\iota)+A$.

The following proposition relies elements $\bar{\mathcal{F}}[\iota]$ and the
elements $\bar{\mathcal{F}}_{\gamma,-N}$ from Section 4.3.

######  Proposition 4.7.

If $\iota$ is given in notation (4.1) then

$\bar{\mathcal{F}}[\iota
p^{n}]=\sigma^{M(\iota)+n}\bar{\mathcal{F}}^{0}_{\gamma(\iota),-(M(\iota)+n)}\,.$

###### Proof.

We have

$\sigma^{-(M(\iota)+A+n)}\bar{\mathcal{F}}[\iota
p^{n}]=\sum_{\begin{subarray}{c}1\leqslant s_{1}\leqslant s<p\\\
(\bar{a},\bar{n})\end{subarray}}p^{A}\eta^{o}(1,\dots,s_{1}-1)\eta(s_{1},\dots,s)\gamma_{[s_{1},s]}^{*}(\bar{a},\bar{n})D_{\bar{a},\bar{n}}\,,$

where the sum is taken for $(\bar{a},\bar{n})$ with $M(\bar{n})=A$,
$\bar{n}\geqslant-(M(\iota)+n)$ and
$\gamma(\bar{a},\bar{n})=\gamma(\iota)=p^{A}\alpha$. By Lemma 4.2,

$\eta^{o}(1,\dots,s_{1}-1)=(-1)^{s_{1}-1}\eta(s_{1}-1,\dots,1)\,$

and we obtain

$\sum_{\begin{subarray}{c}1\leqslant s_{1}\leqslant s<p\\\
(\bar{a},\bar{n})\end{subarray}}(-1)^{s_{1}-1}(B_{s_{1}-1}(1,\dots,s)+B_{s_{1}}(1,\dots,s))p^{A}\gamma^{*}_{[s_{1},s]}(\bar{a},\bar{n})D_{(\bar{a}\bar{n})}=$

$\sum_{\begin{subarray}{c}1\leqslant s_{1}\leqslant s<p\\\
(\bar{a},\bar{n})\end{subarray}}(-1)^{s_{1}-1}B_{s_{1}}(1,\dots,s)p^{A}(\gamma^{*}_{[s_{1},s]}(\bar{a},\bar{n})-\gamma^{*}_{[s_{1}+1,s]}(\bar{a},\bar{n})){D}_{(\bar{a},\bar{n})}=$

$\sum_{\begin{subarray}{c}1\leqslant s<p\\\
(\bar{a},\bar{n})\end{subarray}}\sum_{1\leqslant s_{1}\leqslant
s}(-1)^{s_{1}-1}B_{s_{1}}(1,\dots,s)p^{A}a_{s_{1}}p^{n_{s_{1}}^{*}}{D}_{(\bar{a}\bar{n})}=$

$\sum_{\begin{subarray}{c}1\leqslant s<p\\\
(\bar{a},\bar{n})\end{subarray}}\sum_{\begin{subarray}{c}1\leqslant
s_{1}\leqslant s\\\
\pi\in\Phi_{ss_{1}}\end{subarray}}(-1)^{s_{1}-1}\eta({\pi(1)},\dots,{\pi(s)})p^{A}a_{s_{1}}p^{n_{s_{1}}^{*}}D_{(\bar{a},\bar{n})}=$

$\sum_{\begin{subarray}{c}1\leqslant s<p\\\
(\bar{a},\bar{n})\end{subarray}}\eta(1,\dots,s)a_{1}p^{A}\sum_{\begin{subarray}{c}1\leqslant
s_{1}\leqslant s\\\
\pi\in\Phi_{ss_{1}}\end{subarray}}(-1)^{s_{1}-1}D_{a_{\pi^{-1}(1)}n_{\pi^{-1}(1)}}\dots
D_{a_{\pi^{-1}(s)}n_{\pi^{-1}(s)}}=$ $\sum_{\begin{subarray}{c}1\leqslant
s<p\\\
(\bar{a},\bar{n})\end{subarray}}\eta(1,\dots,s)a_{1}p^{n_{1}}[\dots[D_{a_{1}n_{1}},D_{a_{2}n_{2}}],\dots,D_{a_{s}n_{s}}]=\bar{\mathcal{F}}^{0}_{\gamma(iota),-(M(\iota)+n)}\,.$

because $M(\bar{n})=n_{1}$. The proposition is proved. ∎

######  Corollary 4.8.

All $\bar{\mathcal{F}}[\iota p^{m}]$ and $\bar{\mathcal{F}}[\iota p^{m}]_{0}$
belong to $\bar{\mathcal{L}}_{W_{M}(k)}$.

This finishes the proof of Theorem 4.1.

###  4.8. Description of $\bar{\mathcal{L}}^{[v_{0}]}$ in [3]

Now we use Theorem 4.1 to recover the description of
$\bar{\mathcal{L}}^{[v_{0}]}$ obtained in [3].

######  Theorem 4.9.

Let $\widetilde{N}=N^{*}-1$. Then the ideal $\bar{\mathcal{L}}^{[v_{0}]}$ is
the minimal ideal in $\bar{\mathcal{L}}$ such that
$\bar{\mathcal{L}}_{W_{M}(k)}^{[v_{0}]}$ contains all
$\bar{\mathcal{F}}^{0}_{\alpha,-\widetilde{N}}$ with $\gamma\geqslant v_{0}$.

###### Proof.

By Theorem 4.1, $\bar{\mathcal{L}}^{[v_{0}]}$ is the minimal ideal in
$\bar{\mathcal{L}}$ such that $\bar{\mathcal{L}}^{[v_{0}]}_{W_{M}(k)}$
contains all $\bar{\mathcal{F}}[\iota p^{r_{\iota}}]$, where
$\iota\in\mathfrak{A}_{1}^{+}(p)$ and $r_{\iota}\geqslant 0$ is such that
$\iota p^{r_{\iota}+1}\notin\mathfrak{A}^{0}$.

By (4.1),

$\iota=p^{m}(qp^{M-1}\alpha-p^{M-1-A}b^{*})=p^{\widetilde{M}(\iota)+M-1}(q\gamma(\iota)-b^{*})\in\mathfrak{A}_{1}^{+}(p)\,,$

where $\gamma(\iota)=p^{A}\alpha$, $\alpha\in A[p-1,m]$ and
$\widetilde{M}(\iota)=m-A$. Note that $\iota\in\mathfrak{A}^{+}_{1}(p)$ means
also that $\iota\leqslant p^{M-1}(p-1)b^{*}$ and we have the following two
cases (cf. Prop. 4.10 below): either
$v_{p}(\gamma(\iota)+\widetilde{M}\iota)=0$ or $\widetilde{M}(\iota)+M-1=0$
(what is equivalent to $A=M-1$ amd $m=0$).

Then by Prop. 4.7, $\bar{\mathcal{F}}[\iota p^{r_{\iota}}]$ can be replaced by
$\bar{\mathcal{F}}^{0}_{\gamma(\iota),-\widetilde{N}}$ because
$\widetilde{N}=N^{*}-1=\widetilde{M}(\iota)+r_{\iota}$ for some
$r_{\iota}\geqslant r(\iota)$, where as earlier

$r(\iota)=\max\\{r\geqslant 0\ |\ \iota p^{r(\iota)}\in\mathfrak{A}^{0}\\}\,.$

As a result, it remains to verify the following proposition. ∎

######  Proposition 4.10.

Suppose $\gamma\geqslant v_{0}$ is such that for any
$\iota\in\mathfrak{A}^{+}_{1}(p)$, $\gamma\neq\gamma(\iota)$ and
$v_{p}(\gamma)\geqslant-\widetilde{N}$. Then
$\bar{\mathcal{F}}^{0}_{\gamma,-\widetilde{N}}\in\bar{\mathcal{L}}^{[v_{0}]}_{W_{M}(k)}$.

###### Proof.

As it was mentioned above, we have the two following cases:

1) $p^{M-1-v_{p}(\gamma)}(q\gamma-b^{*})>p^{M-1}(p-1)b^{*}$;

2) $v_{p}(\gamma)\geqslant 0$ and $q\gamma-b^{*}>p^{M-1}(p-1)b^{*}$.

The expression for $\bar{\mathcal{F}}^{0}_{\gamma,-\widetilde{N}}$ is a linear
combination of the terms

$p^{n_{1}}[\dots[D_{a_{1}n_{1}},D_{a_{2}n_{2}}],\dots,D_{a_{r}n_{r}}]\,,$

where $M>n_{1}\geqslant n_{2}\geqslant\ldots\geqslant
n_{r}\geqslant-\widetilde{N}$ and
$p^{n_{1}}a_{1}+\dots+p^{n_{r}}a_{r}=\gamma$.

We can assume that all $a_{i}<(p-1)v_{0}$, because otherwise
${\mathcal{F}}^{0}_{\gamma,-\widetilde{N}}\in\mathcal{L}(p)_{W_{M}(k)}$ and
$\bar{\mathcal{F}}^{0}_{\gamma,-\widetilde{N}}=0$.

Consider the first case.

From definitions it follows that
$\bar{\mathcal{F}}^{0}_{\gamma,-\widetilde{N}}$ is a linear combination of the
terms of the form

$[\dots[\bar{\mathcal{F}}^{0}_{\gamma^{\prime},-(\widetilde{N}-1)},D_{a_{1}^{\prime},-\widetilde{N}}],\dots,D_{a_{r}^{\prime},-\widetilde{N}}]$

where
$\gamma=\gamma^{\prime}+(a_{1}^{\prime}+\dots+a_{r}^{\prime})/p^{\widetilde{N}}$,
$r\geqslant 0$, all $a^{\prime}_{i}\in\mathbb{Z}^{0}(p)$ and
$\gamma^{\prime}\in A[p-1,\widetilde{N}-1]$.

If $a_{1}^{\prime}+\dots+a_{r}^{\prime}\geqslant(p-1)b^{*}/q$, i.e. it is
$\geqslant(p-1)v_{0}$, then the whole term belongs to
$\bar{\mathcal{L}}(p)_{W_{M}(k)}=0$.

Otherwise, the inequality

$b^{*}(p-1)/p^{\widetilde{N}}<q\gamma-b^{*}=q\gamma^{\prime}-b^{*}+q(a_{1}^{\prime}+\dots+a^{\prime}_{r})/p^{\widetilde{N}}$

implies that $q\gamma^{\prime}-b^{*}>0$ and, therefore,
$\gamma^{\prime}\geqslant v_{0}$. If $\gamma^{\prime}=\gamma(\iota^{\prime})$
for a suitable $\iota^{\prime}\in\mathfrak{A}^{+}_{1}(p)$ then
$\bar{\mathcal{F}}^{0}_{\gamma^{\prime},-(\widetilde{N}-1)}\in\bar{\mathcal{L}}^{[v_{0}]}$
and we are done. Otherwise, we should repeat the procedure with
$\gamma^{\prime}$ instead of $\gamma$.

As a result we either prove our proposition or will come to the second case.

Here we proceed similarly using the above calculations with $\widetilde{N}=0$:

$\gamma=\gamma^{\prime}+(a_{1}^{\prime}+\ldots+a^{\prime}_{r})$, $r\geqslant
0$ and $v_{p}(\gamma^{\prime})\geqslant 1$. Then

$q\gamma-b^{*}=q\gamma^{\prime}-b^{*}+q(a_{1}^{\prime}+\ldots+a^{\prime}_{r})>p^{M-1}(p-1)b^{*}\,.$

Again, if $a_{1}^{\prime}+\ldots+a^{\prime}_{r}\geqslant(p-1)v_{0}$ we are in
$\bar{\mathcal{L}}(p)_{W_{M}(k)}$. Otherwise, $q\gamma^{\prime}-b^{*}>0$
implies $\gamma^{\prime}\geqslant v_{0}$ and we have alternative:
$\gamma^{\prime}=\gamma(\iota^{\prime})$ with
$\iota^{\prime}\in\mathfrak{A}_{1}^{+}(p)$ or repeat the procedure. ∎

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