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Ramification filtration and differential forms V. A. Abrashkinab a Department of Mathematical Sciences, University of Durham, United Kingdom b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
Англоязычная версия:
Izvestiya: Mathematics, 2023, Volume 87, Issue 3, Pages DOI: https://doi.org/10.4213/im9322e 
Реферативные базы данных:
     IntroductionLet $L$ be a complete discrete valuation field with finite residue field of characteristic $p$. Let $\Gamma_L$ be the absolute Galois group of $L$. Let $\{\Gamma_L^{(v)}\}_{v\geqslant 0}$ be the filtration of $\Gamma_L$ by the ramification subgroups, [1]. This filtration provides $\Gamma_L$ with additional structure and allows us to introduce various classes of infinite field extensions (arithmetically profinite, deeply ramified etc.), which play an important role in modern arithmetic algebraic geometry. For $\Gamma_L$-modules $H$, the evaluation of $v_0(H)\in\mathbb{Q} $ such that $\Gamma_L^{(v)}$ act trivially on $H$ for $v>v_0(H)$, provides us with good estimates for discriminants of the fields of definition of $h\in H$. Such estimates are used very often to answer various number theoretic questions. However, an explicit description of the structure of the ramification filtration for a very long time was known only at the level of the Galois groups of abelian field extensions. At the time when the structure of the Galois group $\Gamma_L$ was completely described (the case of the maximal $p$-extensions – Shafarevich [2], Demushkin [3], and the general case – Janssen–Wingberg [4]) it became clear that $\Gamma_L$ is a very weak invariant of the field $L$. The situation cardinally changed later when it was established (Mochizuki [5], author [6]) that taking $\{\Gamma_L^{(v)}\}_{v\geqslant 0}$ into account gives us absolute invariant of the field $L$. However, in order to work with this invariant we still need to know an explicit description of the filtration by the groups $\Gamma_L^{(v)}$. I. R. Shafarevich always paid attention to this problem, for example, cf. Introduction to [7]. His motivation was the following: for every prime number $p$ there is only one such filtration and we know almost nothing about its structure. In 1990’s the author developed a nilpotent version of the Artin–Schreier theory and obtained an explicit description of the ramification filtration modulo the subgroup of $p$th commutators of $\Gamma_L$. Such description was obtained, first, in the characteristic $p$ case, [8]–[10], and then developed in the mixed characteristic case, [11]–[13]. These results play a crucial role in this paper, where we study the images of the ramification subgroups $\Gamma_L^{(v)}$ in the group of automorphisms $\operatorname{Aut}_{\mathbb{Z}_p}H$ of finite $\mathbb{Z}_p[\Gamma_L]$-modules $H$. Our main result states that this arithmetic structure can be completely described (under some additional condition) in purely geometric properties of Fontaine’s etale $\phi $-modules $M(H)$. More precisely, if $\operatorname{char}L=p$ we construct differential forms $\widetilde{\Omega} [N]$, $N\in\mathbb{N} $, on an extension of scalars of $M(H)$, and specify the way how the image of the ramification filtration in $\operatorname{Aut}_{\mathbb{Z}_p}H$ can be recovered from these forms. Note that the definition of $\widetilde{\Omega} [N]$ depends only on a natural connection constructed on $M(H)$. If $\operatorname{char}L=0$ we assume that $L$ contains a $p$th root of unity $\zeta_1\ne 1$ and restrict ourselves to the case of Galois $\mathbb{F}_p$-modules. Then we apply the field-of-norms functor to reduce the situation to the characteristic $p$ case and use a characterization of “good” lifts of automorphisms of our cyclic field extension of $L$ from [11], [12]. This characterization uses again the differential forms $\widetilde{\Omega} [N]$ and a power series coming from the $p$-adic period of the formal group $\mathbb{G}_m$. By our opinion this result establishes quite interesting link between the Galois theory of local fields and very popular area of $D$-modules, lifts of Frobenius, Higgs vector bundles etc. § 1. Statement of the main result1.1. General notationEverywhere in the paper $p$ is a fixed prime number. If $E_0$ is a field then $E_0^{\mathrm{sep}}$ is its separable closure in some algebraic closure $E_0^{\mathrm{alg}}$ of $E_0$. If $E$ is a field such that $E_0\subset E\subset E_0^{\mathrm{sep}}$ set $\Gamma_{E}=\operatorname{Gal}(E_0^{\mathrm{sep}}/E)$. The field $E_0^{\mathrm{sep}}$ will be considered as a left $\Gamma_{E_0}$-module, i. e., for any $\tau_1,\tau_2\in\Gamma_{E_0}$ and $o\in E_0^{\mathrm{sep}}$, $(\tau_1\tau_2)o=\tau_1(\tau_2o)$. If $\operatorname{char}E_0=p$ we set for any $a\in E_0^{\mathrm{sep}}$, $\sigma (a)=a^p$. If $V$ is a module over a ring $R$ then $\operatorname{End}_RV$ is the $R$-algebra of $R$-linear endomorphisms of $V$. We always consider $V$ as a left $\operatorname{End}_RV$-module, i. e., if $l_1,l_2\in\operatorname{End}_RV$ and $v\in V$ then $(l_1l_2)v=l_1(l_2(v))$. We also consider $\operatorname{End}_RV$ as a Lie $R$-algebra with the Lie bracket $[l_1,l_2]=l_1l_2-l_2l_1$. If $S$ is an $R$-module then we often denote by $V_S$ the extension of scalars $V\otimes_RS$. 1.2. Functorial system of lifts to characteristic $0$Suppose $K_0=k_0((t_0))$ is the field of formal Laurent series in a (fixed) variable $t_0$ with coefficients in a finite field $k_0$ of characteristic $p$. The uniformiser $t_0$ provides a $p$-basis for any field extension $ E$ of $ K_0$ in $ K_0^{\mathrm{sep}}$, i. e., the set $\{1,t_0,\dots,t_0^{p-1}\}$ is a $E^p$-basis of $ E$. We use this $p$-basis to construct a compatible system of lifts $O(E)$ of the fields $E$ to characteristic $0$. This is a special case of the construction of lifts from [14]; it can be explained as follows. For all $M\in\mathbb{N} $, set $O_{M}(E)=W_{M}(\sigma^{M-1} E)[\overline t_0]$, where $\overline t_0=[t_0]$ is the Teichmuller representative of $t_0$ in the ring of Witt vectors $W_{M}(E)$. The rings $O_M(E)$ are the lifts of $E$ modulo $p^M$, i. e., they are flat $\mathbb{Z} /p^M$-algebras such that $O_M(E)\otimes_{\mathbb{Z} /p^M}\mathbb{Z} /p= E$. Note that the system
$$
\begin{equation*}
\{O_M(E)\mid M\in\mathbb{N},\, K_0\subset E\subset K_0^{\mathrm{sep}}\}
\end{equation*}
\notag
$$
is functorial in $M$ and $ E$. In particular, if $E/K_0$ is Galois then there is a natural action of $\operatorname{Gal}(E/ K_0)$ on $O_M(E)$ and $O_M(E)^{\operatorname{Gal}(E/K_0)}=O_M(K_0)$. The morphisms $W_M(\sigma)$ induce $\sigma $-linear morphisms on $O_M(E)$ which will be denoted again by $\sigma $. In particular, $\sigma (\overline t_0)=\overline t_0^{\,p}$. Introduce the lifts of the above fields $E$ to characteristic $0$ by setting $O(E)=\varprojlim_M O_M(E)$. Then $O_M(E)=O(E)/p^M$ and $O(E)[1/p]$ is a complete discrete valuation field with uniformiser $p$ and the residue field $E$. Clearly, we have the induced morphism $\sigma $ on each $O(E)$. Also, if $ E/ K_0$ is Galois then $\operatorname{Gal}(E/ K_0)$ acts on $O(E)$ and $O(E)^{\operatorname{Gal}(E/ K_0)}=O(K_0)$. Notice that $O(K_0)=\varprojlim_M W_M(k_0)((\overline t_0))$ is the completion of the ring of formal Laurent series $W(k)((\overline t_0))$. Set $O_{\mathrm{sep}}=O(K_0^{\mathrm{sep}})$. The system of lifts $O(E)$, $ E\subset K_0^{\mathrm{sep}}$, can be extended to the system of lifts of all extensions of $ K_0$ in $K_0^{\mathrm{alg}}$. Indeed, note that $K_0^{\mathrm{alg}}=\bigcup_{n\geqslant 0}K_0(t_n)^{\mathrm{sep}}$, where $t_n^{p^n}=t_0$. Then $t_n$ gives the $p$-basis $\{1,t_n,\dots,t_n^{p-1}\}$ for all separable extensions $E$ of $K_0(t_n)$ in $K_0^{\mathrm{alg}}$ and we obtain (as earlier) the corresponding lifts $O(E)$. The system of lifts $O(E)$ is functorial in $E\subset K_0^{\mathrm{alg}}$ (use that any separable extension $ E_n$ of $ K_0(t_n)$ appears uniquely as the composite $ E_0 K_0(t_n)$, where $ E_0/ K_0$ is separable). In particular, consider $K_0^{\mathrm{rad}}=\bigcup_{N\in\mathbb{N}} K_0^{\mathrm{ur}}(t_0^{1/N})$. Then for the above defined lift of $ K_0^{\mathrm{rad}}$ we have $O(K_0^{\mathrm{rad}})=\bigcup_{N\in\mathbb{N}}O(K_0^{\mathrm{ur}})[\overline t_0^{1/N}]$, where $K_0^{\mathrm{ur}}=\overline k_0((t_0))$ is the maximal unramified extension of $ K_0$. 1.3. Equivalence of the categories of $p$-groups and Lie algebras [15]Let $L$ be a finitely generated Lie $\mathbb{Z}_p$-algebra of nilpotent class $<p$, i. e., the ideal of $p$th commutators $C_p(L)$ of $L$ is equal to zero. Let $A$ be an enveloping algebra of $L$. Then the elements of $L\subset A$ generate the augmentation ideal $J$ of $A$. There is a morphism of $\mathbb{Z}_p$-algebras $\Delta\colon A\to A\otimes A$ uniquely determined by the condition $\Delta (l)=l\otimes 1+1\otimes l$ for all $l\in L$. Then the set ${\exp}(L)\ \operatorname{mod}J^p$ is identified with the group of all “diagonal elements modulo degree $p$” consisting of $a\in 1+J\ \operatorname{mod}J^p$ such that $\Delta (a) \equiv a\otimes a\ \operatorname{mod}(J\otimes 1+1\otimes J)^p$. In particular, there is a natural embedding $L\subset A/J^p$ and the identity
$$
\begin{equation*}
\exp(l_1)\cdot \exp(l_2)\equiv \exp(l_1 \circ l_2)\ \operatorname{mod}J^p
\end{equation*}
\notag
$$
induces the Campbell–Hausdorff composition law
$$
\begin{equation*}
(l_1,l_2)\mapsto l_1 \circ l_2= l_1+l_2+\frac{1}{2}[l_1,l_2]+\cdots,\qquad l_1,l_2\in L.
\end{equation*}
\notag
$$
This composition law provides the set $L$ with a group structure. We denote this group by $G(L)$. Clearly $G(L)\simeq {\exp}(L)\ \operatorname{mod}J^p$. With the above notation the functor $L\mapsto G(L)$ determines the equivalence of the categories of finitely generated $\mathbb{Z}_p$-Lie algebras and profinite $p$-groups of nilpotence class $<p$. Note that a subset $I\subset L$ is an ideal in $L$ iff $G(I)$ is a normal subgroup in $G(L)$. 1.4. Lie-conditionFor any finite field extension $ K$ of $ K_0$ in $ K_0^{\mathrm{sep}}$, let $\operatorname{M\Gamma}_{K}$ be the category of finitely generated $\mathbb{Z}_p$-modules $H$ with continuous left action of $\Gamma_{K}$. Each element $h\in H$ is defined over some finite extension $K(h)$ of $K$. In some sense the family of these fields determines “arithmetic” properties of $H$. More detailed information about the fields $K(h)$ can be obtained from the knowledge of the images of the ramification subgroups in upper numbering $\Gamma_{K}^{(v)}$, $v>0$, in $\operatorname{Aut}_{\mathbb{Z}_p} H$. For example, the minimal number $v_0(H)\in\mathbb{Q} $ such that all $\Gamma_{K}^{(v)}$ with $v>v_0(H)$ act trivially on $H$ provides us with upper estimates for the discriminants of the fields of definition of $h\in H$ (cf. [6]). Let $ H_0\in\operatorname{M\Gamma}_{K_0}$ and let $\pi_{H_0}\colon\Gamma_{K_0}\to \operatorname{Aut}_{\mathbb{Z}_p} H_0$ be the group homomorphism which determines the $\Gamma_{K_0}$-module structure on $ H_0$. Consider the full subcategory $\operatorname{M\Gamma}_{K_0}^{\mathrm{Lie}}$ in $\operatorname{M\Gamma}_{K_0}$ which consists of modules $ H_0$ satisfying the following condition. Condition (Lie). The image $I(H_0):=\pi_{H_0}(\mathcal I)\subset \operatorname{Aut}_{\mathbb{Z}_p}(H_0)$ of the wild inertia subgroup $\mathcal I\subset\Gamma_{K_0}$ appears in the form $\exp(L(H_0))$, where $L(H_0)\subset\operatorname{End}_{\mathbb{Z}_p}H_0$ is a Lie subalgebra such that $L(H_0)^p=0$. The condition $L(H_0)^p=\{l_1\cdots l_p\mid l_1,\dots,l_p\in L(H_0)\}=0$ (the product is taken in $\operatorname{End}_{\mathbb{Z}_p} H_0\supset L(H_0)$) implies that $L(H_0)$ is a finitely generated nilpotent $\mathbb{Z}_p$-algebra Lie of nilpotence class $<p$. This gives the group isomorphism $\exp\colon G(L(H_0)) \simeq I(H_0)$. Note that any normal subgroup of $I(H_0)$ appears in the form $\exp G(J)$, where $J$ is a Lie ideal of $L(H_0)$. Remark 1.1. If $p H_0=0$ and $\operatorname{dim}_{\mathbb{F}_p} H_0\leqslant p$ then $ H_0$ is of the Lie type. 1.5. The first main result: the characteristic $p$ caseSuppose $ H_0\,{\in}\operatorname{M\Gamma}_{K_0}^{\mathrm{Lie}}$. Our target is to determine for all $v> 0$, the images $\pi_{H_0}(\Gamma_{K_0}^{(v)})$ of the ramification subgroups $\Gamma_{K_0}^{(v)}$ via an explicit construction of the ideals $L(H_0)^{(v)}\subset L(H_0)$ such that $\exp (L(H_0)^{(v)})= \pi_{H_0}(\Gamma_{K_0}^{(v)})$. Our approach uses Fontaine’s “analytical” description of the Galois modules $H_0\in\operatorname{M\Gamma}_{K_0}^{\mathrm{Lie}}$ in terms of etale $(\phi,O(K_0))$-modules $M(H_0)$. A geometric nature of $M(H_0)$ is supported by the existence of an analogue of the classical connection $\nabla\colon M(H_0)\to M(H_0)\otimes_{O(K_0)}\Omega^1_{O(K_0)}$ (cf. [16]). (This map is uniquely characterized by the condition $\nabla\phi =(\phi \otimes\phi)\cdot \nabla $.) The required information about the behaviour of ramification subgroups can be then extracted from some differential forms
$$
\begin{equation*}
\widetilde{\Omega}[N]\in M(H_0)_{O(K_0^{\mathrm{rad}})} \otimes_{O(K_0)} \Omega^1_{O(K_0)}.
\end{equation*}
\notag
$$
The construction of these differential forms is given completely in terms of the above connection $\nabla $ and can be explained as follows. Let $ K\subset K_0^{\mathrm{sep}}$ be a fixed tamely ramified finite extension of $K_0$ such that $\pi_{H_0}(\Gamma_{K})=I(H_0)$. Then $H:=H_0|_{\Gamma_{K}}$ can be described via an etale $(\phi,O(K))$-module $M(H)=M(H_0)\otimes_{O(K_0)}O(K)$. Recall that $M(H)=(H\otimes_{\mathbb{Z}_p}O_{\mathrm{sep}})^{\Gamma_{K}}$ is a finitely generated $O(K)$-module with a $\sigma $-linear morphism $\phi\colon M(H)\to M(H)$ such that the image $\phi(M(H))$ generates $M(H)$ over $O(K)$. This allows us to identify the elements of $H$ with a set of $O_{\mathrm{sep}}$-solutions of a suitable system of equations with coefficients in $O(K)$. We establish below the construction of $M(H)$ by introducing a $\mathbb{Z}_p$-linear embedding $\mathcal F\colon H\to M(H)$ which induces by extension of scalars the identification $H_{O(K)}\simeq M(H)$. (We denote this identification by the same symbol $\mathcal F$.) Now let $\widetilde{B}$ be (a inique) $O(K)$-linear operator on $M(H)$ such that for any $m\in \mathcal F(H)$, $\nabla (m)=\widetilde{B}(m)d{\overline t}/\overline t$. Then for every $N\in\mathbb{Z}_{\geqslant 0}$ we introduce the differential forms
$$
\begin{equation*}
\widetilde{\Omega}[N]=\phi^N\widetilde{B}\phi^{-N}\, \frac{d\overline t}{\overline t}\in \operatorname{End}M(H)_{O(K^{\mathrm{rad}})}\otimes_{O(K)}\Omega^1_{O(K)}.
\end{equation*}
\notag
$$
Now we can use the identification $\mathcal F\colon H_{O(K)}\simeq M(H)$ to obtain the corresponding differential forms $\Omega [N]$ on $\operatorname{End}(H)_{O(K^{\mathrm{rad}})}$ and to verify that
$$
\begin{equation*}
\Omega [N]\in L(H)_{O(K^{\mathrm{rad}})} \otimes_{O(K)}\Omega^1_{O(K)}= L(H_0)_{O(K_0^{\mathrm{rad}})} \otimes_{O(K_0)}\Omega^1_{O(K_0)}.
\end{equation*}
\notag
$$
Remark 1.2. Our differential forms will usually appear in the form $\Omega =F\cdot d\overline t_0/\overline t_0$, where $F\in L(H)_{K_0^{\mathrm{rad}}}$. Then we set by definition Our first main result can be stated as follows. Theorem 1.1. Suppose $H_0\in\operatorname{M\Gamma}_{K_0}^{\mathrm{Lie}}$ is finite. Then there is $N_0(H_0)\in\mathbb{N}$ such that for any (fixed) $N\geqslant N_0(H_0)$ the following property holds: if $(\operatorname{id}_{L(H)}\otimes\sigma^{-N})\Omega [N]= \sum_{r\in\mathbb{Q}}\overline t_0^{\,-r}l_{r}d\overline t_0/\overline t_0$, where all $l_r\in L(H_0)_{W(\overline{k}_0)}$, then the ideal $L(H_0)^{(v)}$ is the minimal ideal in $L(H_0)$ such that for all $r\geqslant v$, $l_r\in L(H_0)^{(v)}_{W(\overline{k}_0)}$. Corollary 1.1. If $v_0(H_0)=\max\{r\mid l_r\ne 0\}$ then the ramification subgroups $\Gamma_{K_0}^{(v)}$ act trivially on $H_0$ iff $v>v_0(H_0)$. Remark 1.3. The construction of $\Omega [N]$ almost does not depend on the choice of the tamely ramified finite field extension $ K$ of $ K_0$. It depends essentially on the choice of the uniformising element $t_0$ in $ K_0$ and a compatible system of $\alpha (k)\in W(k)$, where $[k:k_0]<\infty $, such that the trace of $\alpha (k)$ in the field extension $W(k)[1/p]/{K}_0$ equals $1$. Remark 1.4. If $H_0$ is not $p$-torsion Theorem 1.1 can be applied to the factors $H_0/p^M$ and our result describes the structure of the images of $L(H_0)^{(v)}$ in all $L(H_0)/p^M$. 1.6. The second main result: the mixed characteristic caseLet $E_0$ be a finite field extension of $\mathbb{Q}_p$ with residue field $k_0$ and a uniformizing element $\pi_0$. Assume that $E_0$ contains a $p$th primitive root of unity $\zeta_1$. Consider the category $\operatorname{M\Gamma}_{E_0,1}^{\mathrm{Lie}}$ of finitely generated $\mathbb{F}_p[\Gamma_{E_0}]$-modules which satisfy a direct analog of the Lie condition from Section 1.4. Take the infinite arithmetically profinite field extension $\widetilde{E}_0$ obtained from $E_0$ by joining all $p$-power roots of $\pi_0$. Then the theory of the field-of-norms functor $X$ provides us with the complete discrete valuation field of characteristic $p$, $X(\widetilde{E}_0)\,{=}\,K_0$, which has the same residue field and the uniformizing element $t_0$ obtained from the $p$-power roots of $\pi_0$. The functor $X$ also provides us with the identification of Galois groups $\Gamma_{K_0}=\Gamma_{\widetilde{E}_0}\subset \Gamma_{E_0}$. If $H_{E_0}\in\operatorname{M\Gamma}^{\mathrm{Lie}}_{E_0,1}$ then we obtain $H_0:=H_{E_0}|_{\Gamma_{K_0}} \in\operatorname{M\Gamma}_{K_0}^{\mathrm{Lie}}$. As earlier, take a finite tamely ramified extension $K$ of $K_0$ (it corresponds to a unique tamely ramified extension $E$ of $E_0$ with uniformizer $\pi $ such that $\pi^{e_0}=\pi_0$, where $e_0$ is the ramification index of $E/E_0$, and construct $(\phi,O(K))$-module $M(H)$. This module inherits the action of $\operatorname{Gal}(E(\sqrt[p]{\pi})/E)= \langle\tau_0\rangle^{\mathbb{Z} /p}$. (Here $\tau_0$ is such that $\tau_0(\sqrt[p]{\pi})= \zeta_1\sqrt[p]{\pi}$.) This situation was considered in all details in the papers [11], [12]. In particular, in those papers we gave a characterization of the “good” lifts $\widehat\tau_0$ of $\tau_0$. By definition, $\widehat\tau_0\in\Gamma_{E}$ is “good” if its restriction to $H_{E_0}$ belongs to the image of the ramification subgroup $\Gamma_{E}^{(e^*)}$. Here $e^*:=pe/(p-1)$ and $e=e(E/\mathbb{Q}_p)$ is the ramification index of $E/\mathbb{Q}_p$. (This makes sense because $\tau_0\in \operatorname{Gal}(E(\sqrt[p]{\pi}/E)^{(e^*)})$.) Note that the field-of-norms functor is compatible with ramification filtrations on $\Gamma_{E_0}$ and $\Gamma_{K_0}$. Therefore, the knowledge of “good lifts” $\widehat\tau_0$ together with Theorem 1.1 gives a complete description of the image of the ramification filtration of $\Gamma_{E_0}$ in $\operatorname{Aut}H_{E_0}$. In [12] we proved that the action of $\widehat\tau_0$ appears from an action of a formal group scheme of order $p$. As a result, the lift $\widehat\tau_0$ is completely determined by the value $d\widehat\tau_0(0)\in L(H)$ of its differential at $0$, and we can use the characterization of differentials of “good” lifts from [12; Theorem 5.1]. Namely, let us first specify our $p$th root of unity
$$
\begin{equation*}
\zeta_1=1+\sum_{j\geqslant 0}[\beta_j] \pi^{(e^*/p)+j}\ \operatorname{mod}p
\end{equation*}
\notag
$$
(here all $[\beta_j]$ are the Teichmuller representatives of elements from the residue field of $E$). Then we introduce the power series $\omega (t)\in O(K)$ such that
$$
\begin{equation*}
1+\sum_{j\geqslant 0}\beta_j^pt^{e^*+pj}= \widetilde{\exp}(\omega (t)^p)
\end{equation*}
\notag
$$
(here $\widetilde{\exp}$ is the truncated exponential). The series $\omega (t)^p$ is a kind of approximation of the $p$-adic period of the formal multiplicative group, which appears usually in explicit formulas for the Hilbert symbol (e.g. [17]). In other words, we obtain another geometric condition characterizing “arithmetic” of the $\Gamma_{E_0}$-module $H_{E_0}$. Remark 1.5. Notice that the power series $\omega (t)^p$ has non-zero coefficients only for the powers $t^{e^*+pj}$ and all these exponents $\geqslant e^*$. Therefore, the differential forms $\Omega [m]$ contribute to the right-hand side only via the images of $\mathcal F^0_{e^*+pj,-m}t^{-(e^*+pj)}$ in $L(H)_k$. But for $m\gg 0$, these images belong to the images of the ramification ideals $\mathcal L^{(e^*)}_k$ and, therefore, disappear modulo $L(H)_{k}^{(e^*)}$, and the sum in the right-hand side is, as a matter of fact, finite. § 2. $\phi$-module $M(H)$2.1. Specification of $\log \pi_{H}\colon \Gamma_{K}\to G(L(H))$As earlier, $H_0\in\operatorname{M\Gamma}_{K_0}^{\mathrm{Lie}}$, $K$ is a finite tamely ramified extension of $K_0$ in $K_0^{\mathrm{sep}}$ such that $\pi_{H_0}(\Gamma_{K})=I(H_0)$, $H=H_0|_{\Gamma_{K}}$. Set $L(H)=L(H_0)$, $\pi_{H}=\pi_{H_0}|_{\Gamma_{K}}$. Consider the continuous group epimorphism $l_{H}:=\log(\pi_{H})\colon \Gamma_{K}\to G(L(H))$. Since the $p$-group $G(L(H))$ has nilpotence class $<p$ this epimorphism can be described in terms of the covariant version of the nilpotent Artin–Schreier theory from [9]. Namely, there are $e\in L(H)_{O(K)}$ and $f\in L(H)_{O_{\mathrm{sep}}}$ such that $(\operatorname{id}_{L(H)}\otimes\,\sigma)(f)= e \circ f$ and for any $\tau\in\Gamma_{K}$, $l_{H}(\tau)=(-f) \circ (\operatorname{id}_{L(H)}\otimes\tau)f$. It could be easily verified that $l_{H}$ is a group homomorphism. Indeed,
$$
\begin{equation*}
\begin{aligned} \, l_{H}(\tau_1\tau_2) &= (-f) \circ (\operatorname{id}_{L(H)}\otimes \tau_1\tau_2)f \\ &=(-f) \circ (\operatorname{id}_{L(H)}\otimes\tau_1)f \circ (-f) \circ (\operatorname{id}_{L(H)}\otimes\tau_2)f= l_{H}(\tau_1)\circ l_{H}(\tau_2), \end{aligned}
\end{equation*}
\notag
$$
because $(\operatorname{id}_{L(H)}\otimes\, \tau_1)l_{H}(\tau_2)=l_{H}(\tau_2)$. Notation. We will use below the following notation: $\sigma_H=\operatorname{id}_H\otimes\,\sigma $ and $\sigma_{L(H)}=\operatorname{id}_{L(H)}\otimes\, \sigma $. For example, if $u=\sum_{\alpha}h_{\alpha}\otimes\, o_{\alpha}$, where all $h_{\alpha}\in H$ and $o_{\alpha}\in O(K)$ then $\sigma_H(u)=\sum_{\alpha}h_{\alpha}\otimes \sigma (o_{\alpha})$. Or, if $X$ is a linear operator on $L(H)_{O(K)}$ then $\sigma_{L(H)}X$ is also a linear operator such that In addition, $\mathcal X:=X\cdot\sigma_H$ is a unique sigma linear operator such that $\mathcal X|_H=X|_H$, and we have the following identity: $\sigma_H \cdot X=\sigma_{LH}(X)\cdot\sigma_H$. If there is no risk of confusion we will use just the notation $\sigma$. Remark 2.1. Originally we developed in [9] the contravariant version of the nilpotent Artin–Schreier theory, cf. the discussion in [18]. The contravariant version uses similar relations $\sigma_{L(H)}(f)= f \circ e$ and the map $l_{H}$ was defined via $\tau\mapsto (\operatorname{id}_{L(H)}\otimes\tau) f \circ (-f)$. In this case $l_{H}$ determines the group homomorphism from $\Gamma_{K}$ to the opposite group $G^0(L(H))$ (this group is isomorphic to $G(L(H))$ via the map $g\mapsto g^{-1}$). The results from the papers [8]–[10] were obtained in terms of the contravariant version, but the results from [11]–[13], [19] used the covariant version. We can easily switch from one theory to another via the automorphism $-\operatorname{id}_{L(H)}$. We consider $O_{\mathrm{sep}}$ as a left $\Gamma_{K}$-module via the action $o\mapsto \tau (o)$ with $o\in O_{\mathrm{sep}}$ and $\tau\in\Gamma_{K}$. This corresponds to our earlier agreement about the left action of the elements of $\Gamma_{K}$ as endomoprhisms of $O_{\mathrm{sep}}$. As a result we obtain the left $\Gamma_{K}$-module structure on $H_{O_{\mathrm{sep}}}$ by the use of the (left) action of $\Gamma_{K}$ on $H$ via $h\mapsto l_{H}(\tau)(h)$. Because $L(H)_{O_{\mathrm{sep}}}\subset \operatorname{End}_{O_{\mathrm{sep}}}(H_{O_{\mathrm{sep}}})$ we can introduce for any $h\in H$, $\mathcal F(h):=\exp (-f)(h)\in H_{O_{\mathrm{sep}}}$. Proof. Suppose $f=\sum_{\alpha} l_{\alpha}\otimes o_{\alpha}$, where all $l_{\alpha}\in L(H)$ and $o_{\alpha}\in O_{\mathrm{sep}}$.
If $\tau\in\Gamma_{K}$ then
$$
\begin{equation*}
\begin{aligned} \, \tau (\mathcal F(h)) &=(\tau\otimes\operatorname{id}_{O_{\mathrm{sep}}})\biggl(\exp \biggl( -\sum_{\alpha}l_{\alpha}\otimes \tau (o_{\alpha})\biggr)(h)\biggr) \\ &=(\tau\otimes\operatorname{id}_{O_{\mathrm{sep}}})\bigl(\exp \bigl( (\operatorname{id}_{L(H)}\otimes\tau)(-f)\bigr)(h)\bigr) \\ &=(\tau\otimes\operatorname{id}_{O_{\mathrm{sep}}})\bigl(\exp \bigl((-l_{H}(\tau)) \circ (-f)\bigr)(h)\bigr) \\ &=(\tau\otimes\operatorname{id}_{O_{\mathrm{sep}}})\bigl(\bigl(\exp (-l_{H}(\tau))\cdot \exp (-f)\bigr)h \bigr) \\ &=(\tau\otimes\operatorname{id}_{O_{\mathrm{sep}}})(\pi_{H}(\tau^{-1})\otimes \operatorname{id}_{O_{\mathrm{sep}}})\mathcal F(h) \\ &=\bigl(\tau \cdot \pi_H(\tau^{-1})\otimes \operatorname{id}_{O_{\mathrm{sep}}} \bigr) \mathcal F(h)=\mathcal F(h). \end{aligned}
\end{equation*}
\notag
$$
The proposition is proved. The elements $e$ and $f$ are not determined uniquely by $l_{H}$. A pair $e'\in L(H)_{O(K)}$ and $f'\in L(H)_{O_{\mathrm{sep}}}$ give the same group epimorphism $l_{H}$ iff there is $x\in L(H)_{O(K)}$ such that $e'=\sigma (x) \circ e \circ (-x)$ and $f'=x \circ f$. 2.2. Special choice of $e\in L(H)_{O(K)}$We can always assume (by replacing, if necessary, $ K$ by its finite unramified extension) that a uniformiser $t$ in $ K$ is such that $t^{e_0}=t_0$, where $e_0$ is the ramification index for $ K/ K_0$. Then $O(K)=\varprojlim_M W_M(k)((\overline t))$, where $\overline t^{\,e_0}=\overline t_0$ and $\overline t$ is the Teichmuller representative of $t$. We denote by $k$ the residue field of $ K$ and fix a choice of $\alpha_0=\alpha (k)\in W(k)$ such that its trace in the field extension $W(k)[1/p]/\mathbb{Q}_p$ equals $1$. Let $\mathbb{Z}^+(p):=\{a\in\mathbb{N} \mid \operatorname{gcd}(a,p)=1\}$ and $\mathbb{Z}^0(p)=\mathbb{Z}^+(p)\cup\{0\}$. Definition 2.1. An element $e\in L(H)_{O(K)}$ is special if $e=\sum_{a\in\mathbb{Z}^0(p)}\overline t^{\,-a}l_{a0}$, where $l_{00}\in\alpha_0L(H)$ and for all $a\in\mathbb{Z}^+(p)$, $l_{a0}\in L(H)_{W(k)}$. Lemma 2.1. Suppose $e\in L(H)_{O(K)}$. Then there is $x\in L(H)_{O(K)}$ such that $\sigma (x) \circ e \circ (-x)$ is special. Proof. Use induction on $s$ to prove lemma modulo the ideals of $s$th commutators $C_s(L(H)_{O(K)})$.
If $s=1$ there is nothing to prove. Suppose lemma is proved modulo $C_s(L(H)_{O(K)})$. Then there is $x\in L(H)_{O(K)}$ such that
$$
\begin{equation*}
\sigma (x) \circ e \circ (-x) = \sum_{a\in\mathbb{Z}^0(p)}\overline t^{\,-a}l_{a0}+l,
\end{equation*}
\notag
$$
where $l\in C_s(L(H)_{O(K)})$. Using that we obtain the existence of $x_s\in C_s(L(H)_{O(K)})$ such that where $l_0\in\alpha_0L$ and all remaining $l_a\in L_{W(k)}$. Then we can take $x'=x-x_s$ to obtain our statement modulo $C_{s+1}(L(H)_{O(K)})$. Lemma 2.2. Suppose $e\in L_{O(K)}$ is special and $x\in L_{O(K)}$. Then the element $\sigma (x) \circ e \circ (-x)$ is special iff $x\in L$ (or, equivalently, if $\sigma x=x$). Proof. Use relation (cf. [20]) where $\operatorname{Ad}(U)(V)=UVU^{-1}$ and $\operatorname{ad}(U)(V)=[U,V]$. Indeed, if $X=x\in L(H)$ and $Y=e$ then $\sum_{n\geqslant 0}\operatorname{ad}^n(x)(e)/n!$ is also special.
When proving the inverse statement we can use induction modulo the ideals $C_s(L(H))_{O(K)}$, $s\geqslant 1$, as follows. Assume the lemma is proved modulo $C_s(L(H)_{O(K)})$. Then using the if part we can assume that $x\in C_s(L(H)_{O(K)})$. Therefore, $e+\sigma (x)-x$ is special modulo $C_{s+1}(L(H))_{O(K)}$, i. e.,
$$
\begin{equation*}
\sigma (x)-x\in \alpha_0C_s(L)+\sum_{a\in\mathbb{Z}^+(p)} t^{-a}C_s(L)_{W(k)}
\end{equation*}
\notag
$$
modulo $C_{s+1}(L(H)_{O(K)})$. By (2.1) this implies the congruence
$$
\begin{equation*}
\sigma (x)\equiv x\ \operatorname{mod}C_{s+1}(L(H))_{O(K)},
\end{equation*}
\notag
$$
i. e., $x\in C_s(L(H))\ \operatorname{mod}C_{s+1}(L(H)_{O(K)})$.
The lemma is proved. 2.3. Construction of the $\phi $-module $M(H)$Note that $\pi_{H}=\exp (l_{H})$, and therefore for all $\tau\in\Gamma_{K}$, it holds
$$
\begin{equation*}
\pi_{H}(\tau)= \exp (-f)\cdot \exp (\operatorname{id}_{L(H)}\otimes \tau)f,
\end{equation*}
\notag
$$
where $f\,{\in}\, L(H)_{O_{\mathrm{sep}}}\,{\subset} \operatorname{End}_{O_{\mathrm{sep}}}(H_{O_{\mathrm{sep}}})$, $\sigma_{L(H)}(f)\,{=}\,e\circ f$ and $\exp f\,{\in}\operatorname{Aut}_{O_{\mathrm{sep}}}(H_{O_{\mathrm{sep}}})$. Let $\operatorname{MF}^{\mathrm{et}}_{K}$ be the category of etale $\phi $-modules over $O(K)$. Recall that its objects are $O(K)$-modules of finite type $M$ together with a $\sigma $-linear morphism $\phi\colon M\to M$ such that its $O(K)$-linear extension $\phi_{O(K)}\colon M\otimes_{O(K),\sigma}O(K)\to M$ is isomorphism. The correspondence $H\mapsto M(H):=(H\otimes_{\mathbb{Z}_p} O_{\mathrm{sep}})^{\Gamma_{K}}$, where $\phi\colon M(H)\to M(H)$ comes from the action of $\sigma $ on $O_{\mathrm{sep}}$, determines the equivalence of the categories $\operatorname{M\Gamma}_{K}$ and $\operatorname{MF}^{\mathrm{et}}_{K}$. Consider the $\mathbb{Z}_p$-linear embedding $\mathcal F\colon H\to H_{O_{\mathrm{sep}}}$ from Section 2.1. Let $M(H)=\mathcal F(H)_{O(K)}$. By extension of scalars we obtain natural isomorphisms (use that $O(K)$ and $O_{\mathrm{sep}}$ are flat $\mathbb{Z}_p$-modules):
$$
\begin{equation*}
\mathcal F\otimes\operatorname{id}_{O_{\mathrm{sep}}}\colon H_{O_{\mathrm{sep}}} \simeq M(H)_{O_{\mathrm{sep}}},\qquad \mathcal F\otimes \operatorname{id}_{O(K)}\colon H_{O(K)}\simeq M(H),
\end{equation*}
\notag
$$
which will be denoted for simplicity just by $\mathcal F$. Note that by Proposition 2.1, $M(H)= (H_{O_{\mathrm{sep}}})^{\Gamma_{K}}$. The $O(K)$-module $M(H)$ is provided with the $\sigma $-linear morphism $\phi \colon M(H)\to M(H)$ uniquely determined for all $h\in H$ via
$$
\begin{equation*}
\phi (\mathcal F(h))=\exp (-\sigma_{L(H)}f)(h) =(\exp (-f) \cdot \exp (-e))(h)=\mathcal F(\exp (-e)(h)).
\end{equation*}
\notag
$$
Consider the $O(K)$-linear operator
$$
\begin{equation*}
A:=\exp (-e)\in \exp (L(H)_{O(K)})\subset \operatorname{Aut}_{O(K)} H_{O(K)}.
\end{equation*}
\notag
$$
Then $\mathcal A:=A\cdot \sigma_H$ will be a unique $\sigma $-linear operator on $L(H)_{O(K)}$ such that $\mathcal A|_{H}= A|_{H}$. Clearly, for any $u\in H_{O(K)}$, and $M(H)$ is etale $\phi $-module associated with the $\mathbb{Z}_p[\Gamma_{K}]$-module $H$. For example, suppose $pH=0$ and $\{h_i\mid 1\leqslant i\leqslant N\}$ is $\mathbb{F}_p$-basis of $H$. Then $\{\mathcal F(h_i)\mid 1\leqslant i\leqslant N\}$ is a $ K$-basis for $M(H)$. If $A(h_i)=\exp (-e)(h_i)=\sum_ja_{ij}h_j$ with all $a_{ij}\in K$ then $\phi (\mathcal F(h_i))=\sum_ja_{ij}\mathcal F(h_j)$, and $((a_{ij}))$ appears as the corresponding “Frobenius matrix”. It can be easily seen also that if $\{h_i\mid 1\leqslant i\leqslant N\}$ is a minimal system of $\mathbb{Z}_p$-generators in $H$ then $\{\mathcal F(h_i)\mid 1\leqslant i\leqslant N\}$ is a minimal system of $O(K)$-generators in $M(H)$. 2.4. The connection $\nabla $ on $M(H)$The $(\phi, O(K))$-module $M(H)$ can be provided with a connection $\nabla \colon M(H)\to M(H)\otimes_{O(K)}\Omega^1_{O(K)}$, [16]. This is an additive map uniquely determined by the properties: a) for any $o\in O(K)$ and $m\in M(H)$, $\nabla (mo)=\nabla (m)o+m\otimes d(o)$; b) $\nabla\cdot \phi =(\phi\otimes\phi) \cdot \nabla$. By a), $\nabla $ is uniquely determined by its restriction to $\mathcal F(H)$ (use that $M(H)=\mathcal F(H)_{O(K)}$). Let $\widetilde{B}$ be a unique $O(K)$-linear operator on $M(H)$ such that for any $m\in\mathcal F(H)$, $\nabla (m)=\widetilde{B}(m)d\overline t/\overline t$. Consider the $O(K)$-linear operator $B\in\operatorname{End}H_{O(K)}$ such that for all $u\in H_{O(K)}$, $\widetilde{B}(\mathcal F(u))=\mathcal F(B(u))$. Obviously, $\widetilde B$ and $B$ can be recovered one from another. Note that for any $\mathbb{Z}_p$-module $\mathcal C$, the elements $c\in \mathcal C_{O(K)}$ appear uniquely in the form $c=\sum_n c_n\otimes \overline t^{\,n}$ with all $c_n\in \mathcal C_{W(k)}$. Therefore, the map $\operatorname{id}_{\mathcal C}\otimes\, \partial_{\overline t}\colon \mathcal C_{O(K)} \to \mathcal C_{O(K)}$ such that $c\mapsto \sum_n c_n\otimes n\overline t^{\,n}$ is well-defined. If $\mathcal C\subset \mathcal C_1$ is an embedding of $\mathbb{Z}_p$-modules then we have $(\operatorname{id}_{\mathcal C_1}\otimes\, \partial_{\overline t})|_{\mathcal C_{O(K)}}=\operatorname{id}_{\mathcal C}\otimes\, \partial_{\overline t}$. With the above notation: 1) for all $m\in M(H)$, $\nabla (m)=(\widetilde{B} +\operatorname{id}_{\mathcal F(H)}\otimes\, \partial_{\overline t}) (m)d\overline t/\overline t$; 2) for all $u\in H_{O(K)}$ and $X\in \operatorname{End}H_{O(K)}$,
$$
\begin{equation*}
(\operatorname{id}_H\otimes\, \partial_{\overline t})(X (u))= (\operatorname{id}_{\operatorname{End} H}\otimes\, \partial_{\overline t})(X)(u) + X \bigl((\operatorname{id}_{H}\otimes\, \partial_{\overline t})u\bigr).
\end{equation*}
\notag
$$
Proposition 2.2. Let $C=-(\operatorname{id}_{\operatorname{End}H}\otimes\, \partial_{\overline t})A A^{-1}$ and let for any $n\geqslant 1$, $D^{(n)}=A \cdot \sigma_{\operatorname{End}H}(A) \cdot \ldots \cdot \sigma_{\operatorname{End}H}^{n-1}(A)$. Then Remark 2.2. In Section 4 we will prove that $C,B\in L(H)_{O(K)}\subset\operatorname{End}H_{O(K)}$. In particular, $\sigma_{\operatorname{End}H}C{=}\,\sigma_{L(H)}C$ and the correspondence $u{\kern1pt}{\mapsto}{\kern1pt} (B{+}\operatorname{id}_{L(H)}\!{\otimes}\, \partial_{\overline t})(u)d\overline t/\overline t$ gives a connection on $L(H)_{O(K)}$. Proof of Proposition 2.2. Indeed, for any $u\in H_{O(K)}$, it holds
$$
\begin{equation*}
\begin{aligned} \, (\nabla \cdot \phi)(\mathcal F(u)) &=(\nabla \cdot \mathcal F\cdot \mathcal A)u=\bigl((\widetilde B+\operatorname{id}_{\mathcal F(H)}\otimes\, \partial_{\overline t}) \cdot \mathcal F \cdot \mathcal A\bigr)(u)\, \frac{d\overline t}{\overline t} \\ &=\bigl(\mathcal F\cdot (B+\operatorname{id}_H\otimes\, \partial_{\overline t})\cdot \mathcal A\bigr)(u)\, \frac{d\overline t}{\overline t}. \end{aligned}
\end{equation*}
\notag
$$
On the other hand,
$$
\begin{equation*}
\begin{aligned} \, (\phi\otimes\phi)(\nabla (\mathcal F(u))) &= \phi \bigl(\mathcal F(B+\operatorname{id}_H\otimes\, \partial_{\overline t})u\bigr)\phi \biggl(\frac{d\overline t}{\overline t}\biggr) \\ &= p\bigl(\mathcal F\cdot \mathcal A \cdot (B+\operatorname{id}_H\otimes\, \partial_{\overline t})\bigr)(u)\, \frac{d\overline t}{\overline t}. \end{aligned}
\end{equation*}
\notag
$$
Equivalently, we have the following identity on $H_{O(K)}$,
$$
\begin{equation*}
(B\cdot A+(\operatorname{id}_H\otimes\, \partial_{\overline t}) \cdot A) \cdot \sigma_H= pA \cdot \sigma_H \cdot (B+\operatorname{id}_H \otimes\, \partial_{\overline t}).
\end{equation*}
\notag
$$
Rewrite this equality as follows
$$
\begin{equation*}
(B \cdot A-pA \cdot \sigma_{\operatorname{End}H}(B)) \cdot \sigma_H= \bigl(-(\operatorname{id}_H\otimes\, \partial_{\overline t}) \cdot A+ A\cdot \sigma_H\cdot (\operatorname{id}_H\otimes\, p\partial_{\overline t}) \cdot \sigma_H^{-1}\bigr) \cdot \sigma_H.
\end{equation*}
\notag
$$
Notice that on $H_{O(K)}$ we have $\sigma_H \cdot (\operatorname{id}_H\otimes\, p\partial_{\overline t}) \cdot \sigma_H^{-1}= \operatorname{id}_H\otimes\, \partial_{\overline t}$. As a result, the right-hand side equals $-(\operatorname{id}_{\operatorname{End}H}\otimes\, \partial_{\overline t})(A)\cdot \sigma_H$. From $\sigma_H|_H=\operatorname{id}_H$ it follows that by restriction on $H$ we have
$$
\begin{equation*}
B \cdot A-pA \cdot \sigma_{\operatorname{End}H}(B) =-(\operatorname{id}_{\operatorname{End}H}\otimes\, \partial_{\overline t})A.
\end{equation*}
\notag
$$
By $O(K)$-linearity this identity holds on the whole $H_{O(K)}$. As a result, our identity appears in the form It remains to recover $B$ using that§ 3. Ramification filtration modulo $p$th commutatorsRecall that $K=k((t))\subset K_0^{\mathrm{tr}}$, $\pi_{H}(\Gamma_{K})=\exp (L(H))=I(H)\subset\operatorname{Aut}_{\mathbb{Z}_p}(H)$. Note also that $O(K)=W(k)((\overline t))$, where ${\overline t}^{\,e_0}=\overline t_0$ and $e_0$ is the ramification index of $K/K_0$. 3.1. Lie algebra $\mathcal L$ and identification $\eta_{<p}$Let $ K_{<p}$ be the maximal $p$-extension of $ K$ in $ K_0^{\mathrm{sep}}$ with the Galois group of nilpotent class $<p$. Then $\mathcal G_{<p}:=\operatorname{Gal}(K_{<p}/ K)= \varprojlim_M \Gamma_{K}/\Gamma_{K}^{p^M}C_p(\Gamma_{K})$. Let $\widetilde{\mathcal L}_{W(k)}$ be a profinite free Lie $W(k)$-algebra with the set of topological generators $\{D_{0}\}\cup\{D_{an}\mid a\in\mathbb{Z}^+(p), \, n\in\mathbb{Z}/N_0\}$. Let $\mathcal L_{W(k)}=\widetilde{\mathcal L}_{W(k)}/C_p(\widetilde{\mathcal L}_{W(k)})$, where $C_p(\widetilde{\mathcal L}_{W(k)})$ is the ideal of $p$th commutators. Define the $\sigma $-linear action on $ \mathcal L_{W(k)}$ via $D_{an}\mapsto D_{a,n+1}$ and $D_0\mapsto D_0$, denote this action by the same symbol $\sigma $, and set $ \mathcal L= \mathcal L_{W(k)}|_{\sigma =\operatorname{id}}$. Fix $\alpha_0=\alpha_0(k)\in W(k)$ such that the trace of $\alpha_0$ in the field extension $W(k)[1/p]/\mathbb{Q}_p$ equals $1$. For any $n\in\mathbb{Z} /N_0$, set $D_{0n}=(\sigma^n\alpha_0)D_0$. We are going to apply the profinite version of the covariant nilpotent Artin–Schreier theory to the Lie algebra $\mathcal L$ and the special element $e_{<p}{\kern1pt}{=}\sum_{a\in\mathbb{Z}^0(p)} \overline t^{\,-a}D_{a0}{\in} \mathcal L\mathbin{\widehat\otimes} O(K)$. In other words, if we fix
$$
\begin{equation*}
f_{<p}\in \{f\in \mathcal L\mathbin{\widehat\otimes} O_{\mathrm{sep}}\mid \sigma_{\mathcal L}(f)= e \circ f\}\ne\varnothing,
\end{equation*}
\notag
$$
then the map $\eta_{<p}:=\pi_{f_{<p}}(e_{<p})$ given by the correspondence $\tau\mapsto (-f_{<p}) \circ (\operatorname{id}_{\mathcal L}\otimes\tau)f_{<p}$ induces the group isomorphism $\overline\eta_{<p}\colon\Gamma_{<p}\simeq G(\mathcal L)$. The following property is an easy consequence of the above construction. Proposition 3.1. Suppose $e\in L(H)_{O(K)}$ is special and given with notation from the definition from Section 2.2. Then the map $\log\pi_{H}\colon \mathcal G_{<p}\to G(L(H))$ is given via the correspondences $D_{a0}\mapsto l_{a0}$ (and $D_{an}\mapsto \sigma_{L(H)}^n(l_{a0})$) for all $a\in\mathbb{Z}^0(p)$. 3.2. The ramification ideals $\mathcal L^{(v)}$For $v\geqslant 0$, denote by $\mathcal G^{(v)}_{<p}$ the image of $\Gamma_{K}^{(v)}$ in $\mathcal G_{<p}$. Then $\overline{\eta}_{<p}(\mathcal G^{(v)}_{<p})=G(\mathcal L^{(v)})$, where $\mathcal L^{(v)}$ is an ideal in $\mathcal L$. The images $\mathcal L^{(v)}(M)$ of the ideals $\mathcal L^{(v)}$ in the quotients $\mathcal L/p^M\mathcal L$ for all $M\in\mathbb{N}$ were explicitly described in [10]. By going to the projective limit on $M$ this description can be presented as follows. Definition 3.1. Let $\overline n=(n_1,\dots,n_s)$ with $s\geqslant 1$. Suppose there is a partition $0=i_0<i_1<\dots <i_r=s$ such that for $i_j<u\leqslant i_{j+1}$, it holds $n_u=m_{j+1}$ and $m_1>m_2>\dots >m_r$. Then set If such a partition does not exist we set $\eta (\overline n)=0$. For $s\in\mathbb{N} $, $\overline a=(a_1,\dots,a_s)\in\mathbb{Z}^0(p)^s$ and $\overline n=(n_1,\dots,n_s)\in\mathbb{Z}^s$, set
$$
\begin{equation*}
[D_{\overline a,\overline n}]=[\dots, [D_{a_1n_1},D_{a_2n_2}],\dots,D_{a_sn_s}].
\end{equation*}
\notag
$$
For $\alpha\geqslant 0$ and $N\in\mathbb{Z}_{\geqslant 0} $, introduce $\mathcal F^0_{\alpha,-N}\in{L}_{W(k)}$ such that
$$
\begin{equation*}
\mathcal F^0_{\alpha,-N}=\sum_{\substack {1\leqslant s<p\\ \gamma (\overline a, \overline n)=\alpha}} a_1\eta (\overline n)p^{n_1}[D_{\overline a,\overline n}].
\end{equation*}
\notag
$$
Here $n_1\geqslant 0$, all $n_i\geqslant -N$ and $\gamma (\overline a,\overline n)= a_1p^{n_1}+a_2p^{n_2}+\dots +a_sp^{n_s}$. Note that the non-zero terms in the above expression for $\mathcal F^0_{\alpha, -N}$ can appear only if $n_1\geqslant n_2\geqslant\dots \geqslant n_s$ and $\alpha $ has at least one presentation in the form $\gamma (\overline a,\overline n)$. Denote by $\mathcal I^{(v)}[N]$ the minimal closed ideal in $\mathcal L$ such that its extension of scalars $\mathcal I^{(v)}[N]_{W(k)}$ contain all $\mathcal F^0_{\alpha,-N}$ with $\alpha\geqslant v$. Our result from [10] about explicit generators of the ideal ${\mathcal L}^{(v)}$ can be stated in the following form. Theorem 3.1. For any $v>0$ and $M\in\mathbb{N} $, there is $\widetilde{N}(v,M)\in\mathbb{N} $ such that if $N\geqslant \widetilde{N}(v,M)$, then the images of the ideals $\mathcal L^{(v)}$ and $\mathcal I^{(v)}[N]$ in $\mathcal L/p^M$ coincide. 3.3. Some relationsLet $A(\mathcal L)$ be the enveloping algebra of $ \mathcal L$ and $\widetilde{A}(\mathcal L)= A(\mathcal L)/J(\mathcal L)^p$, where $J(\mathcal L)$ is the augmentation ideal in $A(\mathcal L)$. Note that there is a natural embedding of $\mathbb{Z}_p$-modules $\mathcal L\subset \widetilde{A}(\mathcal L)$. Let $A_{<p}=\exp (-e_{<p})\in \widetilde{A}(\mathcal L)_{O(K)}$ and $C_{<p}=- (\operatorname{id}_{\widetilde{A}(\mathcal L)}\otimes\, \partial_{\overline t})A_{<p} \cdot A_{<p}^{-1}$. For $s\geqslant 1$, set $\overline 0_s={\displaystyle(\,\underbrace{0,\dots,0}_{s\text{ times}}\,)}$. Proposition 3.2. Let $D^{(m)}_{<p}:=A_{<p} \cdot \sigma_{\widetilde{A}(\mathcal L)}(A_{<p})\cdot \ldots \cdot \sigma_{\widetilde{A}(\mathcal L)}^{m-1}(A_{<p})$, where $m\,{\geqslant}\, 1$. Then we have the following relations: Proof. For (3.1) use (cf. [20; Theorem 4.22]), to obtain
$$
\begin{equation*}
d\exp (-e_{<p}) \cdot \exp (e_{<p})= \sum_{k\geqslant 1}\frac{1}{k!} (-\operatorname{ad} e_{<p})^{k-1}(-de_{<p})
\end{equation*}
\notag
$$
and note that
$$
\begin{equation*}
\begin{aligned} \, (-\operatorname{ad}e_{<p})^{k-1}(de_{<p}) &= (-1)^{k-1}\bigl[\underbrace{e_{<p},\dots, [e_{<p}}_{k-1 \text{ times}},de_{<p}],\dots\bigr] \\ &=\bigl[\dots, [de_{<p},\underbrace{e_{<p}],\dots, e_{<p}}_{k-1\text{ times}}\bigr]. \end{aligned}
\end{equation*}
\notag
$$
For (3.2) we need the following relation (cf. [20; Section 4.4])
$$
\begin{equation*}
\exp (X) \cdot Y \cdot \exp (-X)= \sum_{n\geqslant 0}\frac{1}{n!}\operatorname{ad}^n(X)(Y).
\end{equation*}
\notag
$$
After applying this relation to the summand with $m=1$ we obtain
$$
\begin{equation*}
\begin{aligned} \, &p\operatorname{Ad}D_{<p}^{(1)}(\sigma_{\mathcal L}C_{<p}) = \exp (-e_{<p})\cdot \sigma_{\mathcal L}(C_{<p}) \cdot \exp (e_{<p}) \\ &\qquad=\sum_{s\geqslant 0}\eta (\overline 0_s)(-1)^s\operatorname{ad}^s(e_{<p})(C_{<p}) = \sum_{\overline n\geqslant 0, \overline a}\,\sum_{n_1=0}^{n_1=1} \eta (\overline n)p^{n_1}[D_{\overline a,\overline n}] \overline t^{\,-\gamma (\overline a, \overline n)}. \end{aligned}
\end{equation*}
\notag
$$
Repeating this procedure we obtain relation (3.2).
Similar calculations prove the remaining item (3.3). Proposition 3.2 is proved. § 4. Proof of Theorem 1.1Recall briefly what we’ve already achieved. The field $K=k((t))$ is tamely ramified extension of $K_0=k_0((t_0))$, where $t^{e_0}\,{=}\, t_0$, $H:=H_0|_{\Gamma_{K}}$ and the corresponding group epimorphism $\pi_H\colon \Gamma_K\to I(H)\subset \operatorname{Aut}_{\mathbb{Z}_p}H$ is such that $I(H)=\exp (L(H))$, where $L(H)\subset\operatorname{End}_{\mathbb{Z}_p}H$ and $L(H)^p=0$. Applying formalism of nilpotent Artin–Schreier theory we obtained a special $e=\sum_{a\in\mathbb{Z}^0(p)}\overline t^{\,-a}l_{a0}\in L(H)_{O(K)}$ and $f\in L(H)_{O_{\mathrm{sep}}}$ such that and for any $\tau \in \Gamma_{K}$, $\pi_H(\tau) =\exp (-f) \cdot \exp(\operatorname{id}_{L(H)}\otimes \tau)f$.We used $O(K)$-linear operator $\mathcal F\,{=}\exp (-f)$ to introduce $O(K)$-module $M(H):=\mathcal F(H_{O(K)})$. Let $A=\exp (-e)$ and let ${\mathcal A}$ be a unique $\sigma $-linear operator on $H_{O(K)}$ such that for any $h\in H$, $\mathcal A (h)=A(h)$. Then the $\sigma $-linear $\phi \colon M(H)\to M(H)$ is such that for any $u\in L(H)_{O(K)}$, $\phi (\mathcal F(u))=\mathcal F(\mathcal A(u))$. As a result, we obtain the structure of etale $(\phi, O(K))$-module on $M(H)$ related to the $\Gamma_K$-module $H$. Let $\nabla $ be the connection on $M(H)$ from Section 2.4, and let $\widetilde{B}$ be the $O(K)$-linear operator on $M(H)$ uniquely determined by the condition: for any $m\in\mathcal F(H)$, $\nabla (m)=\widetilde{B}(m)d\overline t/\overline t$. Then for any $u\in M(H)$,
$$
\begin{equation*}
\nabla (u)=(\widetilde{B}+\operatorname{id}_{\mathcal F(H)}\otimes\, \partial_{\overline t})(u)\, \frac{d\overline t}{\overline t}
\end{equation*}
\notag
$$
and we introduce the differential forms
$$
\begin{equation*}
\widetilde{\Omega}[N]= \phi^N\widetilde{B}\phi^{-N}\, \frac{d\overline t}{\overline t} \in \operatorname{End} M(H)_{O(K^{\mathrm{rad}})}\otimes\Omega^1_{O(K)}.
\end{equation*}
\notag
$$
Finally, define the $O(K)$-linear operator $B$ on $H$ by setting for any $u\in H_{O(K)}$, $\mathcal F(B(u))=\widetilde{B}(\mathcal F(u))$, and transfer $\widetilde{\Omega}[N]$ to $\operatorname{End}H_{O(K^{\mathrm{rad}})}$ in the following form
$$
\begin{equation*}
\Omega [N]= \operatorname{Ad}({\mathcal A}^N)(B)\, \frac{d\overline t}{\overline t} \in \operatorname{End}(H)_{O(K^{\mathrm{rad}})}\otimes _{O(K)}\Omega^1_{O(K)}.
\end{equation*}
\notag
$$
Recall that $\mathcal A=A\cdot \sigma_H$, where $A=\exp(-e)$. Lemma 4.1. If $\mathcal D^{(N)}=\sigma_{\operatorname{End}H}^{-N}(A)\cdot \ldots \cdot \sigma_{\operatorname{End}H}^{-1}(A)$, then Proof. Use induction on $N\geqslant 0$. If $N=0$ there is nothing to prove. Suppose lemma is proved for $N\geqslant 0$. Then
$$
\begin{equation*}
\begin{aligned} \, &\bigl(\sigma_{\operatorname{End}H}^{-(N+1)}\cdot \operatorname{Ad}(\mathcal A^{N+1})\bigr)(B)= \sigma_{\operatorname{End}H}^{-(N+1)}\bigl(\mathcal A \cdot \operatorname{Ad}(\mathcal A^N) (B) \cdot \mathcal A^{-1}\bigr) \\ &\qquad =\sigma_{\operatorname{End}H}^{-(N+1)}\bigl(\mathcal A \cdot (\sigma_{\operatorname{End}H}^N \cdot \operatorname{Ad}(\mathcal D^{(N)})(B))\cdot \mathcal A^{-1}\bigr) \\ &\qquad=\sigma_{\operatorname{End}H}^{-(N+1)}\bigl(A \cdot \sigma_H \cdot \bigl(\sigma_{\operatorname{End}H}^{N}\operatorname{Ad}(\mathcal D^{(N)})(B)\bigr)\cdot \sigma_H^{-1} \cdot A^{-1}\bigr) \\ &\qquad=\sigma_{\operatorname{End}H}^{-(N+1)}(A \cdot \bigl(\sigma_{\operatorname{End}H}^{N+1}\operatorname{Ad}(\mathcal D^{(N)})(B)\bigr)\cdot A^{-1}) \\ &\qquad=\sigma_{\operatorname{End}H}^{-(N+1)}(A) \cdot \operatorname{Ad}(\mathcal D^{(N)})(B) \cdot \sigma_{\operatorname{End}H}^{-(N+1)}(A^{-1})= \operatorname{Ad}(\mathcal D^{(N+1)})(B). \end{aligned}
\end{equation*}
\notag
$$
The lemma is proved. Under the projection $\log\overline{\pi}_H\colon \mathcal G_{<p}\to G(L(H))$ we have:
$$
\begin{equation*}
\begin{gathered} \, D_{an}\mapsto l_{an}=\sigma_{L(H)}^nl_{a0},\quad e_{<p}\mapsto e,\quad f_{<p}\mapsto f,\quad A_{<p}\mapsto A,\quad C_{<p}\mapsto C, \\ B_{<p}\mapsto B,\quad\text{and}\quad \sigma^{-m}_{\widetilde{\mathcal A}(\mathcal L)}D^{(m)}_{<p}\mapsto \mathcal D^{(m)}. \end{gathered}
\end{equation*}
\notag
$$
Remark 4.2. Because $\log \overline\pi_H(\mathcal L_{<p})=L(H)$ we obtain the statement from Remark 2.2. As a result, our differential form appears as the image of $\sum \mathcal F^0_{\alpha,-N}\overline t^{\,-\alpha}\, d\overline t/\overline t$. It remains to notice that when getting back to the field $K_0$, we have $d\overline t/\overline t=e_0^{-1}\, d\overline t_0/\overline t_0$, $\overline t^{\,-\alpha}=\overline t^{\,-\alpha /e_0}$, $\Gamma_K^{(\alpha)}=\Gamma_{K_0}^{(\alpha /e_0)}$ and $\pi_H|_{\mathcal I}=\pi_{H_0}|_{\mathcal I}$. Theorem 1.1 is proved. Remark 4.3. a) The conjugacy class of the differential form $\Omega [N]$ does not depend on a choice of a special form for $e$. b) It would be very interesting to verify whether our results could be established in the case of $\Gamma_{K}$-modules which do not satisfy the $\operatorname{Lie}$ condition, e. g., for the $\Gamma_{K}$-module from [21] (the case $n=p$ in the notation of that paper). § 5. Mixed characteristicLet $ E_0$ be a complete discrete valuation field of characteristic $0$ with finite residue field $k_0$ of characteristic $p$. Let $\overline E_0$ be an algebraic closure of $E_0$ and for any field $E$ such that $E_0\subset E\subset \overline E_0$, set $\operatorname{Gal}(\overline E_0/E)=\Gamma_E$. Suppose that $E_0$ contains a primitive $p$th root of unity $\zeta_1$. We are going to develop an analog of the above characteristic $p$ theory in the context of finite $\mathbb{F}_p[\Gamma_{E_0}]$-modules $H_{E_0}$ satisfying an analogue of the Lie condition from Section 2.1: if $\pi_{H_{E_0}}\colon \Gamma_{E_0}\to\operatorname{Aut}_{\mathbb{F}_p}(H_{E_0})$ determines a $\Gamma_{E_0}$-action on $H_{E_0}$ then there is a Lie $\mathbb{F}_p$-subalgebra $L(H_{E_0})\subset \operatorname{End}_{\mathbb{F}_p}(H_{E_0})$ such that $L(H_{E_0})^p=0$ and $\exp (L(H_{E_0}))=\pi_{H_{E_0}}(I)$, where $I$ is the wild ramification subgroup in $\Gamma_{E_0}$. Remark 5.1. Contrary to the characteristic $p$ case we restrict ourselves to the Galois modules killed by $p$ because the theory from [11], [12] is developed recently only under that assumption. Fix a choice of a uniformising element $\pi_0$ in $E_0$. Let $\widetilde{E}_0=E_0(\{\pi^{(n)}_0)\mid n\in\mathbb{Z}_{\geqslant 0}\}) \subset \overline E_0$, where $\pi^{(0)}_0=\pi_0$ and for all $n\in\mathbb{N} $, $\pi^{(n)p}_0=\pi^{(n-1)}_0$. The field-of-norms functor $X$ provides us with: – a complete discrete valuation field $X(\widetilde{E}_0)=K_0$ of characteristic $p$ with residue field $k_0$ and a fixed uniformizer $t_0=\varprojlim \pi^{(0)}_n$; – an identification of $\Gamma_{K_0}= \operatorname{Gal} (K_0^{\mathrm{sep}}/K_0)$ with ${\Gamma}_{\widetilde{E}_0} \subset\Gamma_{E_0}$. Let $E$ be a finite tamely ramified extension of $E_0$ in $\overline E_0$ such that $\pi_{H_{E_0}} (\Gamma_E)=I(H_0)$. By replacing $E$ with a suitable finite unramified extension we can assume that $E$ has uniformiser $\pi $ such that $\pi^{e_0}=\pi_0$. Let $k$ be the residue field of $E$. It is easy to see that the field $\widetilde{E}:=E\widetilde{E}_0$ appears in the form $E(\{\pi^{(n)}\mid n\geqslant 0\})$, where $\pi^{(0)}=\pi $, $\pi^{(n)p}=\pi^{(n-1)}$ and for all $n$, $\pi^{(n)e_0}=\pi^{(n)}_0$. In particular, $K:=X(\widetilde{E})=k((t))$, where $t=\varprojlim \pi^{(n)}$ is uniformiser such that $t^{e_0}=t_0$. Let $\mathcal G_{<p}=\Gamma_K/\Gamma_K^pC_p(\Gamma_K)$ and $\Gamma_{<p}=\Gamma_E/\Gamma_E^pC_p(\Gamma_E)$. According to [11], [12] we have the following natural exact sequence:
$$
\begin{equation*}
\mathcal G_{<p}\to \Gamma_{<p}\to \langle \tau_0 \rangle^{\mathbb{Z} /p}\to 1,
\end{equation*}
\notag
$$
where $\tau_0\in\operatorname{Gal}(E(\pi^{(1)})/E)$ is such that $\tau_0(\pi^{(1)})=\zeta_1\pi^{(1)}$. We can use the identification $\overline\eta_{<p}\colon \mathcal G_{<p}\simeq G(\mathcal L)$ from Section 3.1 obtained via the special element $e_{<p}$ and the corresponding $f_{<p}$ such that $\sigma_{L(H)}(f_{<p})=e_{<p}\circ f_{<p}$. Then we can use the equivalence of categories from Section S1.3 to identify $\Gamma_{<p}$ with $G(L)$ where $L$ is a profinite Lie $\mathbb{F}_p$-algebra included into the following exact sequence: When studying the structure of (5.1) in [12] we proved that $\tau_0$ can be replaced by a suitable $h_0\in\operatorname{Aut}K$. More precisely, suppose
$$
\begin{equation*}
\zeta_1\equiv 1+\sum_{j\geqslant 0}[\beta_j]\pi_0^{(e_0^*/p)+j}\ \operatorname{mod}p
\end{equation*}
\notag
$$
with Teichmüller representatives $[\beta_j]$ of $\beta_j\in k$ and $e^*=ep/(p-1)$, where $e$ is the ramification index for $E/\mathbb{Q}_p$. Then $h_{0}$ can be defined as follows: $h_{0}|_k=\operatorname{id}_k$ and
$$
\begin{equation*}
h_{0}(t)=t\biggl(1+\sum_{j\geqslant 0}\beta_j^pt^{e^*+pj}\biggr)= t\,\widetilde{\mathrm{exp}}(\omega (t)^p),
\end{equation*}
\notag
$$
where $\widetilde{\exp}$ is the truncated exponential and $\omega (t)\in t^{e^*/p}k[[t]]^*$. This allowed us to apply formalism of the nilpotent Artin–Schreier theory to specify “good” lifts $\tau_{<p}$ of $\tau_0$ to $L$ (what is equivalent to specifying “good lifts” $h_{<p}$ of $h_0$). In particular, we obtained in [11], [12] the following description of the image $\overline L\subset L$ of $\mathcal L$ from exact sequence (5.1). Introduce the weight function $\operatorname{wt}$ on $\mathcal L_{k}$ by setting $\operatorname{wt}(D_{an})=s\in\mathbb{N} $ iff $(s-1)e^*\leqslant a<se^*$. Then
$$
\begin{equation*}
\operatorname{Ker}(\mathcal L\to L)=\mathcal L(p)=\{l\in \mathcal L\mid \operatorname{wt}(l)\geqslant p\},
\end{equation*}
\notag
$$
$\overline L=\mathcal L/\mathcal L(p)$ and $\mathcal L\to \overline L$ is the natural projection. If $h_{<p}$ is a lift of $h_0$ to $K_{<p}$ then it is uniquely determined by $c=c(h_{<p})\in L_{K}$ such that
$$
\begin{equation*}
(\operatorname{id}_{\mathcal L_{<p}}\otimes h_{<p})f=c\circ \bigl(\operatorname{Ad}(h_{<p})\otimes\operatorname{id}_{K_{<p}}\bigr)f.
\end{equation*}
\notag
$$
This allowed us to describe the corresponding action of the group $\langle h_{<p}\rangle^{\mathbb{Z} /p}$ on $f$ as an action of an infinitesimal group scheme of order $p$. The differential of this action is given by the “linear part” $c_1\in \overline L_{K}$ of $c$ which could be described by a suitable recurrent procedure. Finally, we proved that $c_1(0)\in \overline L_{k}$ (where $c_1=\sum_{n\in\mathbb{Z}}c_1(n)t^n$ with all $c_1(n)\in\overline L_{k}$) is an absolute invariant of the lift $h_{<p}$. Consider the expansion $\omega (t)^p=\sum_{j\geqslant 0}A_jt^{e^*+pj}$, $A_j\in k$. Denote by $\overline L^{(e^*)}\subset \overline L$ the image of the ramification subgroup $\mathcal L^{(e^*)}$ in $L$ (cf. (5.1)). Then [12; Theorem 5.1] states: $\bullet$ the lift $\tau_{<p}$ of $\tau_0$ is “good” iff the value $c_1(0)\in \mathcal L_{k}$ of the differential $d\tau_{<p}$ at $0$ satisfies the following congruence:
$$
\begin{equation*}
c_1(0)\equiv \sum_{j\geqslant 0} \sum_{i\geqslant 0}\sigma^i(A_j \mathcal F^0_{e^*+pj,-i}) \ \operatorname{mod}\overline{L}_k^{(e^*)}.
\end{equation*}
\notag
$$
It remains to note that for $i,j\gg 0$, $\mathcal F^0_{e^*+pj,-i}\in \overline L_k^{(e^*)}$ and the right-hand double sum contains only finitely many nonzero terms modulo $\overline L_{k}^{(e^*)}$. It can be rewritten also in the following form:
$$
\begin{equation*}
\sum_{i,j\geqslant 0}\operatorname{Res}\bigl( \sigma^i(A_jt^{e^*+pj} \cdot \sigma^{-i}\Omega_{<p}[i])\bigr),
\end{equation*}
\notag
$$
and the image of this expression in $L(H)_k$ equals
$$
\begin{equation*}
\sum_{i\geqslant 0} \operatorname{Res}\bigl(\sigma^{i+1}\omega (t) \cdot \Omega [i]\bigr).
\end{equation*}
\notag
$$
The author expresses his deep gratitude to Prof. E. Khukhro for helpful discussions.
Цитирование в формате AMSBIB
\RBibitem{Abr23} |
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