Ramification filtration and differential forms
V. A. Abrashkin^{ab} ^{a} *Department of Mathematical Sciences, University of Durham, United Kingdom*
^{b} *Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia*
**Abstract:**
Let

$L$ be a complete discrete valuation field of prime characteristic

$p$ with finite residue field. Denote by

$\Gamma_{L}^{(v)}$ the ramification subgroups of $\Gamma_{L}=\operatorname{Gal}(L^{\mathrm{sep}}/L)$. We consider the category

$\operatorname{M\Gamma}_{L}^{\mathrm{Lie}}$ of finite

$\mathbb{Z}_p[\Gamma_{L}]$-modules

$H$, satisfying some additional (Lie)-condition on the image of

$\Gamma_L$ in

$\operatorname{Aut}_{\mathbb{Z}_p}H$. In the paper it is proved that all information about the images of the groups

$\Gamma_L^{(v)}$ in

$\operatorname{Aut}_{\mathbb{Z}_p}H$ can be explicitly extracted from some differential forms

$\widetilde{\Omega} [N]$ on the Fontaine etale

$\phi $-module

$M(H)$ associated with

$H$. The forms

$\widetilde{\Omega}[N]$ are completely determined by a canonical connection

$\nabla $ on

$M(H)$. In the case of fields

$L$ of mixed characteristic, which contain a primitive

$p$th root of unity, we show that a similar problem for

$\mathbb{F}_p[\Gamma_L]$-modules also admits a solution. In this case we use the field-of-norms functor to construct the corresponding

$\phi $-module together with the action of the Galois group of a cyclic extension

$L_1$ of

$L$ of degree

$p$. Then our solution involves the characteristic

$p$ part (provided by the field-of-norms functor) and the condition for a “good” lift of a generator of

$\operatorname{Gal}(L_1/L)$. Apart from the above differential forms the statement of this condition uses the power series coming from the

$p$-adic period of the formal group

$\mathbb{G}_m$.

**Keywords:**
*local field, Galois group, ramification filtration.*
**UDC:**
512.625
**MSC:** 11S20,

11S15,

11R32 **Received:** 10.02.2022

**Revised:** 02.11.2022

**Language:** English

**DOI:**
10.4213/im9322