Ramification filtration and differential forms

TL;DR

This paper links ramification filtration in local fields of characteristic p to differential forms on Fontaine modules, providing explicit descriptions of ramification subgroups through differential forms and connections.

Contribution

It introduces a method to explicitly extract ramification subgroup information from differential forms on Fontaine modules, extending to mixed characteristic fields with roots of unity.

Findings

Explicit extraction of ramification subgroup data from differential forms.

Extension of methods to mixed characteristic fields with roots of unity.

Use of field-of-norms functor and formal group periods in the analysis.

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 $\Gamma _{L}=\operatorname{Gal}(L^{sep}/L)$. We consider the category $\operatorname{M\Gamma }_{L}^{Lie}$ 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 ramification subgroups $\Gamma _L^{(v)}$ can be explicitly extracted from some differential forms $\Omega [N]$ on the Fontaine etale $\phi $-module $M(H)$ associated with $H$. The forms $\Omega [N]$ are completely determined by a connection $\nabla $ on $M(H)$. In the case of fields $L$ of mixed characteristic containing a primitive $p$-th root of unity we show that the 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 coresponding $\phi $-module together with the action of a cyclic group of order $p$ coming from a cyclic extension of $L$. Then the solution involves the characteristic $p$ part (provided by the field-of-norms functor) and the condition for a "good" lift of a generator of the involved cyclic group of order $p$. Apart from the above differential forms the statement of this condition also uses a power series coming from the $p$-adic period of the formal group $\mathbb{G}_m$.