Ramification filtration and differential forms

Let $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by ${\mathrm{\Gamma }}_L^{(v)}$ the ramification subgroups of ${\mathrm{\Gamma }}_L=\mathrm{Gal}(L^{sep}\mathrm{/}L)$. We consider the category ${\mathrm{M\Gamma }}_L^{Lie}$ of finite $$\mathbb{Z}$_p[\Gamma _{L}]$-modules$H$,satisfyingsomeadditional(Lie)-conditionontheimageof$\Gamma L$in$\operatorname{Aut}{$\mathbb{Z}$_p}H$. In the paper it is proved that all information about the images of the ramification subgroups $\Gamma _L^{(v)}$canbeexplicitlyextractedfromsomedifferentialforms$\Omega [N]$ontheFontaineetale$\phi $-module$M(H)$associatedwith$H$\mathrm{.}Theforms$\Omega [N]$arecompletelydeterminedbyaconnection$\nabla $on$M(H)$. In the case of fields $L$ of mixed characteristic containing a primitive $p$-th root of unity we show that the similar problem for $$\mathbb{F}$_p[\Gamma _L]$-modules also admits a solution. In this case we use the field-of-norms functor to construct the coresponding $\varphi$-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 ${\mathbb{G}}_m$.