Ramification filtration and differential forms

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 $\mathbb{G}_m$.