Let $K$ be a finite extension of ${\mathbf{Q}}_p$. The field of norms of a $p$-adic Lie extension $K_{\infty }/K$ is a local field of characteristic $p$ which comes equipped with an action of $\mathrm{Gal}(K_{\infty }/K)$. When can we lift this action to characteristic $0$, along with a compatible Frobenius map? In this note, we formulate precisely this question, explain its relevance to the theory of $(\varphi ,\Gamma )$-modules, and give a condition for the existence of certain types of lifts.

Let $K$ be a field of characteristic $p\>0$, $C$ a proper, smooth, geometrically connected curve over $K$, and 0 and $\infty$ two $K$-rational points on $C$. We show that any representation of the local Galois group at $\infty$ extends to a representation of the fundamental group of $C-\{0,\infty \}$ which is tamely ramified at 0, provided either that $K$ is separately closed or that $C$ is ${\mathbf{P}}^1$. In the latter case, we show there exists a unique such extension, called “canonical”, with the property that the image of the geometric fundamental group...