Let $L/K$ be an abelian extension of $p$-adic fields, and let $𝒪$ denote the valuation ring of $K$. We study ideals of the valuation ring of $L$ as integral representations of the Galois group $\mathrm{Gal}(L/K)$. Assuming $K$ is absolutely unramified we use techniques from the theory of factorisability to investigate which ideals are isomorphic to an $𝒪$-order in the group algebra $K\left[\mathrm{Gal}\right(l/K\left)\right]$. We obtain several general and also explicit new results.

We continue the examination of the stable reduction and fields of moduli of $G$-Galois covers of the projective line over a complete discrete valuation field of mixed characteristic $(0,p)$, where $G$ has a cyclic$p$-Sylow subgroup $P$ of order $p^n$. Suppose further that the normalizer of $P$ acts on $P$ via an involution. Under mild assumptions, if $f:Y\to {\mathbb{P}}^1$ is a three-point $G$-Galois cover defined over $\overline{\mathbb{Q}}$, then the $n$th higher ramification groups above $p$ for the upper numbering of the (Galois closure of the) extension $K/\mathbb{Q}$ vanish,...

This paper considers some refined versions of the Inverse Galois Problem. We study the local or global behavior of rational specializations of some finite Galois covers of ${\mathbb{P}}_{\mathbb{Q}}^1$.

Let $K$ be a $p$-adic local field with residue field $k$ such that $[k:k^p]=p^e\<+\infty$ and $V$ be a $p$-adic representation of $\text{Gal}(\overline{K}/K)$. Then, by using the theory of $p$-adic differential modules, we show that $V$ is a Hodge-Tate (resp. de Rham) representation of $\text{Gal}(\overline{K}/K)$ if and only if $V$ is a Hodge-Tate (resp. de Rham) representation of $\text{Gal}(\overline{K^{\text{pf}}}/K^{\text{pf}})$ where $K^{\text{pf}}/K$ is a certain $p$-adic local field with residue field the smallest perfect field $k^{\text{pf}}$ containing $k$.

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...