Let $R$ be a 2-dimensional normal excellent henselian local domain in which $2$ is invertible and let $L$ and $k$ be its fraction field and residue field respectively. Let ${\Omega }_R$ be the set of rank 1 discrete valuations of $L$ corresponding to codimension 1 points of regular proper models of $SpecR$. We prove that a quadratic form $q$ over $L$ satisfies the local-global principle with respect to ${\Omega }_R$ in the following two cases: (1) $q$ has rank 3 or 4; (2) $q$ has rank $\ge 5$ and $R=A\left[\right[y\left]\right]$, where $A$ is a complete discrete valuation ring with...

Soient $X$ un espace analytique affinoïde réduit sur un corps $K$ complet pour une valeur absolue non archimédienne, $r:X\to \widehat{X}$ sa réduction canonique et $p\in \widehat{X}$ un point de la variété algébrique affine $\widehat{X}$. Ce travail décrit la singularité du point $p$ à l’aide d’objets associés à l’espace $X$: la localisation formelle ${𝒪}_{X,\left(p\right)}$ qui est une $K$-algèbre noethérienne de spectre maximal $r^{-1}\left(p\right)$ et dont la réduction est ${𝒪}_{\widehat{X},\left(p\right)}$ ; un complété formel ${𝒪}_{X,\left(p\right)}$ qui est une $K$-algèbre noethérienne dont la réduction est ${𝒪}_{\widehat{X},\left(p\right)}$. Les résultats essentiels sont obtenus...

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...