Frobenius et ramification logarithmique
Résumé
Given $K_{\mathfrak{p}}$ a finite extension of $\mathbb{Q}_{p}$, we study the $\ell$-adification of the multiplicative group of our local field $\mathcal{R}_{K_{\mathfrak{p}}}$ endowed with the logarithmic valuation introduced by Jaulent. Instead of considering the maximal, abelian unramified pro-$\ell$-extension of $K_{\mathfrak{p}}$ we study here the ${\mathbb{Z}_{\ell}}$-cyclotomic one, and the usual valuation is replaced by the logarithmic one. We show that it is possible to use Neukich's abstract theory in this context. Thus, it allows to define a logarithmic local symbol, a global one and a logarithmic Frobenius. The interesting point is that usual and logarithmic Frobenius coincide when usual and logarithmic ramification are the same.
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