## Artin conductor of a de Rham representation. (Conducteur d’Artin d’une représentation de de Rham.)(French. English summary)Zbl 1168.11052

Berger, Laurent (ed.) et al., Représentation $$p$$-adiques de groupes $$p$$-adiques I. Représentations galoisiennes et $$(\varphi, \Gamma)$$-modules. Paris: Société Mathématique de France (ISBN 978-2-85629-256-3/pbk). Astérisque 319, 187-212 (2008).
Author’s summary: We endow Fontaine’s ring $$\mathbf{B}^+_{\text{dR}}$$ with a filtration defined by means of “valuation of convergence”. This filtration is stable under the action of $$G_K = \text{Gal}(\overline{K}/K)$$ and its restriction to $$\overline{K}$$ coincides with the filtration induced by the filtration of $$G_K$$ by ramification subgroups. If $$V$$ is a de Rham representation, this filtration induces an increasing filtration on $$\mathbf{D}_{\text{dR}} (V)$$ and we show that the natural numerical invariant attached to this filtration coincides, if $$V$$ is potentially semi-stable, with the Artin conductor of the representation $$\mathbf{D}_{\text{pst}}(V)$$ of the Weil-Deligne group of $$K$$.
For the entire collection see [Zbl 1156.14002].

### MSC:

 11S37 Langlands-Weil conjectures, nonabelian class field theory 22E50 Representations of Lie and linear algebraic groups over local fields 14F30 $$p$$-adic cohomology, crystalline cohomology