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

Colmez, Pierre

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].

Reviewer: Elmar Große-Klönne (Berlin)

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

Keywords:

Fontaine’s ring; filtration; valuation of convergence; Artin conductor; Swan conductor; de Rham representation; filtration by ramification subgroups; \(p\)-adic representation