Construction of \((\phi,\Gamma)\)-modules: \(p\)-adic representations and \(B\)-pairs. (Construction de \((\phi,\Gamma)\)-modules: représentations \(p\)-adiques et \(B\)-paires.) (English) Zbl 1219.11078

Berger, Laurent

Algebra Number Theory 2, No. 1, 91-120 (2008).

Summary: Let \(B_e = B_{\text{cris}}^{\varphi=1}\). We study the category of \(B\)-pairs \((W_e,W_{dR}^+)\) where \(W_e\) is a free \(B_e\)-module with a semilinear and continuous action of \(G_K\) and where \(W_{dR}^+\) is a \(G_K\)-stable \(B_{dR}^+\)-lattice in \(B_{dR}\otimes_{B_e}W_e\). This category contains the category of \(p\)-adic representations and is naturally equivalent to the category of all \((\varphi, \Gamma)\)-modules over the Robba ring.

MSC:

11F80	Galois representations
11F85	\(p\)-adic theory, local fields
14F30	\(p\)-adic cohomology, crystalline cohomology
11S25	Galois cohomology
11S20	Galois theory

Keywords:

\(p\)-adic Hodge theory; \((\varphi; \Gamma)\)-modules; Frobenius slopes; B-pairs