Overconvergent \(p\)-adic representations. (Représentations \(p\)-adiques surconvergentes.) (French) Zbl 0928.11051

Cherbonnier, F.; Colmez, P.

Invent. Math. 133, No. 3, 581-611 (1998).

Let \(K\) be a local \(p\)-adic field, \({\mathcal G}_K\) its absolute Galois group, J.-M. Fontaine’s theory of \((\varphi,\Gamma)\)-modules associates to each \(p\)-adic representation \(V\) of \({\mathcal G}_k\) a module \(D(V)\) over a certain local field of dimension 2, with natural and commuting actions of the Frobenius \(\varphi\) and the Galois group \(\Gamma=\text{Gal} (K( \mu_{p^\infty} /K)\).
The main point is that, starting from \(D(V)\), one can reconstruct the representation \(V\), and one hopes to describe in this way all the classical invariants of \(V\), for instance its Galois cohomology (Laurent Herr’s thesis, 1995) [see L. Herr, Bull. Soc. Math. Fr. 126, 563–600 (1998; Zbl 0967.11050)].
In this paper, the authors show that any \(p\)-adic representation of \({\mathcal G}_K\) is “overconvergent”. This notion, introduced in the first author’s thesis (1996), is too technical to be recalled here, but it is the essential tool which allows to recover from \(D(V)\) the more subtle invariants of \(V\) built upon the rings \(B_{\text{cris}}\), \(B_{\text{dR}}\) etc.
The proof, using the action of \(\Gamma\), is inspired from Shankar Sen’s proof that any \(p\)-adic representation of \({\mathcal G}_K\) has a generalized Hodge-Tate decomposition [Invent. Math. 62, 89–116 (1980; Zbl 0463.12005)], but of course, it must overcome technical difficulties which appear because the rings involved are more complicated than \(\mathbb C_n\).
In another paper [“Théorie d’Iwasawa des représentations \(p\)-adiques d’un corps local”, J. Am. Math. Soc. 12, 241–268 (1999; Zbl 0933.11056)] the authors have given applications to Iwasawa theory of local fields and explicit reciprocity laws.

Reviewer: Thong Nguyen Quang Do (Besançon)

MSC:

11S20	Galois theory
14F30	\(p\)-adic cohomology, crystalline cohomology
11S25	Galois cohomology

Keywords:

overconvergent \(p\)-adic representation; Iwasawa theory of \(p\)-adic representations; \(p\)-adic representation; Fontaine’s theory; generalized Hodge-Tate decomposition

Citations:

Zbl 0463.12005; Zbl 0933.11056; Zbl 0967.11050