\(p\)-adic monodromy conjecture. (Les conjectures de monodromie \(p\)-adiques.) (French) Zbl 1127.12301

Colmez, Pierre

Séminaire Bourbaki. Volume 2001/2002. Exposés 894–908. Paris: Société Mathématique de France (ISBN 2-85629-149-X/pbk). Astérisque 290, 53-101, Exp. No. 897 (2003).

The paper is a survey of a variety of subjects related to the conjecture about the quasi-unipotence of differential modules over the Robba ring possessing the Frobenius structure. The conjecture was proved by different methods by Y. André [Invent. Math. 148, No. 2, 285–317 (2002; Zbl 1081.12003)], Z. Mebkhout [Invent. Math. 148, No. 2, 319–351 (2002; Zbl 1071.12004)], and K. Kedlaya [Ann. Math. (2) 160, No. 1, 93–184 (2004; Zbl 1088.14005)].
The author discusses \(p\)-adic differential equations, \(\varphi\)-modules, \(p\)-adic Galois representations including the hierarchy of representations introduced by [J.-M. Fontaine [Périodes \(p\)-adiques. Séminaire de Bures-sur-Yvette, France, 1988, Astérisque 223, 113–184 (1994; Zbl 0865.14009)].
For the entire collection see [Zbl 1050.00006].

Reviewer: Anatoly N. Kochubei (Kyïv)

MSC:

12H25	\(p\)-adic differential equations
11S20	Galois theory
11S25	Galois cohomology
11F80	Galois representations
14F30	\(p\)-adic cohomology, crystalline cohomology

Keywords:

differential module; Robba ring; Frobenius structure; \(\varphi\)-module; Galois representation

Citations:

Zbl 1081.12003; Zbl 1071.12004; Zbl 1088.14005; Zbl 0865.14009