Groupes p-divisibles, groupes finis et modules filtr\'es

TL;DR

This paper extends the classification of p-divisible groups and finite flat p-groups from perfect fields to more general complete discrete valuation rings with unequal characteristic, using generalized filtered modules, and applies this to Galois representations.

Contribution

It generalizes Fontaine and Laffaille's classification to arbitrary complete discrete valuation rings with unequal characteristic using generalized filtered modules.

Findings

Extended classification to arbitrary complete discrete valuation rings.

Proved that certain crystalline Galois representations contain Tate modules of p-divisible groups.

No restriction on ramification index in the classification.

Abstract

Let k be a perfect field of characteristic p>0. When p>2, Fontaine and Laffaille have classified p-divisibles groups and finite flat p-groups over the Witt vectors W(k) in terms of filtered modules. Still assuming p>2, we extend these classifications over an arbitrary complete discrete valuation ring A with unequal characteristic (0,p) and residue field k by using "generalized" filtered modules. In particular, there is no restriction on the ramification index. In the case k is included in \bar{F}_p (and p>2), we then use this new classification to prove that any crystalline representation of the Galois group of Frac(A) with Hodge-Tate weights in {0,1} contains as a lattice the Tate module of a p-divisible group over A.