Représentations p-adiques et équations différentielles

Abstract.

In this paper, we associate to every p-adic representation V a p-adic differential equation D ^† _rig(V), that is to say a module with a connection over the Robba ring. We do this via the theory of Fontaine’s (ϕ,Γ _K )-modules.¶This construction enables us to relate the theory of (ϕ,Γ _K )-modules to p-adic Hodge theory. We explain how to construct D _cris(V) and D _st(V) from D ^† _rig(V), which allows us to recognize semi-stable or crystalline representations; the connection is then unipotent or trivial on D ^† _rig(V)[1/t].¶In general, the connection has an infinite number of regular singularities, but V is de Rham if and only if those are apparent singularities. A structure theorem for modules over the Robba ring allows us to get rid of all singularities at once, and to obtain a “classical” differential equation, with a Frobenius structure.¶Using this, we construct a functor from the category of de Rham representations to that of classical p-adic differential equations with Frobenius structure. A recent theorem of Y. André gives a complete description of the structure of the latter object. This allows us to prove Fontaine’s p-adic monodromy conjecture: every de Rham representation is potentially semi-stable.¶As an application, we can extend to the case of arbitrary perfect residue fields some results of Hyodo (H ¹ _g =H ¹ _st ), of Perrin-Riou (the semi-stability of ordinary representations), of Colmez (absolutely crystalline representations are of finite height), and of Bloch and Kato (if the weights of V are ≥2, then Bloch-Kato’s exponential exp _V is an isomorphism).