Repr{é}sentations p-adiques et {é}quations diff{é}rentielles

Abstract

In this paper, we associate to every $p$-adic representation $V$ a $p$-adic differential equation ${\mathbf{D}}_{\mathrm{rig}}^{†}(V)$, that is to say a module with a connection over the Robba ring. We do this via the theory of Fontaine's $(\varphi ,{\Gamma }_K)$-modules. This construction enables us to relate the theory of $(\varphi ,{\Gamma }_K)$-modules to $p$-adic Hodge theory. We explain how to construct $mathbf{D}_{mathrmcris}(V)$ and ${\mathbf{D}}_{\mathrm{st}}(V)$ from ${\mathbf{D}}_{\mathrm{rig}}^{†}(V)$, which allows us to recognize semi-stable or crystalline representations; the connection is then either unipotent or trivial on ${\mathbf{D}}_{\mathrm{rig}}^{†}(V)[1/t]$. In general, the connection has an infinite number of regular singularities, but we show that $V$ 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. A recent theorem of Y. André gives a complete description of the structure of such an 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 ($H_g^1=H_{st}^1$), 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 $\ge 2$, then Bloch-Kato's exponential ${\exp }_V$ is an isomorphism).

71 pages. In French. Uses Xypic. 3rd Version: this revised version includes a proof of Fontaine's monodromy conjecture and some applications. Submitted for publication

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