Equations différentielles $p$-adiques et Séries Gevrey arithmétiques

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

In this paper, we associate to every $p$-adic representation $V$ a $p$-adic differential equation $\mathbf{D}^{\dagger}_{\mathrm{rig}}(V)$, that is to say a module with a connection over the Robba ring. We do this via the theory of Fontaine's $(ϕ,Γ_K)$-modules. This construction enables us to relate the theory of $(ϕ,Γ_K)$-modules to $p$-adic Hodge theory. We explain how to construct $mathbf{D}_{mathrm{cris}}(V)$ and $\mathbf{D}_{\mathrm{st}}(V)$ from $\mathbf{D}^{\dagger}_{\mathrm{rig}}(V)$, which allows us to recognize semi-stable or crystalline representations; the connection is then either…