## $$p$$-adic representations and differential equations. (Représentations $$p$$-adiques et équations différentielles.)(French)Zbl 1113.14016

Summary: We associate to every $$p$$-adic representation $$V$$ a $$p$$-adic differential equation $${\mathbf D}^\dagger_{\text{rig}}(V)$$, that is to say a module with a connection over the Robba ring. We do this via the theory of Fontaine’s $$(\Phi, \Gamma_K)$$-modules.
This construction enables us to relate the theory of $$(\Phi,\Gamma_K)$$-modules to $$p$$-adic Hodge theory. We explain how to construct $${\mathbf D}_{\text{cris}}(V)$$ and $${\mathbf D}_{\text{st}}(V)$$ from $${\mathbf D}^\dagger_{\text{rig}}(V)$$, which allows us to recognize semi-stable or crystalline representations; the connection is then unipotent or trivial on $${\mathbf D}^\dagger_{\text{rig}}(V)[1/t]$$.
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 [Invent. Math. 148, No. 2, 285–317 (2002; Zbl 1081.12003)] 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 $$(H^1_g=H^1_{\text{st}})$$, 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 $$\geq 2$$, then Bloch-Kato’s exponential $$\exp_V$$ is an isomorphism).

### MSC:

 14F30 $$p$$-adic cohomology, crystalline cohomology 11S20 Galois theory 12H25 $$p$$-adic differential equations 14F40 de Rham cohomology and algebraic geometry 14G20 Local ground fields in algebraic geometry

Zbl 1081.12003
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