# Representations p-adiques et equations differentielles

Abstract : 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 $(\varphi,\Gamma_K)$-modules. This construction enables us to relate the theory of $(\varphi,\Gamma_K)$-modules to $p$-adic Hodge theory. We explain how to construct $\mathbf(D)_(\mathrm(cris))(V)$ 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 unipotent or trivial. Along with techniques from the theory of $p$-adic differential equations, the study of $\mathbf(D)^(\dagger)_(\mathrm(rig))(V)$ allows us to give a new proof of Sen's theorem characterizing $\mathbf(C)_p$-admissible representations. Finally we can use the previous results to 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 $r \gg 0$, then Bloch-Kato's exponential $\exp_(V(r))$ is an isomorphism), whose proofs (for a finite residue field) relied on the study of dimensions of Galois cohomology groups.
Mots-clés :
Document type :
Theses
Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2001. French
Domain :

https://tel.archives-ouvertes.fr/tel-00003571
Contributor : Laurent Berger <>
Submitted on : Tuesday, October 14, 2003 - 7:56:29 PM
Last modification on : Tuesday, October 14, 2003 - 7:56:29 PM

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• HAL Id : tel-00003571, version 1

### Citation

Laurent Berger. Representations p-adiques et equations differentielles. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2001. French. <tel-00003571>

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