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**Analogue \(p\)-adique du théorème de Turrittin et le théorème de la monodromie \(p\)-adique.**
*(French)*
Zbl 1071.12004

The aim of this paper is to prove the basic structure theorem about \(p\)-adic differential equations. Another proof of the same theorem is available in [Y. André, “Filtrations de type Hasse-Arf et monodromie \(p\)-adique.” Invent. Math. 148, No. 2, 285–317 (2002; Zbl 1081.12003)]. Even if both proofs are based on the same properties (mainly on the integrality of the \(p\)-adic irregularity) they are very distinct. The one given here follows rather faithfully (with many extra difficulties) the classical proof of Turrittin theorem for differential equations over a formal power series field. It is then as effective as possible, even if it uses several tools, for instance the decomposition theorem, which are far from being effective.

Let \(K\) be a discretely valued complete field of characteristic \(0\) with residue field \(k\) of characteristic \(p\) and let \(\mathcal{R}=\mathcal{R}_{K,x}\) be the so called Robba ring (namely the ring of power series \(\sum_{n\in \mathbb{Z}} a_n\,x^n\), \((a_n\in K)\) that converges in some annulus \(1-\varepsilon<| x| <1\)). Any finite separable extension of \(k((x))\) can be written \(k'((t))\), for some finite extension \(k'\) of \(k\) and with \(x\) in \(t\,k'[t]\). It can then be lifted in an extension \(\mathcal{R}_{K',t}\) of \(\mathcal{R}\). The \(p\)-adic analog of Turrittin theorem says that for any \(\mathcal{R}\)-differential module (namely \(\mathcal{R}[d/dx]\)-module, free of finite rank as \(\mathcal{R}\)-module) \(\mathcal{M}\) which is endowed with a Frobenius structure (namely such that, for \(\varphi(x)=x^p\), \(\mathcal{M}\) and \(\varphi^{*}(\mathcal{M}): = \mathcal{R}_{\;\vrule height 9pt width0pt\llap{}_\varphi}\)\nwarrow _\cal{R}

Let \(K\) be a discretely valued complete field of characteristic \(0\) with residue field \(k\) of characteristic \(p\) and let \(\mathcal{R}=\mathcal{R}_{K,x}\) be the so called Robba ring (namely the ring of power series \(\sum_{n\in \mathbb{Z}} a_n\,x^n\), \((a_n\in K)\) that converges in some annulus \(1-\varepsilon<| x| <1\)). Any finite separable extension of \(k((x))\) can be written \(k'((t))\), for some finite extension \(k'\) of \(k\) and with \(x\) in \(t\,k'[t]\). It can then be lifted in an extension \(\mathcal{R}_{K',t}\) of \(\mathcal{R}\). The \(p\)-adic analog of Turrittin theorem says that for any \(\mathcal{R}\)-differential module (namely \(\mathcal{R}[d/dx]\)-module, free of finite rank as \(\mathcal{R}\)-module) \(\mathcal{M}\) which is endowed with a Frobenius structure (namely such that, for \(\varphi(x)=x^p\), \(\mathcal{M}\) and \(\varphi^{*}(\mathcal{M}): = \mathcal{R}_{\;\vrule height 9pt width0pt\llap{}_\varphi}\)\nwarrow _\cal{R}

Reviewer: Gilles Christol (Paris)

### MSC:

12H25 | \(p\)-adic differential equations |

14F30 | \(p\)-adic cohomology, crystalline cohomology |

11F80 | Galois representations |