## 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 $$\varphi^{*}(\mathcal{M}): = \mathcal{R}_{_\varphi\nwarrow _\mathcal{R}} \otimes\ \mathcal{M}$$ are isomorphic as $$\mathcal{R}[d/dx]$$-modules) there exists a finite separable extension $$k'((t))$$ of $$k((x))$$ such that the $$\mathcal{R}_{K',t}$$-differential module $$\mathcal{R}_{K',t}\otimes_{\mathcal{R}}\mathcal{M}$$ is unipotent (namely, can be obtained by successive extensions from differential modules isomorphic to $$\mathcal{R}_{K',t}$$ itself).
The proof is very technical. Let sum up its two main steps. Starting with an irreducible differential module $$\mathcal{M}$$, the author begins tensorizing it by a well chosen exponential. The exponentials $$\exp(\lambda x^{-n})$$ that are used in the formal case must be replaced, in the $$p$$-adic case, by analog but more involved functions, namely the Robba’s exponentials. It is proved that by tensorizing by such an exponential, possibly after some ramification $$x=z^d$$ of order $$d$$ prime to $$p$$, either the module becomes reducible either its $$p$$-adic slope decreases. This enables to do a recursion on (rank, slope) and allows to conclude when the rank of $$\mathcal{M}$$ is prime to $$p$$.
The case when the rank $$r$$ of $$\mathcal{M}$$ is divisible by $$p$$ cannot be obtained directly but uses a very beautiful trick. The basic observation is that the $$\mathcal{R}$$-module End$$(\mathcal{M})$$ has a natural differential structure and a canonical decomposition $$\mathcal{R}\,$$Id$$\oplus \mathcal{Q}$$ as differential module so that $$\mathcal{Q}$$ is a differential module of rank ($$r^2-1$$) prime to $$p$$ and hence has an irreducible sub-differential module of rank prime to $$p$$ at which the first step can be applied. This trick needs to go from $$\mathcal{M}$$ to its endomorphisms and, by recursion, to go further and further into the set of differential modules that can be obtained from $$\mathcal{M}$$ by algebraic constructions and their sub-quotients. Actually, such a travel seems to be unavoidable.
Let notice that in the $$p$$-adic case the conclusion is simpler than in the formal case because, for each differential module of rank one, the Frobenius structure provides a separable extension that trivialize it.

### MSC:

 12H25 $$p$$-adic differential equations 14F30 $$p$$-adic cohomology, crystalline cohomology 11F80 Galois representations

### Keywords:

$$p$$-adic monodromy

Zbl 1081.12003
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