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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 \(\varphi^{*}(\mathcal{M}): = \mathcal{R}_{_\varphi\nwarrow _\mathcal{R}} \otimes\ \mathcal{M}\) 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.

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.

Reviewer: Gilles Christol (Paris)

### MSC:

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

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

11F80 | Galois representations |