Hasse-Arf filtrations and \(p\)-adic monodromy.
(Filtrations de type Hasse-Arf et monodromie \(p\)-adique.)

*(French)*Zbl 1081.12003The 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 Z. Mebkhout , “Analogue \(p\)-adique du théorème de Turrittin.” [Invent. Math. 148, No. 2, 319–351 (2002; Zbl 1071.12004)]. Even if both proofs are based on the same properties (the analog of Hasse-Arf theorem for \(p\)-adic differential equations) they are very distinct. The one given here is based on the (differential) Galois theory. Actually, the author brings out the notion of filtred Tannakian category of Hasse-Arf type. For the filtrations under consideration, \(F^\lambda M\otimes N=0\) as soon as \(F^\lambda M=F^\lambda N=0\). Hence they are entirely distinct from the \(\otimes\)-filtrations of [N. Saavedra Rivano, Categories tannakiennes. Lecture Notes in Mathematics. 265. (Berlin-Heidelberg-New York: Springer-Verlag) (1972; Zbl 0241.14008)]. For each object, the (decreasing) filtration provides a Newton polygon. The Hasse-Arf assumption, is that the vertices of this polygon have integral coordinates.

Let \(K\) be a discretely valued complete field with perfect residue field \(k\) of characteristic \(p\) and let \(G_K\) be the Galois group of its separable closure. The Hasse-Arf theorem says precisely that the category of representations of \(G_K\) endowed with the ramification filtration is ... of Hasse-Arf type. Another example is given by the \(\ell\)-adic representations of \(G_K\). However this paper is mainly concerned with the category of \(p\)-adic differential equations (namely \(\mathcal{R}[d/dx]\)-module, free of finite rank as \(\mathcal{R}\)-module where \(\mathcal{R}\) is the ring of power series \(\sum_{n\in \mathbb{Z}} a_n\,x^n\), \((a_n\in K)\) supposed of characteristic \(0\) that converges in some annulus \(1-\varepsilon<| x| <1\)) endowed with a Frobenius structure. From [G. Christol, Z. Mebkhout, “Sur le théorème de l’indice des équations différentielles \(p\)-adiques. IV”. Invent. Math. 143, No. 3, 629–672 (2001; Zbl 1078.12501)], it is known to have a filtration “by the slopes” and to be of Hasse-Arf type. Moreover, one now knows that this category is equivalent to a subcategory of the category of \(p\)-adic representations of \(G_K\) namely the De Rham ones. [L. Berger, “Représentations \(p\)-adiques et équations différentielles”. Invent. Math. 148, No. 2, 219–284 (2002; Zbl 1113.14016)].

The Tannakian category of \(p\)-adic differential equations has a fiber functor (it not so obvious). This enables to define its Galois group \(G\). The main result of the paper is that \(G\) is isomorphic to \(\mathfrak{I}\times \mathbb{G}_a\) where \(\mathfrak{I}\) is the inertia subgroup of \(G_{k((x))}.\) In this isomorphism, the slope filtration on \(G\) becomes the ramification filtration (in above numbering) of \(G_{k((x))}.\) As a direct application one obtains the existence, for any \(p\)-adic differential equation \(\mathcal{M}\) endowed with a Frobenius structure, of an “étale” extension \(\mathcal{R}_{K',t}\) of \(\mathcal{R}\) such that the \(\mathcal{R}_{K',t}\)-module \(\mathcal{R}_{K',t}\otimes_{\mathcal{R}}\mathcal{M}\) is unipotent (namely, it can be obtained by successive extensions from the \(\mathcal{R}_{K',t}[d/dt]\)-module \(\mathcal{R}_{K',t}\)).

Explicit computations over an example, very usefull to understand the general proof, can be found in [Y. André, Représentations galoisiennes et opérateurs de Bessel \(p\)-adiques. Ann. Inst. Fourier 52, No. 3, 779–808 (2002; Zbl 1014.12007)]

Let \(K\) be a discretely valued complete field with perfect residue field \(k\) of characteristic \(p\) and let \(G_K\) be the Galois group of its separable closure. The Hasse-Arf theorem says precisely that the category of representations of \(G_K\) endowed with the ramification filtration is ... of Hasse-Arf type. Another example is given by the \(\ell\)-adic representations of \(G_K\). However this paper is mainly concerned with the category of \(p\)-adic differential equations (namely \(\mathcal{R}[d/dx]\)-module, free of finite rank as \(\mathcal{R}\)-module where \(\mathcal{R}\) is the ring of power series \(\sum_{n\in \mathbb{Z}} a_n\,x^n\), \((a_n\in K)\) supposed of characteristic \(0\) that converges in some annulus \(1-\varepsilon<| x| <1\)) endowed with a Frobenius structure. From [G. Christol, Z. Mebkhout, “Sur le théorème de l’indice des équations différentielles \(p\)-adiques. IV”. Invent. Math. 143, No. 3, 629–672 (2001; Zbl 1078.12501)], it is known to have a filtration “by the slopes” and to be of Hasse-Arf type. Moreover, one now knows that this category is equivalent to a subcategory of the category of \(p\)-adic representations of \(G_K\) namely the De Rham ones. [L. Berger, “Représentations \(p\)-adiques et équations différentielles”. Invent. Math. 148, No. 2, 219–284 (2002; Zbl 1113.14016)].

The Tannakian category of \(p\)-adic differential equations has a fiber functor (it not so obvious). This enables to define its Galois group \(G\). The main result of the paper is that \(G\) is isomorphic to \(\mathfrak{I}\times \mathbb{G}_a\) where \(\mathfrak{I}\) is the inertia subgroup of \(G_{k((x))}.\) In this isomorphism, the slope filtration on \(G\) becomes the ramification filtration (in above numbering) of \(G_{k((x))}.\) As a direct application one obtains the existence, for any \(p\)-adic differential equation \(\mathcal{M}\) endowed with a Frobenius structure, of an “étale” extension \(\mathcal{R}_{K',t}\) of \(\mathcal{R}\) such that the \(\mathcal{R}_{K',t}\)-module \(\mathcal{R}_{K',t}\otimes_{\mathcal{R}}\mathcal{M}\) is unipotent (namely, it can be obtained by successive extensions from the \(\mathcal{R}_{K',t}[d/dt]\)-module \(\mathcal{R}_{K',t}\)).

Explicit computations over an example, very usefull to understand the general proof, can be found in [Y. André, Représentations galoisiennes et opérateurs de Bessel \(p\)-adiques. Ann. Inst. Fourier 52, No. 3, 779–808 (2002; Zbl 1014.12007)]

Reviewer: Gilles Christol (Paris)

##### MSC:

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

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

11F80 | Galois representations |