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A $$p$$-adic local monodromy theorem. (English) Zbl 1088.14005
Let $$p$$ be a prime number. This paper is devoted to the proof of the $$p$$-adic local monodromy theorem, also known as R. Crew’s conjecture [Ann. Sci. Éc. Norm. Supér., IV. Sér. 31, No. 6, 717–763 (1998; Zbl 0943.14008)]. This conjecture was simultaneously and independently proved by other methods by G. Christol and Z. Mebkhout [Invent. Math. 148, No.2, 319–351 (2002; Zbl 1071.12004)] and by Y. André [Invent. Math. 148, No. 2, 285–317 (2002; Zbl 1081.12003)]. Let us first explain what the statement of the conjecture is.
Let $$K$$ be a complete field for a $$p$$-adic valuation and of residue field $$k$$. Let $$\Gamma_{\text{an,con}}$$ be the Robba ring over a complete $$p$$-adic field $$K$$ (usually denoted by $$\eusm{R}_K$$). Let us recall that $$\Gamma_{\text{an,con}}$$, (resp. $$\Gamma_{\text{con}}$$, usually denoted by $$\eusm{E}^{\dagger}_K$$) is the set of analytic functions with coefficients in $$K$$, converging over some annulus $$1-\varepsilon < | x| <1$$ (where $$\varepsilon$$ depends on the function), (resp. is the set of analytic functions with coefficients in $$K$$, which converge and are bounded over some annulus $$1-\varepsilon < | x| <1$$). These rings can be endowed with a Frobenius morphism $$\sigma$$, which is a lifting of the map $$x\mapsto x^p$$ over $$k((T))$$. Let $$\partial$$ be a continuous derivation of $$\Gamma_{\text{con}}$$. A $$(\sigma,\nabla)$$-module $$M$$ over $$\Gamma_{\text{an,con}}$$ is a finite locally free module $$M$$ equipped with an $$\Gamma_{\text{an,con}}$$-linear isomorphism $$F$$ : $$\sigma^*M \rightarrow M$$ and with a connection $$\nabla$$ compatible with $$F$$. Such a module is called quasi-unipotent if, after tensoring $$\Gamma_{\text{an,con}}$$ over $$\Gamma_{\text{con}}$$ with a finite extension of $$\Gamma_{\text{con}}$$, the module admits a filtration by $$(\sigma,\nabla)$$-submodules such that each successive quotient admits a basis of elements in the kernel of $$\nabla$$. The main statement of the article is:
Theorem (Local monodromy theorem). Let $$\sigma$$ be any Frobenius for the Robba ring $$\Gamma_{\text{an,con}}$$. Then every $$(\sigma,\nabla)$$-module over $$\Gamma_{\text{an,con}}$$ is quasi-unipotent.
The proof of the theorem is very difficult and is based upon a structure theorem for $$\sigma$$-modules. One key tool is the Dieudonné-Manin classification of $$\sigma$$-modules over a complete discrete valuation ring of mixed characteristic $$(0,p)$$ with algebraically closed residue field, that is used over auxiliary rings (note that $$\Gamma_{\text{an,con}}$$ is not a discrete valuation ring). The proof uses also a previous result of N. Tsuzuki who proved the local monodromy conjecture for unit-root $$(\sigma,\nabla)$$-module over $$\Gamma_{\text{con}}$$ (with 0 slopes) [Am. J. Math. 120, No. 6, 1165–1190 (1998; Zbl 0943.14007)].
This result is fundamental in the theory of $$p$$-adic cohomology. It lead to many applications. Let us mention the following application due to L. Berger [Invent. Math. 148, No. 2, 219–284 (2002; Zbl 1113.14016)] that proves a conjecture of J-M. Fontaine: every de Rham representation is semi-stable.

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 12H25 $$p$$-adic differential equations 11G25 Varieties over finite and local fields
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