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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.

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.

Reviewer: Christine Noot-Huyghe (Strasbourg)