Propriétés du groupe tannakien des structures de Hodge \(p\)-adiques et torseur entre cohomologies cristalline et étale. (Properties of the Tannakian group of \(p\)-adic Hodge structures and torsor between crystalline and étale cohomologies.)
The paper consists of an introduction, notations and conventions, and four chapters: \((1)\) Construction of a fibre functor over \({\mathbb Q}_p\) on the category of weakly admissible filtered \((\phi,N)\)-modules; \((2)\) Weakly admissible filtered \((\phi,N)\)-modules and isogenies; \((3)\) Simple connectivity of the semi-simple part of the group \(\mathbf{H}_c\); \((4)\) Consequences for the torsor between the fibre functors \(\omega_c\) and \(\omega_{\text{Gal}}\). The introduction gives a very readible overview of this complicated matter. Let \(p\) be a prime number, \(K\) a local field of zero characteristic with algebraically closed residue field \(k\) of characteristic \(p\), \(W=W(k)\) the ring of Witt vectors with fraction field \(K_0\). Write \(\sigma\) for the Frobenius on \(k\), \(W\) and \(K_0\). \(K\) is a finite extension of \(K_0\) with algebraic closure \(\overline{K}\). One also writes \(G_K=\text{Gal}(\overline{K}/K)\) and \(G_{{\mathbb Q}_p}=\text{Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p)\). J.-M. Fontaine introduced filtered \((\phi,N)\)-modules over \(K\) as finite dimensional \(K_0\)-vector spaces equipped with a bijective \(\sigma\)-linear map \(\phi\) and a linear nilpotent map \(N\), and after extension of scalars to \(K\), a filtration. These \((\phi,N)\)-modules may satisfy a kind of stability condition in the sense of geometric invariant theory. In this case they are called \textit{weakly admissible}, and they form a Tannakian category \(\underline{\text{MF}}^{fa}_K (\phi,N)\) over \({\mathbb Q}_p\). This category can be considered as a \(p\)-adic analogue of the category of complex Hodge structures. On the other hand, one has the Tannakian category \(\mathbf{Rep}_{{\mathbb Q}_p}(G_K)\) of \(p\)-adic representations of \(G_K\), and its Tannakian subcategory \(\mathbf{Rep}_{{\mathbb Q}_p}^{ss}(G_K)\) of semi-stable representations. Fontaine constructed a \(\otimes\)-equivalence between \(\mathbf{Rep}_{{\mathbb Q}_p}^{ss}(G_K)\) and a Tannakian subcategory \(\underline{\text{MF}}^{a}_K(\phi,N)\) of \textit{admissible} filtered \((\phi,N)\)-modules. Fontaine conjectured that weakly admissible filtered \((\phi,N)\)-modules are admissible. This can be proved in several special situations. The present paper deals with consequences of this conjecture \(\text{Conj}(fa\Rightarrow a)_K\). In particular, there must be a fibre functor \(\omega\) over \({\mathbb Q}_p\) giving an equivalence between \(\underline{\text{MF}}^{fa}_K(\phi,N)\) and \(\mathbf{Rep}_{{\mathbb Q}_p}(\text\textbf{H})\) for some affine group \(\mathbf{H}\) over \({\mathbb Q}_p\). It is shown that there exists \(c\in\mathbf{H}(K_0)\) giving fibre functors \(\omega_c\) over \({\mathbb Q}_p\) such that, for such a \(c\), the group of \(\otimes\)-automorphisms \(\mathbf{H}_c\) of \(\omega_c\) induces an equivalence \(\underline{\text{MF}}^{fa}_K(\phi,N){\buildrel\sim\over\longrightarrow}\mathbf{Rep}_{{\mathbb Q}_p}(\text\textbf{H}_c)\). The proof is based on the construction of suitable \(\phi\)-isocrystals of slope zero. The \(\mathbf{H}_c\) play a key role in the whole paper. A second subject in relation with \(\mathbf{H}_c\) is the lifting by an isogeny. This means that if \(H\) is an algebraic quotient of \(\mathbf{H}_c\) and \(\alpha:H'\rightarrow H\) is an isogeny over \({\mathbb Q}_p\), can one lift the morphism \(\mathbf{H}_c\rightarrow H\) to a morphism \(\mathbf{H}_c\rightarrow H'\)? This leads, by \(\text{Conj}(fa\Rightarrow a)_K\), to \(\text{Conj(isogeny)}_K\): If \(\rho:G_K\rightarrow H({\mathbb Q}_p)\) is a semi-stable \(p\)-adic representation and \(\alpha:H'\rightarrow H\) is an isogeny such that the one-parameter subgroup of \(H\) given by the Hodge-Tate decomposition can be lifted, then \(\rho\) can be lifted to a semi-stable \(p\)-adic representation \(\rho':G_K\rightarrow H'({\mathbb Q}_p)\). The third chapter deals with the semi-simple part \(\mathbf{S}_c\) of \(\mathbf{H}_c\). It is shown that \(\mathbf{S}_c\) is the projective limit of simply connected algebraic groups. The last chapter is concerned with the torsor between the fibre functor \(\omega_c\) and the one which associates with an admissible filtered module the underlying vector space of the corresponding semi-stable \(p\)-adic representation. In the geometric situation, i.e.\ if the \((\phi,N)\)-module comes from the cohomology of a smooth proper scheme with semi-stable reduction, one has to do with the torsor between log-crystalline and \(p\)-adic cohomologies. One knows that \(H^1({\mathbb Q}_p,S)\) is trivial for \(S\) semi-simple simply connected, and one shows that any \(\mathbf{H}_c\)-torsor is the projective limit of torsors with a \({\mathbb Q}_p\)-rational point. \(\text{Conj}(fa\Rightarrow a)_K\) implies that this also holds for the above torsor between étale and crystalline cohomologies. This is related to work of Rapoport and Zink.
scientific article; zbMATH DE number 1092695
scientific article