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Potentially crystalline \(p\)-adic representations. (Représentations \(p\)-adiques potentiellement cristallines.) (French) Zbl 0887.11048
The paper is a sequel to J. M. Fontaine [Grothendieck Festschrift Vol. II, Prog. Math. 87, 249-309 (1990; Zbl 0743.11066)], where many of the important notions are introduced. Let \(p\) be a prime, \(K\) a complete discrete valuation field of characteristic zero, absolute ramification index \(e\) and perfect residue field \(k\) of characteristic \(p\). Let \(W\) denote the ring of Witt vectors with coefficients in \(K\) and with \(K_0\) as fraction field, all three of them acted upon by the Frobenius \(\sigma\). \(\overline{K}\) will denote an algebraic closure of \(K\) and \(G_K=\text{ Gal}(\overline{K}/K)\). The paper deals with \(p\)-adic representations of \(G_K\), i.e. the data of a finite dimensional \({\mathbb Q}_p\)-vector space equipped with a continuous action of \(G_K\). These form a category \(\mathbf{Rep}_{{\mathbb Q}_p}(G_K)\). J. M. Fontaine constructed an equivalence between \(\mathbf{Rep}_{{\mathbb Q}_p}(G_K)\) and a suitable category of \((\phi,\Gamma)\)-modules. It would be of importance to find a characterization of \((\phi,\Gamma)\)-modules corresponding to Hodge-Tate, de Rham, semi-stable, crystalline representations of \(G_K\). Here the crystalline case is dealt with.
To state the main result several notions must be introduced. First, one writes \(K_{\infty}\) for the union of the fields \(K_n\subset\overline{K}\) generated over \(K_0\) by the \(p^n\)-th roots of unity. For an algebraic extension \(L\) (\(\subset\overline{K}\)) of \(K_0\) one writes \(G_L=\text{ Gal}(\overline{K}/L)\), \(H_L=G_L\cap\text{ Gal}(\overline{K}/K_{\infty})\), \(\Gamma_L=G_L/H_L\), in particular, \(\Gamma=\Gamma_K\). One also writes, for a \(p\)-adic representation \(V\), \({\mathcal M}=\mathbf{D}_{\mathcal E}^*(V)\), where \(\mathbf{D}_{\mathcal E}^*\) is the contravariant version of Fontaine’s quasi-inverse functor of the equivalence from the category of \((\phi,\Gamma)\)-modules over some complete discrete valuation field \({\mathcal E}\) to \(\mathbf{Rep}_{{\mathbb Q}_p}(G_K)\). \(S\) will be \(W[[\pi]]\), where \(\pi\) is an element of a suitable Witt ring as defined by Fontaine. One has \(S[1/p]=K_0\otimes_WS\). For a \(\Gamma\)-module \(M\) one defines \(\displaystyle{\widehat{M}=\lim_\leftarrow M/\pi^nM}\). One may define a connection \(\nabla:M\rightarrow M\otimes_{K_0((t))}\Omega^1_{K_0((t))/K_0}\) on \(M\) and an induced one \(\nabla_{\widehat{M}}\) on \(\widehat{M}\). The result can now be stated:
Let \(K\subset K_{\infty}\), and let \(V\) be a \(p\)-adic representation of dimension \(d\) and finite height. Let \(M=j_*({\mathcal M})\) denote the union of the finite type \(S\)-submodules of \(\mathcal M\) that are stable under Frobenius. It is a \((\phi,\Gamma)\)-module over \(S[1/p]\). Then the following are equivalent statements: \((1)\) \(V\) is potentially crystalline, \((2)\) there is a finite extension \(L\) of \(K\), contained in \(K_{\infty}\), such that the induced representation of \(G_L\) is crystalline, \((3)\) \(V\) is a de Rham representation, \((4)\) there is an integer \(r\) and an \(S\)-submodule \(N\) of \(M\) free of rank \(d\), \(\Gamma\)-stable and such that the action of \(\Gamma\) on \((N/\pi N)(-r)\) is finite, \((5)\) the connection \(\nabla_{\widehat{M}}\) is trivial. Furthermore, if this is the case, \(V\) is crystalline as a \(G_L\)-representation if and only if \(\Gamma_L\) acts trivially on \(\text{ Ker} \nabla_{\widehat{M}}\).
The greater part of the paper is taken by the proof of the above result, split into the implications \((4)\Rightarrow(2)\), \((3)\Rightarrow(5)\) and \((5)\Rightarrow(4)\), the others following from previous remarks.

MSC:
11S31 Class field theory; \(p\)-adic formal groups
14F30 \(p\)-adic cohomology, crystalline cohomology
14L05 Formal groups, \(p\)-divisible groups
14F40 de Rham cohomology and algebraic geometry
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