Wach, Nathalie Potentially crystalline \(p\)-adic representations. (Représentations \(p\)-adiques potentiellement cristallines.) (French) Zbl 0887.11048 Bull. Soc. Math. Fr. 124, No. 3, 375-400 (1996). 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. Reviewer: W.W.J.Hulsbergen (Haarlem) Cited in 19 Documents 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 Keywords:Galois representation; crystalline representation Citations:Zbl 0743.11066 × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML Link References: [1] FONTAINE (J.-M.) . - Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate , Journées arithmétiques de Rennes III, Astérisque, t. 65, 1979 , p. 3-80. MR 82k:14046 | Zbl 0429.14016 · Zbl 0429.14016 [2] FONTAINE (J.-M.) . - Sur certains types de représentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate , Ann. of Math., t. 115, 1982 , p. 529-577. MR 84d:14010 | Zbl 0544.14016 · Zbl 0544.14016 · doi:10.2307/2007012 [3] FONTAINE (J.-M.) . - Représentations p-adiques des corps locaux, The Grothendieck Festschrift , vol. II. - Birkhäuser, Boston, 1991 , p. 249-309. MR 92i:11125 | Zbl 0743.11066 · Zbl 0743.11066 [4] FONTAINE (J.-M.) . - Le corps des périodes p-adiques, exposé II, séminaire I.H.E.S 1988 , Astérisque, t. 223, 1994 , p. 59-111. MR 95k:11086 | Zbl 0940.14012 · Zbl 0940.14012 [5] FONTAINE (J.-M.) et ILLUSIE (L.) . - p-adic periods : a survey, exposé Bombay 1989 , Proceedings of the Indo-French Conference on Geometry, NBHM Hindustan Book Agency, Delhi, 1993 , p. 57-93. MR 95e:14013 | Zbl 0836.14010 · Zbl 0836.14010 [6] FONTAINE (J.-M.) et LAFFAILLE (G.) . - Construction de représentations p-adiques , Ann. Scient. École Normale Sup., t. 15, 1982 , p. 547-608. Numdam | MR 85c:14028 | Zbl 0579.14037 · Zbl 0579.14037 [7] SEN (S.) . - Continuous cohomology and p-adic Galois representations , Inv. Math., t. 62, 1980 , p. 89-116. MR 82e:12018 | Zbl 0463.12005 · Zbl 0463.12005 · doi:10.1007/BF01391665 [8] SERRE (J.-P.) . - Classes des corps cyclotomiques, d’après Iwasawa . - Séminaire Bourbaki, 1958 . Numdam | Zbl 0119.27603 · Zbl 0119.27603 [9] SERRE (J.-P.) . - Corps locaux , 2e éd. - Hermann, Paris, 1968 . MR 50 #7096 [10] WACH (N.) . - Représentations de torsion , à paraître dans Compositio Math. · Zbl 0902.11051 [11] WINTENBERGER (J.-P.) . - Le corps des normes de certaines extensions infinies de corps locaux; applications , Ann. Scient. École Normale Sup., t. 16, 4e série, 1983 , p. 59-89. Numdam | MR 85e:11098 | Zbl 0516.12015 · Zbl 0516.12015 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.