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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
##### Keywords:
Galois representation; crystalline representation
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##### References:
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