Let \(p\) be a prime number, and let \(K_0\) be a field of characteristic zero, complete with respect to a discrete valuation, absolutely non-ramified with perfect residue field \(k\) of characteristic \(p\). Fix an algebraic closure \(\overline{K}\) of \(K_0\) and let \(G_{K_0}=\text{Gal}(\overline{K}/K_0)\). One would like to describe \(p\)-adic representations of \(G_{K_0}\), in particular, one may ask when these come from algebraic geometry, and if so, can they be described by simpler objects. Here the question for crystalline representations in terms of Fontaine’s \((\varphi,\Gamma)\)-modules is answered.
For an integer \(n\geq 1\), one writes \(K_n\subset\overline{K}\) for the subfield generated over \(K_0\) by the \(p^n\)-th roots of unity, (\(p\neq 2\)). Write \(K_{\infty}=\cup_{n\geq 1}K_n\), \(\Gamma_{K_0}=\text{Gal}(\overline{K}/K_{\infty})\) and \(\Gamma_f=\Gamma_{K_0,\text{tors}}\). Define \(\Gamma_0=\Gamma/\Gamma_f\). \(\Gamma_0\) can be identified with the Galois group of the cyclotomic \({\mathbb{Z}}_p\)-extension of \(K_0\) contained in \(K_{\infty}\), \(p\neq 2\). For \(p=2\) one defines \(\Gamma_0=\Gamma\). J.-M. Fontaine defined an action of \(\Gamma_0\) on a certain ring \(S_0\) [The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 249-309 (1991; Zbl 0743.11066)]. Then, for \(h\in{\mathbb{N}}\), one may define the full subcategory \(\Gamma_0\Phi{\mathcal M} ^h_{S_0}\) of the category of \((\varphi,\Gamma_0)\)-modules over \(S_0\), and one writes \(\Gamma_0\Phi{\mathcal M}^h_{S_0,\text{tors}}\) for the full subcategory whose objects are of \(p\)-torsion.
Let \(\Gamma_0\Phi {\mathcal M}^+_{S_0}=\cup_{h\in{\mathbb{N}}}\Gamma_0\Phi{\mathcal M}^h_{S_0}\) and define the cr-height of an object \(M\) of \(\Gamma_0\Phi{\mathcal M}^+_{S_0}\) as the smallest \(h\) such that \(M\) is an object of \(\Gamma_0\Phi{\mathcal M}^h_{S_0}\). One constructs an exact and faithful functor \[ V^*_{\text{cr}}:\Gamma_0\Phi{\mathcal M}^h_{S_0,\text{tors}}\rightarrow \text{Rep}_{{\mathbb{Z}}_p,\text{tors}}(G_{K_0}) \] whose essential image is denoted \(\text{Rep}^h_{{\mathbb{Z}}_p,\text{tors,cr}}(G_{K_0})\), and is called the category of representations of finite cr-height \(\leq h\).
The main result of the paper is: For an integer \(h\) with \(0\leq h\leq p-1\) and a representation \(T\) in \(\text{Rep}_{{\mathbb{Z}}_p, \text{tors}}(G_{K_0})\), one has that \(T\) is in \(\text{Rep}^h_{{\mathbb{Z}}_p, \text{tors,cr}} (G_{K_0})\) iff \(T\) is isomorphic to a subquotient of a crystalline representation with Hodge-Tate weights in \([0,h]\). A \(p\)-adic representation \(V\) of \(G_{K_0}\) is said to be of cr-height \(\leq h\) if it is “isomorphic” with some object \(M\) of \(\Gamma_0\Phi{\mathcal M}^h_{S_0}\) (this can be made precise). One also has the notion of finite height for a \(p\)-adic representation \(V\) of \(G_{K_0}\). Then, if \(V\) is of finite cr-height, it is of finite height. One obtains: Let \(r,h\in{\mathbb{Z}}\), \(0\leq h\leq p-1\), and let \(V\) be a \(p\)-adic representation of \(G_{K_0}\). Then \(V\) is crystalline with Hodge-Tate weights in \([r,r+h]\) iff \(V(r)\) is a representation of finite cr-height \(\leq h\). Moreover, \(V\) is of finite height.
The proof of the main result is surprisingly long and technical. First, a detailed analysis of crystalline torsion representations is given to obtain the exact faithfulness of a functor \(V^*_{\text{cris}}\) restricted to a category of filtered modules \({\mathcal {MF}}^h_{W,\text{tors}}\) over the Witt ring \(W\) of \(k\), of height \(\leq h\), \(h\leq p-1\), with values in \(\text{Rep}_{{\mathbb{Z}}_p,\text{tors}} (G_{K_0})\). For \(h\leq p-2\), \(V^*_{\text{cris}}\) is even fully faithful. If \(\Lambda\) in \({\mathcal {MF}}^h_{\text{tors}}\) has finite length, then \(\text{length}_W\Lambda=\text{length}_{{\mathbb{Z}}_p}V^*_{\text{cris}}(\Lambda)\), and if \(\Lambda\) is free over \(W\), \(V^*_{\text{cris}}(\Lambda)\) is free over \({\mathbb{Z}}_p\) and \(\text{rank}_W(\Lambda)=\text{rank}_{{\mathbb{Z}}_p}V^*_{\text{cris}}(\Lambda)\). As a matter of fact, a construction of the rings of \(p\)-adic periods \(A_{\text{cris}}\) and \(B^+_{\text{cris}}\) à la Fontaine-Laffaille is given in a slightly different way. The next step leads to the construction of a filtration on objects of \(\Gamma_0\Phi{\mathcal M}^h_{S_0}\), \(h\leq p-1\), giving rise to an additive functor \(i^*:\Gamma_0\Phi{\mathcal M}^h_{S_0}\rightarrow{\mathcal {MF}}^h_W\), which is shown to be exact. It is faithful for \(h\leq p-2\). For \(h=p-1\) one must impose an additional condition. Actually, \(i^*\) induces an equivalence between \(\Gamma_0\Phi{\mathcal M}^h_{S_0}\) and \({\mathcal {MF}}^h_W\). Finally, one must “identify” the two representations induced by \(V^*_{\text{cr}}\) and \(V^*_{\text{cris}}\).