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Torsion \(p\)-adic Galois representations and a conjecture of Fontaine. (English) Zbl 1163.11043
Let \(K\) be a finite extension of \(\mathbb{Q}_p\) and \(T\) a finite free \(\mathbb{Z}_p\)-representation of \(\mathrm{Gal}(\overline{K}/K)\). The author proves that \(\mathbb{Q}_p \otimes_{\mathbb{Z}_p} T\) is semi-stable (resp. crystalline) with Hodge-Tate weights in \(\{0,\dots,r\}\) if and only if, for all \(n\), \(T/p^n T\) is torsion semi-stable (resp. crystalline) with Hodge-Tate weights in \(\{0,\dots,r\}\), thereby proving a conjecture of Fontaine (some cases of which had been established by Fontaine-Laffaille, Breuil and Berger). The proof uses Kisin’s results about Breuil’s analogue of \((\varphi,\Gamma)\)-modules, for which all semi-stable representations are of “finite height”.

MSC:
11F80 Galois representations
11F85 \(p\)-adic theory, local fields
11S20 Galois theory
14F30 \(p\)-adic cohomology, crystalline cohomology
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