Liu, Tong Torsion \(p\)-adic Galois representations and a conjecture of Fontaine. (English) Zbl 1163.11043 Ann. Sci. Éc. Norm. Supér. (4) 40, No. 4, 633-674 (2007). 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”. Reviewer: Laurent Berger (Lyon) Cited in 3 ReviewsCited in 38 Documents MSC: 11F80 Galois representations 11F85 \(p\)-adic theory, local fields 11S20 Galois theory 14F30 \(p\)-adic cohomology, crystalline cohomology Keywords:\(p\)-adic Hodge theory; semi-stable representations × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] Berger L. , Colmez P. , Familles de représentations de de Rham et monodromie p -adique , Preprint, available at, http://www.ihes.fr/ lberger/ . · Zbl 1168.11020 [2] Berger L. , Limites de représentations cristallines , Compos. Math. 140 ( 6 ) ( 2004 ) 1473 - 1498 . MR 2098398 | Zbl 1071.11067 · Zbl 1071.11067 · doi:10.1112/S0010437X04000879 [3] Bondarko M.V. , Finite flat commutative group schemes over complete discrete valuation rings iii: classification, tangent spaces, and semistable reduction of Abelian varieties , Preprint, available at, http://arxiv.org/abs/math/0412521 . 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