Berger, Laurent Limits of crystalline representations. (Limites de représentations cristallines.) (French. English summary) Zbl 1071.11067 Compos. Math. 140, No. 6, 1473-1498 (2004). Let \(F\) be the fraction field of the ring of Witt vectors over a perfect field of characteristic \(p\). Let \(G_F\) denote the absolute Galois group of \(F\). The author proves the following conjecture of Fontaine: a \(p\)-adic representation of \(G_F\), which is a limit of subquotients of crystalline representations with Hodge-Tate weights in an interval \([a,b]\), is itself crystalline with Hodge-Tate weights in \([a,b]\) (Theorem 1).In a case \(b-a\leq p-1\) this result follows immediately from the construction of J.-M. Fontaine and G. Laffaille [Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 547–608 (1982; Zbl 0579.14037)]. If \(b-a> p-1\), then the theory of Fontaine-Laffaille doesn’t apply, and in order to prove Theorem 1 the author studies \((\varphi,\Gamma)\)-modules attached to crystalline representations. He improves some results of Fontaine, Wach and Colmez concerning characterization of crystalline representations in terms of \((\varphi,\Gamma)\)-modules (Theorems 2 and 3). He also gives new proofs of some results of Fontaine and Laffaille (V.2). Reviewer: Andrzej Dąbrowski (Szczecin) Cited in 5 ReviewsCited in 51 Documents MSC: 11S25 Galois cohomology 11F80 Galois representations 11R23 Iwasawa theory 13K05 Witt vectors and related rings (MSC2000) 14F30 \(p\)-adic cohomology, crystalline cohomology Keywords:\(p\)-adic representation; crystalline representation; Wach module; Hodge-Tate representation; ring of \(p\)-adic periods; deformation theory Citations:Zbl 0579.14037 × Cite Format Result Cite Review PDF Full Text: DOI