Limits of crystalline representations. (Limites de représentations cristallines.) (French. English summary) Zbl 1071.11067

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).


11S25 Galois cohomology
11F80 Galois representations
11R23 Iwasawa theory
13K05 Witt vectors and related rings (MSC2000)
14F30 \(p\)-adic cohomology, crystalline cohomology


Zbl 0579.14037
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