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Existence of \(F\)-crystals with supplementary structures. (Existence de \(F\)-cristaux avec structures supplémentaires.) (French. English summary) Zbl 1104.14013
Let \(k\) be an algebraically closed field of characteristic \(p\), let \(W(k)\) be the Witt vectors over \(k\), and let \(\sigma\) be the absolute Frobenius. A crystal is a \(W(k)\)-module \(M\) free of finite rank with a \(\sigma\)-semi-linear bijective map \(\phi : M[1/p] \to M[1/p]\). One can associate to such an object two polygons : the Newton polygon (coming from the Dieudonné-Manin decomposition of \(M[1/p]\)) and the Hodge polygon (coming from the elementary divisors of \(\phi(M) \subset M\)). It is a theorem of B. Mazur [Bull. Am. Math. Soc. 78, 653–667 (1972; Zbl 0258.14006)] that the Hodge polygon lies below the Newton polygon and that they have the same endpoints. Conversely, given two polygons satisfying Mazur’s condition, Kottwitz and Rapoport have shown that there exists a crystal having those two polygons as Newton and Hodge polygons.
The article under review is about the generalization of Kottwitz and Rapoport’s theorem to so-called \(G\)-isocrystals, where \(G\) is a connected reductive group over \(\mathbb{Q}_p\). The analogue of Mazur’s criterion in this situation is a theorem of M. Rapoport and M. Richartz [Compos. Math. 103, No. 2, 153–181 (1996; Zbl 0874.14008)], and the problem of the existence of isocrystals satisfying the criterion then becomes a problem about certain affine Deligne-Lusztig varieties being non-empty. The author states two conjectures of Kottwitz and Rapoport in the introduction, and proceeds to prove various cases of them. Some results had previously been obtained by R. Kottwitz and M. Rapoport [Comment. Math. Helv. 78, No. 1, 153–184 (2003; Zbl 1126.14023)] and J.-M. Fontaine and M. Rapoport [Bull. Soc. Math. Fr. 133, No. 1, 73–86 (2005; Zbl 1073.14025)] and C. Lucarelli (Leigh) [J. Inst. Math. Jussieu 3, No. 2, 165–183 (2004; Zbl 1054.14059); preprint
url{arXiv:math.NT/0211327}].

MSC:
14F30 \(p\)-adic cohomology, crystalline cohomology
20G25 Linear algebraic groups over local fields and their integers
20G40 Linear algebraic groups over finite fields
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