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Tame inertia weights of certain crystalline representations. (Poids de l’inertie modérée de certaines représentations cristallines.) (French. English summary) Zbl 1223.14022
Let $$V$$ be a $$p$$-adic representation of $$\mathrm{Gal}(K^{\mathrm{alg}}/K)$$ where $$K$$ is a $$p$$-adic field, and let $$\overline{V}$$ be the “reduction modulo $$p$$” of $$V$$. In their article “Hodge and Newton and tame inertia polygons of semi-stable representations” [Math. Ann. 343, No. 4, 773–789 (2009; Zbl 1248.11092)], the authors explain how to attach a polygon to $$\overline{V}$$, constructed from the weights of the tame inertia subgroup of $$\mathrm{Gal}(K^{\mathrm{alg}}/K)$$ acting on $$\overline{V}$$. They also proved that under certain conditions, this polygon is above the Hodge polygon which can be constructed from the Hodge-Tate weights of $$V$$, with the same endpoints.
The question then arises as to which polygons, lying above the Hodge polygon and having the same endpoints, can actually occur as the tame inertia polygon. In the article under review, the authors determine these polygons for a family of crystalline representations of dimension $$2$$ of $$\mathrm{Gal}(K^{\mathrm{alg}}/K)$$, where $$K$$ is allowed some ramification. In particular, they observe that the tame inertia polygon and the Hodge polygon need not be the same.

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 11S20 Galois theory 11F80 Galois representations
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##### References:
 [1] C. Breuil, Représentations $$p$$-adiques semi-stables et transversalité de Griffiths. Math. Annalen 307 (1997), 191-224. · Zbl 0883.11049 [2] C. Breuil, Integral $$p$$-adic Hodge theory. Advanced studies in pure mathematics 36 (2002), 51-80. · Zbl 1046.11085 [3] C. Breuil, A. Mézard, Multiplicités modulaires et représentations de $$\text{GL}_2(\mathbb{Z}_p)$$ et de $$\text{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$$ en $$ℓ = p$$. Duke math. J. 115 (2002), 205-310. · Zbl 1042.11030 [4] X. Caruso, Représentations semi-stables de torsion dans le cas $$er < p-1$$. J. reine angew. Math. 594 (2006), 35-92. · Zbl 1134.14013 [5] X. Caruso, D. Savitt, Polygones de Hodge, de Newton et de l’inertie modérée des représentations semi-stables. À paraître dans Math. Ann. · Zbl 1248.11092 [6] J. M. Fontaine, G. Laffaille, Construction de représentations $$p$$-adiques. Ann. Sci. École Norm. Sup. 15 (1982), 547-608. · Zbl 0579.14037 [7] N. Katz, Slope filtration of $$F$$-crystals. Astérisque 63 (1979), 113-164. · Zbl 0426.14007
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