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Rank two filtered $$(\varphi ,N)$$-modules with Galois descent data and coefficients. (English) Zbl 1209.11050
Let $$K$$ and $$E$$ be a finite extension of $$\mathbb{Q}_p$$ and $$\rho:G_K\rightarrow\text{GL}_n(E)$$ be any continuous $$n$$-dimensional representation of $$G_K=\text{Gal}(\overline{\mathbb{Q}}_p/K)$$. For a finite Galois extension $$L/K$$, the representation is called $$L$$-semistable if the restriction of $$\rho$$ to $$G_L$$ is semistable. The main goal of the current paper under review is to classify two-dimensional $$L$$-semistable $$E$$-representations of $$G_K$$. To do so, the author uses a (variant of the) work of P. Colmez and J.-M. Fontaine [Invent. Math. 140, No.1, 1–43 (2000; Zbl 1010.14004)], in which they prove the equivalence of the following two categories:
(i) $$L$$-semistable $$E$$-representations of $$G_K$$ with Hodge-Tate weights within the range $$\{0,1,\dots, k-1\}$$.
(ii) Category of weakly admissible filtered $$(\varphi, N, L/K, E)$$-modules $$D$$ such that $$\text{Fil}^0(L\otimes_{L_0}D)=L\otimes_{L_0}D$$, and $$\text{Fil}^k(L\otimes_{L_0}D)=0$$.
Here $$L_0$$ is the maximal unramified extension of $$\mathbb{Q}_p$$ inside $$L$$.
The author uses his classification theorem to study crystalline representations. He proves that, when $$K\neq \mathbb{Q}_p$$, there exists disjoint infinite families of irreducible $$2$$-dimensional crystalline $$E$$-representations of $$G_K$$ that have the same characteristic polynomial and filtration.

MSC:
 11F80 Galois representations
Full Text:
References:
 [1] Laurent Berger, An introduction to the theory of \?-adic representations, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 255 – 292 (English, with English and French summaries). · Zbl 1118.11028 [2] Laurent Berger, Représentations \?-adiques et équations différentielles, Invent. Math. 148 (2002), no. 2, 219 – 284 (French, with English summary). · Zbl 1113.14016 · doi:10.1007/s002220100202 · doi.org [3] Christophe Breuil, Sur quelques représentations modulaires et \?-adiques de \?\?$$_{2}$$(\?_\?). II, J. Inst. Math. Jussieu 2 (2003), no. 1, 23 – 58 (French, with French summary). · Zbl 1165.11319 · doi:10.1017/S1474748003000021 · doi.org [4] Christophe Breuil and Ariane Mézard, Multiplicités modulaires et représentations de \?\?$$_{2}$$(\?_\?) et de \?\?\?(\?_\?/\?_\?) en \?=\?, Duke Math. J. 115 (2002), no. 2, 205 – 310 (French, with English and French summaries). With an appendix by Guy Henniart. · Zbl 1042.11030 · doi:10.1215/S0012-7094-02-11522-1 · doi.org [5] Christophe Breuil and Peter Schneider, First steps towards \?-adic Langlands functoriality, J. Reine Angew. Math. 610 (2007), 149 – 180. · Zbl 1180.11036 · doi:10.1515/CRELLE.2007.070 · doi.org [6] Pierre Colmez and Jean-Marc Fontaine, Construction des représentations \?-adiques semi-stables, Invent. Math. 140 (2000), no. 1, 1 – 43 (French). · Zbl 1010.14004 · doi:10.1007/s002220000042 · doi.org [7] Brian Conrad, Fred Diamond, and Richard Taylor, Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12 (1999), no. 2, 521 – 567. · Zbl 0923.11085 [8] Dousmanis, Gerasimos On reductions of families of two-dimensional crystalline Galois representations. http://arxiv.org/abs/0805.1634 arXiv:0805.1634v3 · Zbl 1246.11109 [9] Dousmanis, Gerasimos On reductions of families of two-dimensional crystalline Galois representations part II. http://arxiv.org/abs/0905.0080 arXiv:0905.0080v1 · Zbl 1246.11109 [10] Jean-Marc Fontaine, Le corps des périodes \?-adiques, Astérisque 223 (1994), 59 – 111 (French). With an appendix by Pierre Colmez; Périodes \?-adiques (Bures-sur-Yvette, 1988). · Zbl 0940.14012 [11] Jean-Marc Fontaine, Représentations \?-adiques potentiellement semi-stables, Astérisque 223 (1994), 321 – 347 (French). Périodes \?-adiques (Bures-sur-Yvette, 1988). · Zbl 0873.14020 [12] Jean-Marc Fontaine and Barry Mazur, Geometric Galois representations, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993) Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 41 – 78. · Zbl 0839.14011 [13] Fontaine, Jean-Marc; Ouyang, Yi Theory of $$p$$-adic Galois Representations. Forthcoming Springer book. · Zbl 1142.11335 [14] Eknath Ghate and Ariane Mézard, Filtered modules with coefficients, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2243 – 2261. · Zbl 1251.11044 [15] David Savitt, On a conjecture of Conrad, Diamond, and Taylor, Duke Math. J. 128 (2005), no. 1, 141 – 197. · Zbl 1101.11017 · doi:10.1215/S0012-7094-04-12816-7 · doi.org [16] Maja Volkov, Les représentations \?-adiques associées aux courbes elliptiques sur \Bbb Q_\?, J. Reine Angew. Math. 535 (2001), 65 – 101 (French, with English summary). · Zbl 1024.11038 · doi:10.1515/crll.2001.046 · doi.org
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