×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.