×

zbMATH — the first resource for mathematics

Filtered modules with coefficients. (English) Zbl 1251.11044
Let \(E\) be a finite extension of \(\mathbb Q_p\). Arithmetic geometry, and in particular elliptic curves and modular forms, is a source of continuous representations \(\rho : \mathrm{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p) \to \mathrm{GL}_2(E)\). Some of these representations are potentially semi-stable in the sense of Fontaine and are then classified by \(E\)-linear filtered \((\varphi,N,G_{K/\mathbb Q_p})\)-modules (see J.-M. Fontaine [Representations \(p\)-adiques semi-stables, Asterisque 223, 113–184 (1994; Zbl 0865.14009)]). These objects are much simpler than the original representation \(\rho\), but even in dimension \(2\) it can be tricky to classify them all. The purpose of the article under review is to write down a list of these objects and hence to “essentially classify [them]”. The list is given in terms of the type of the representation. This type can be special, ramified principal series, unramified supercuspidal or ramified supercuspidal. In each case, the corresponding modules are given explicitly.

MSC:
11F80 Galois representations
PDF BibTeX XML Cite
Full Text: DOI
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] L. Berger. Représentations modulaires de \( \operatorname{GL}_2(\mathbf{Q}_p)\) et représentations galoisiennes de dimension \( 2\), To appear in Astérisque.
[3] L Berger and C. Breuil. Sur quelques représentations potentiellement cristallines de \( \operatorname{GL}_2(\mathbf{Q}_p)\), To appear in Astérisque.
[4] C. Breuil. Lectures on \( p\)-adic Hodge theory, deformations and local Langlands, Volume 20, Advanced course lecture notes. Centre de Recerca Matemàtica, Barcelona (2001).
[5] 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
[6] Christophe Breuil, Invariant \Bbb L et série spéciale \?-adique, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 4, 559 – 610 (French, with English and French summaries). · Zbl 1166.11331 · doi:10.1016/j.ansens.2004.02.001 · doi.org
[7] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over \?: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843 – 939. · Zbl 0982.11033
[8] C. Breuil and M. Emerton. Représentations \( p\)-adiques ordinaires de \( \operatorname{GL}_2(\mathbf{Q}_p)\) et compatibilité local-global, To appear in Astérisque.
[9] 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
[10] C. Breuil and A. Mézard. Représentations semi-stables de \( \operatorname{Gl}_2(\mathbf{Q}_p)\), demi-plan \( p\)-adique et réduction modulo \( p\), To appear in Astérisque.
[11] P. Colmez. Une correspondance de Langlands locale \( p\)-adique pour les représentations semi-stables de dimension \( 2\), To appear in Astérisque.
[12] P. Colmez. Série principale unitaire pour \( \operatorname{GL}_2(\mathbf{Q}_p)\) et représentations triangulines de dimension \( 2\), To appear in Astérisque.
[13] P. Colmez. La série principale unitaire de \( \operatorname{GL}_2(\mathbf{Q}_p)\), To appear in Astérisque.
[14] 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
[15] Matthew Emerton, \?-adic \?-functions and unitary completions of representations of \?-adic reductive groups, Duke Math. J. 130 (2005), no. 2, 353 – 392. · Zbl 1092.11024
[16] Jean-Marc Fontaine, Représentations \?-adiques semi-stables, Astérisque 223 (1994), 113 – 184 (French). With an appendix by Pierre Colmez; Périodes \?-adiques (Bures-sur-Yvette, 1988). · Zbl 0865.14009
[17] 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
[18] 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
[19] Eknath Ghate and Vinayak Vatsal, On the local behaviour of ordinary \Lambda -adic representations, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 7, 2143 – 2162 (2005) (English, with English and French summaries). · Zbl 1131.11341
[20] E. Ghate and V. Vatsal. Locally indecomposable Galois representations, To appear in Canad. J. Math. · Zbl 1277.11055
[21] Kenkichi Iwasawa, Local class field theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Mathematical Monographs. · Zbl 0604.12014
[22] B. Mazur, On monodromy invariants occurring in global arithmetic, and Fontaine’s theory, \?-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991) Contemp. Math., vol. 165, Amer. Math. Soc., Providence, RI, 1994, pp. 1 – 20. · Zbl 0846.11039 · doi:10.1090/conm/165/01599 · doi.org
[23] Toshitsune Miyake, Modular forms, Springer-Verlag, Berlin, 1989. Translated from the Japanese by Yoshitaka Maeda. · Zbl 0701.11014
[24] 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
[25] 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.