×

zbMATH — the first resource for mathematics

Construction of semi-stable \(p\)-adic representations. (Construction de représentations \(p\)-adiques semi-stables.) (French) Zbl 0907.14006
Let \(k\) be a perfect field of characteristic \(p > 0\) and let \(W\) denote the ring of Witt vectors of \(k\) with quotient field \(K_0\). In Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 547-608 (1982; Zbl 0579.14037), J.-M. Fontaine and G. Laffaille constructed crystalline representations of the absolute Galois group \(G\) of \(K_0\) via strongly divisible \(W\)-modules. The author extends this approach to semi-stable representations of \(G\) using ideas of G. Faltings [J. Algebr. Geom. 1, No. 1, 61-81 (1992; Zbl 0813.14012)]. In the special case of dimension \(2\) he constructs all \(2\)-dimensional semi-stable \(p\)-adic representations of \(G\) with differences in Hodge-Tate weights not exceeding \(p-2\).

MSC:
14F30 \(p\)-adic cohomology, crystalline cohomology
13K05 Witt vectors and related rings (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] P. BERTHELOT , Slopes of Frobenius in crystalline cohomology , in Algebraic Geometry (Arcata 1974 ) ; Proc. of Symposia in Pure Maths. 29 (Americ. Math. Soc., 1975 , p. 315-328). MR 52 #3167 | Zbl 0323.14008 · Zbl 0323.14008
[2] C. BREUIL , Topologie log-syntomique, cohomologie log-cristalline et cohomologie de Čech (Bull. Soc. math. France, vol. 124, 1996 , p. 587-647). Numdam | MR 98d:14025 | Zbl 0865.19004 · Zbl 0865.19004 · smf.emath.fr · numdam:BSMF_1996__124_4_587_0 · eudml:87752
[3] C. BREUIL , Représentations p-adiques semi-stables et transversalité de Griffiths (Math. Annalen, vol. 307, 1997 , p. 191-224). MR 98b:14016 | Zbl 0883.11049 · Zbl 0883.11049 · doi:10.1007/s002080050031
[4] C. BREUIL , Cohomologie étale de p-torsion et cohomologie cristalline en réduction semi-stable , preprint, Université Paris-Sud, 1997 .
[5] G. FALTINGS , Crystalline cohomology and p-adic Galois representations , Algebraic Analysis, Geometry and Number Theory, John Hopkins Univ. Press, 1989 , p. 25-79. MR 98k:14025 | Zbl 0805.14008 · Zbl 0805.14008
[6] G. FALTINGS , Cristalline cohomology of semi-stable curves, and p-adic Galois representations (Journal of Algebraic Geometry, vol. 1, 1992 , p. 61-82). MR 93e:14025 | Zbl 0813.14012 · Zbl 0813.14012
[7] J.-M. FONTAINE , Le corps des périodes p-adiques (Astérisque, vol. 223, Soc. Math. de France, 1994 , p. 59-111). MR 95k:11086 | Zbl 0940.14012 · Zbl 0940.14012
[8] J.-M. FONTAINE , Représentations p-adiques semi-stables (Astérisque, vol. 223, Soc. Math. de France, 1994 , p. 113-184). MR 95g:14024 | Zbl 0865.14009 · Zbl 0865.14009
[9] J.-M. FONTAINE , Cohomologie de de Rham, cohomologie cristalline et représentations p-adiques (Lecture Notes in Maths, vol. 1016, 1983 , p. 86-108, p. 113-184). MR 85f:14019 | Zbl 0596.14015 · Zbl 0596.14015
[10] J.-M. FONTAINE , Sur certains types de représentations p-adiques du groupe de Galois d’un corps local ; construction d’un anneau de Barsotti-Tate (Ann. of Maths, vol. 115, 1982 , p. 379-409). MR 84d:14010 | Zbl 0544.14016 · Zbl 0544.14016 · doi:10.2307/2007012
[11] J.-M. FONTAINE , Groupes p-divisibles sur les corps locaux (Astérisque, vol. 47-48, Soc. Math. de France, 1977 ). MR 58 #16699 | Zbl 0377.14009 · Zbl 0377.14009
[12] J.-M. FONTAINE et G. LAFFAILLE , Construction de représentations p-adiques (Ann. Scient. E.N.S., vol. 15, 1982 , p. 547-608). Numdam | MR 85c:14028 | Zbl 0579.14037 · Zbl 0579.14037 · numdam:ASENS_1982_4_15_4_547_0 · eudml:82106
[13] J.-M. FONTAINE et B. MAZUR , Geometric Galois representations , in Conference on Elliptic curves and modular forms (Hong-Kong 1993 ), 1995 , p. 41-78. MR 96h:11049 | Zbl 0839.14011 · Zbl 0839.14011
[14] L. ILLUSIE , Cohomologie de de Rham et cohomologie étale p-adique [d’après G. Faltings, J.M. Fontaine et al.] (Séminaire Bourbaki, vol. 726, juin 1990 . Numdam | Zbl 0736.14005 · Zbl 0736.14005 · numdam:SB_1989-1990__32__325_0 · eudml:110129
[15] K. KATO , Semi-stable reduction and p-adic etale cohomology (Astérisque, vol. 223, Soc. Math. de France, 1994 , p. 269-293). MR 95i:14020 | Zbl 0847.14009 · Zbl 0847.14009
[16] G. LAFFAILLE , Groupes p-divisibles et modules filtrés : le cas peu ramifié (Bull. Soc. math. France, vol. 108, 1980 , p. 187-206). Numdam | MR 82i:14028 | Zbl 0453.14021 · Zbl 0453.14021 · numdam:BSMF_1980__108__187_0 · eudml:87369
[17] B. MAZUR , Deforming Galois representations , in Galois Groups over Q, Springer 1989 , p. 385-438. MR 90k:11057 | Zbl 0714.11076 · Zbl 0714.11076
[18] B. MAZUR , On monodromy invariants occuring in global arithmetic, and Fontaine’s theory (Contemporary Mathematics, vol. 165, 1994 , p. 1-20). MR 95e:11069 | Zbl 0846.11039 · Zbl 0846.11039
[19] B. MAZUR , J. TATE et J. TEITELBAUM , On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer (Inv. Math., vol. 84, 1986 , p. 1-48). MR 87e:11076 | Zbl 0699.14028 · Zbl 0699.14028 · doi:10.1007/BF01388731 · eudml:143332
[20] B. PERRIN-RIOU , Représentations p-adiques ordinaires (Astérisque, vol. 223, Soc. Math. de France, 1994 , p. 185-220). MR 96h:11063 | Zbl 1043.11532 · Zbl 1043.11532
[21] T. SAITO , Modular forms and p-adic Hodge theory (Inv. Math., vol. 129, 1997 , p. 607-620). MR 98g:11060 | Zbl 0877.11034 · Zbl 0877.11034 · doi:10.1007/s002220050175
[22] J.-P. SERRE , Propriétés galoisiennes des points d’ordre fini des courbes elliptiques (Inv. Math., vol. 15, 1972 , p. 259-331). MR 52 #8126 | Zbl 0235.14012 · Zbl 0235.14012 · doi:10.1007/BF01405086 · eudml:142133
[23] T. TSUJI , P-adic-étale cohomology and crystalline cohomology in the semi-stable reduction case , preprint, Kyoto, 1996 .
[24] N. WACH , Représentations cristallines de torsion (Comp. Math., vol. 108, 1997 , p. 185-240). MR 98j:11112 | Zbl 0902.11051 · Zbl 0902.11051 · doi:10.1023/A:1000108818774
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.