Some new directions in p-adic Hodge theory
Journal de Théorie des Nombres de Bordeaux, Volume 21 (2009) no. 2, pp. 285-300.

We recall some basic constructions from p-adic Hodge theory, then describe some recent results in the subject. We chiefly discuss the notion of B-pairs, introduced recently by Berger, which provides a natural enlargement of the category of p-adic Galois representations. (This enlargement, in a different form, figures in recent work of Colmez, Bellaïche, and Chenevier on trianguline representations.) We also discuss results of Liu that indicate that the formalism of Galois cohomology, including Tate local duality, extends to B-pairs.

Nous rappelons quelques constructions fondamentales de la théorie de Hodge p-adique, et décrivons ensuite quelques résultats nouveaux dans ce domaine. Nous traitons principalement la notion de B-paire, introduite récemment par Berger, qui fournit une extension naturelle de la catégorie des représentations Galoisiennes p-adiques. (Sous une autre forme, cette extension figure dans les travaux récents de Colmez, Bellaïche et Chenevier sur les représentations triangulables.) Nous discutons aussi quelques résultats de Liu qui indiquent que le formalisme de la cohomologie Galoisienne, y compris la dualité locale de Tate, se prolonge aux B-paires.

Published online:
DOI: 10.5802/jtnb.671
Kiran S. Kedlaya 1

1 Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139, USA
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Kiran S. Kedlaya. Some new directions in $p$-adic Hodge theory. Journal de Théorie des Nombres de Bordeaux, Volume 21 (2009) no. 2, pp. 285-300. doi : 10.5802/jtnb.671. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.671/

[1] J. Bellaïche, G. Chenevier, p-adic families of Galois representations and higher rank Selmer groups. Astérisque, to appear; arXiv preprint math/0602340v2 (2007).

[2] L. Berger, Représentations p-adiques et équations différentielles. Invent. Math. 148 (2002), 219–284. | MR: 1906150 | Zbl: 1113.14016

[3] L. Berger, An introduction to the theory of p-adic representations. Geometric aspects of Dwork theory. Vol. I, de Gruyter, Berlin, 2004, 255–292. | MR: 2023292 | Zbl: 1118.11028

[4] L. Berger, Construction de (φ,Γ)-modules: représentations p-adiques et B-paires. Alg. and Num. Theory 2 (2008), 91–120. | MR: 2377364 | Zbl: pre05529319

[5] R. Coleman, B. Mazur, The eigencurve. Galois representations in arithmetic algebraic geometry (Durham, 1996), Cambridge Univ. Press, 1998, 1–113. | MR: 1696469 | Zbl: 0932.11030

[6] P. Colmez, Représentations triangulines de dimension 2. Astérisque 319 (2008), 213–258. | MR: 2493219 | Zbl: pre05541498

[7] P. Colmez, Représentations de GL 2 ( p ) et (φ,Γ)-modules. Preprint (2007) available at . | MR: 2482309

[8] P. Colmez, La série principale unitaire de GL 2 ( p ). Preprint (2007) available at . | MR: 902277

[9] P. Colmez, (φ,Γ)-modules et représentations du mirabolique de GL 2 ( p ). Preprint (2007) available at . | MR: 2482309

[10] L. Herr, Sur la cohomologie galoisienne des corps p-adiques. Bull. S.M.F. 126 (1998), 563–600. | Numdam | MR: 1693457 | Zbl: 0967.11050

[11] H. Hida, Galois representations into GL 2 ( p X) attached to ordinary cusp forms. Invent. Math. 85 (1986), 545–613. | MR: 848685 | Zbl: 0612.10021

[12] K.S. Kedlaya, Slope filtrations for relative Frobenius. Astérisque 319 (2008), 259–301. | MR: 2493220 | Zbl: pre05541499

[13] M. Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture. Invent. Math. 153 (2003), 373–454. | MR: 1992017 | Zbl: 1045.11029

[14] M. Kisin, Crystalline representations and F-crystals. Algebraic geometry and number theory, Progress in Math. 253, Birkhäuser, Boston, 2006, 459–496. | MR: 2263197 | Zbl: pre05234060

[15] R. Liu, Cohomology and duality for (φ,Γ)-modules over the Robba ring. Int. Math. Res. Notices 2008, article ID rnm150 (32 pages). | MR: 2416996 | Zbl: pre05495297

[16] J.S. Milne, Arithmetic duality theorems. BookSurge, 2006. | MR: 2261462 | Zbl: 1127.14001

[17] J. Pottharst, Triangulordinary Selmer groups. ArXiv preprint 0805.2572v1 (2008).

[18] W. Schmid, Variation of Hodge structure: the singularities of the period map. Invent. Math. 22 (1973), 211–319. | MR: 382272 | Zbl: 0278.14003

[19] C.A. Weibel, An introduction to homological algebra. Cambridge Univ. Press, 1994. | MR: 1269324 | Zbl: 0797.18001

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