Some new directions in p -adic Hodge theory

Kiran S. Kedlaya[1]

  • [1] Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139, USA

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 2, page 285-300
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Kedlaya, Kiran S.. "Some new directions in $p$-adic Hodge theory." Journal de Théorie des Nombres de Bordeaux 21.2 (2009): 285-300. <http://eudml.org/doc/10881>.

@article{Kedlaya2009,
abstract = {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.},
affiliation = {Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139, USA},
author = {Kedlaya, Kiran S.},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {2},
pages = {285-300},
publisher = {Université Bordeaux 1},
title = {Some new directions in $p$-adic Hodge theory},
url = {http://eudml.org/doc/10881},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Kedlaya, Kiran S.
TI - Some new directions in $p$-adic Hodge theory
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 2
SP - 285
EP - 300
AB - 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.
LA - eng
UR - http://eudml.org/doc/10881
ER -

References

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  2. L. Berger, Représentations p -adiques et équations différentielles. Invent. Math. 148 (2002), 219–284. Zbl1113.14016MR1906150
  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. Zbl1118.11028MR2023292
  4. L. Berger, Construction de ( φ , Γ )-modules: représentations p -adiques et B -paires. Alg. and Num. Theory 2 (2008), 91–120. Zbl1219.11078MR2377364
  5. R. Coleman, B. Mazur, The eigencurve. Galois representations in arithmetic algebraic geometry (Durham, 1996), Cambridge Univ. Press, 1998, 1–113. Zbl0932.11030MR1696469
  6. P. Colmez, Représentations triangulines de dimension 2. Astérisque 319 (2008), 213–258. Zbl1168.11022MR2493219
  7. P. Colmez, Représentations de GL 2 ( ℚ p ) et ( φ , Γ ) -modules. Preprint (2007) available at http://www.math.jussieu.fr/~colmez/. MR2482309
  8. P. Colmez, La série principale unitaire de GL 2 ( ℚ p ) . Preprint (2007) available at http://www.math.jussieu.fr/~colmez/. MR902277
  9. P. Colmez, ( φ , Γ ) -modules et représentations du mirabolique de GL 2 ( ℚ p ) . Preprint (2007) available at http://www.math.jussieu.fr/~colmez/. MR2482309
  10. L. Herr, Sur la cohomologie galoisienne des corps p -adiques. Bull. S.M.F. 126 (1998), 563–600. Zbl0967.11050MR1693457
  11. H. Hida, Galois representations into GL 2 ( ℤ p 〚 X 〛 ) attached to ordinary cusp forms. Invent. Math. 85 (1986), 545–613. Zbl0612.10021MR848685
  12. K.S. Kedlaya, Slope filtrations for relative Frobenius. Astérisque 319 (2008), 259–301. Zbl1168.11053MR2493220
  13. M. Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture. Invent. Math. 153 (2003), 373–454. Zbl1045.11029MR1992017
  14. M. Kisin, Crystalline representations and F -crystals. Algebraic geometry and number theory, Progress in Math. 253, Birkhäuser, Boston, 2006, 459–496. Zbl1184.11052MR2263197
  15. R. Liu, Cohomology and duality for ( φ , Γ ) -modules over the Robba ring. Int. Math. Res. Notices 2008, article ID rnm150 (32 pages). Zbl1248.11093MR2416996
  16. J.S. Milne, Arithmetic duality theorems. BookSurge, 2006. Zbl1127.14001MR2261462
  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. Zbl0278.14003MR382272
  19. C.A. Weibel, An introduction to homological algebra. Cambridge Univ. Press, 1994. Zbl0797.18001MR1269324

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