Skip to main content
SpringerLink
Log in

Syntomic regulators andp-adic integration I: Rigid syntomic regulators

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We construct a new version of syntomic cohomology, called rigid syntomic cohomology, for smooth schemes over the ring of integers of ap-adic field. This version is more refined than previous constructions and naturally maps to most of them. We construct regulators fromK-theory into rigid syntomic cohomology. We also define a “modified” syntomic cohomology, which is better behaved in explicit computations yet is isomorphic to rigid syntomic cohomology in most cases of interest.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. Besser and R. de Jeu,The syntomic regulator for K-theory of fields, in preparation, 1999.

  2. A. A. Beilinson,Higher regulators and values of L-functions, Journal of Soviet Mathematics30 (1985), 2036–2070.

    Article  MATH  Google Scholar 

  3. A. A. Beilinson,Notes on absolute Hodge cohomology, inApplications of Algebraic K-theory to Algebraic Geometry and Number Theory, Part I, II (Boulder, Colo., 1983),Contemporary Mathematics, Vol. 55, American Mathematical Society, Providence, RI, 1986, pp. 35–68.

    Chapter  Google Scholar 

  4. P. Berthelot,Cohomologie rigide et cohomologie rigide a support propres, premier partie, Preprint 96-03 of the University of Rennes, available at http://www.maths.univ-rennes1.fr/~berthelo/, 1996.

  5. P. Berthelot,Finitude et pureté cohomologique en cohomologie rigide, Inventiones Mathematicae128 (1997), 329–377, with an appendix in English by A. J. de Jong.

    Article  MathSciNet  Google Scholar 

  6. A. Besser,A generalization of Coleman’s integration theory, Preprint available at http://www.math.bgu.ac.il/~bessera/padic/coleman.html, 1999.

  7. A. Besser,Syntomic regulators and p-adic integration II: K 2 of curves, this volume.

  8. R. Coleman and E. de Shalit,p-adic regulators on curves and special values of p-adic L-functions, Inventiones Mathematicae93 (1988), 239–266.

    Article  MathSciNet  MATH  Google Scholar 

  9. B. Chiarellotto,Weights in rigid cohomology applications to unipotent F-isocrystals, Annales Scientifiques de l’École Normale Supérieure31 (1998), 683–715.

    Article  MathSciNet  MATH  Google Scholar 

  10. B. Chiarellotto and B. le Stum,Sur la pureté de la cohomologie cristalline, Comptes Rendus de l’Académie des Sciences, Paris, Série I326 (1998), 961–963.

    Google Scholar 

  11. R. Coleman,Torsion points on curves and p-adic abelian integrals, Annals of Mathematics121 (1985), 111–168.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Deligne,Théorie de Hodge. II, Publications Mathématiques de l’Institut des Hautes Études Scientifiques40 (1971), 5–57.

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Deligne,Théorie de Hodge. III, Publications Mathématiques de l’Institut des Hautes Études Scientifiques44 (1974), 5–77.

    Article  MathSciNet  MATH  Google Scholar 

  14. R. De Jeu,Zagier’s conjecture and wedge complexes in algebraic K-theory, Compositio Mathematica96 (1995), 197–247.

    MathSciNet  MATH  Google Scholar 

  15. J. M. Fontaine and W. Messing,p-adic periods and p-adic étale cohomology, inCurrent Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985), Contemporary Mathematics, Vol. 67, American Mathematical Society, Providence, RI, 1987, pp. 179–207.

    Chapter  Google Scholar 

  16. M. Gros,Régulateurs syntomiques et valeurs de fonctions L p-adiques. I, Inventiones Mathematicae99 (1990), 293–320, with an appendix by Masato Kurihara.

    Article  MathSciNet  Google Scholar 

  17. M. Gros,Régulateurs syntomiques et valeurs de fonctions L p-adiques. II, Inventiones Mathematicae115 (1994), 61–79.

    Article  MathSciNet  Google Scholar 

  18. A. Huber,Mixed motives and their realization in derived categories, Lecture Notes in Mathematics1604, Springer-Verlag, Berlin, 1995.

    MATH  Google Scholar 

  19. U. Jannsen,Continuous étale cohomology, Mathematische Annalen280 (1988), 207–245.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Kolster and T. Nguyen Quang Do,Syntomic regulators and special values of p-adic l-functions, Inventiones Mathematicae133 (1998), 417–447.

    Article  MathSciNet  MATH  Google Scholar 

  21. P. Monsky and G. Washnitzer,Formal cohomology. I, Annals of Mathematics88 (1968), 181–217.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Nekovář,Syntomic cohomology and p-adic regulators, 1998. Preprint available from http://can.dpmms.cam.ac.uk/~nekovar/papers/

  23. W. Nizioł,Cohomology of crystalline representations, Duke Mathematical Journal71 (1993), 747–791.

    Article  MathSciNet  MATH  Google Scholar 

  24. W. Nizioł,On the image of p-adic regulators, Inventiones Mathematicae127 (1997), 375–400.

    Article  MathSciNet  MATH  Google Scholar 

  25. W. Nizioł,Cohomology of crystalline smooth sheaves, Compositio Mathematica, to appear, 1999.

  26. A. Ogus,The convergent topos in characteristic p, inThe Grothendieck Festschrift, Vol. III, Progress in Mathematics, Vol. 88, Birkhäuser, Boston, 1990, pp. 133–162.

    Chapter  Google Scholar 

  27. B. Perrin-Riou,Fonctions L p-adiques des représentations p-adiques, Astérisque229 (1995), 1–198.

    MathSciNet  MATH  Google Scholar 

  28. B. Saint-Donat,Exp. v bis> :Techniques de descente cohomologique, inThéorie des topos et cohomologie étale des schémas Tome 2, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Lecture Notes in Mathematics270, Springer-Verlag, Berlin, 1972, pp. 83–162.

    Chapter  Google Scholar 

  29. M. Somekawa,Log-syntomic regulators and p-adic polylogarithms, K-Theory17 (1999), 265–294.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. van der Put,The cohomology of Monsky and Washnitzer, Mémoires de la Société Mathématique de France (N.S.)23 (1986), 33–59; Introductions aux cohomologiesp-adiques, Luminy, 1984.

    MathSciNet  MATH  Google Scholar 

  31. M. van der Put and P. Schneider,Points and topologies in rigid geometry, Mathematische Annalen302 (1995), 81–103.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Author notes

  1. Amnon Besser

    Present address: Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, 84105, Be’er-Sheva, Israel

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Durham Science Laboratories, South Road, DH1 3LE, Durham, England

    Amnon Besser

Authors
  1. Amnon Besser
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amnon Besser.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Besser, A. Syntomic regulators andp-adic integration I: Rigid syntomic regulators. Isr. J. Math. 120, 291–334 (2000). https://doi.org/10.1007/BF02834843

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02834843

Keywords

Search

Navigation