Skip to main content
SpringerLink
Log in

Realisations of Kummer–Chern–Eisenstein-motives

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

Inspired by work of G. Harder we construct via the motive of a Hilbert modular surface an extension of a Tate motive by a Dirichlet motive. We compute the realisation classes and indicate how this is linked to the Hodge-1-motive of the given Hilbert modular surface.

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. Ash, A., Mumford, D., Rapoport, M., Tai, Y.: Smooth Compactification of locally symmetric Varieties. Lie Groups: History, Frontiers and Applications, vol. IV. Math. Sci. Press, Brookline (1975)

  2. Barbieri-Viale, L., Rosenschon, A., Saito, M.: Deligne’s conjecture on 1-motives. Ann. Math. (2) 158(2), 593–633 (2003)

    Google Scholar 

  3. Carlson J.A. (1985): The one-motive of an algebraic surface. Compositio Mathematica 56, 271–314

    MathSciNet  Google Scholar 

  4. Caspar, A.: Kummer–Chern–Eisenstein motives via Hilbert modular surfaces. Dissertation, Rheinische Friedrich-Wilhelms-Universität, Bonn, 2003. Bonner Mathematische Schriften [Bonn Mathematical Publications], 358. Universität Bonn, Mathematisches Institut, Bonn, 2003

  5. Deligne P. (1974): Théorie de Hodge III. Inst. Hautes Études Sci. Publ. Math. 44, 5–77

    MathSciNet  Google Scholar 

  6. Deligne, P.: Varietés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques. Automorphic forms, representations and L-functions. In: Proceedings Symposium Pure Mathematical, Oregon State University, Corvallis, Ore., 1977, Part 2, pp. 247–289. American Mathematical Society, Providence, R.I. (1979)

  7. Deligne, P.: Valeurs de Fonctions L et Périodes d’intégrales. Automorphic forms, representations and L-functions. In: Proceedings Symposium Pure Mathematical, Oregon State University, Corvallis, Ore., 1977, Part 2, pp. 313–346. American Mathematical Society, Providence, R.I. (1979)

  8. van der Geer, G.: Hilbert Modular Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3. Folge) Band 16, Springer, Berlin Heidelberg New York (1988)

  9. Harder, G.: On the Cohomology of \({SL(2,\mathcal{O})}\) . Lie Groups and their Representations. In: Proceedings Summer School on Group Representations of the Bolyai János Math. Soc., Budapest, 1971, pp. 139–150. Halsted, New York (1975)

  10. Harder G., Langlands R., Rapoport M. (1986): Algebraische Zyklen auf Hilbert-Blumenthal-Flächen. Journal für die Reine und Angewandte Mathematik 366, 53–120

    MathSciNet  Google Scholar 

  11. Harder G. (1987): Eisenstein Cohomology of arithmetic Groups. The Case GL2. Inventiones Mathematicae 89(1): 37–118

    Article  MathSciNet  Google Scholar 

  12. Harder, G.: Kohomologie Arithmetischer Gruppen. Lecture Notes, Bonn (1987–1988)

  13. Harder, G.: Eisenstein Cohomology of Arithmetic Groups and its Application to Number Theory. In: Proceedings of the ICM, Kyoto, 1990, pp. 779–790. Mathematical Society, Tokyo, Japan (1991)

  14. Harder G. (1993): Eisensteinkohomologie und die Konstruktion gemischter Motive. Lecture Notes in Mathematics, 1562. Springer, Berlin Heidelberg New York

    Google Scholar 

  15. Harder, G.: Eisenstein Cohomology, Extension of Motives and special Values of L - functions (an Example). Notes

  16. Harder, G.: Modular construction of mixed motives II. Preprint, http://www.math. uni-bonn.de/people/harder/Manuscripts/Eisenstein/ MixMot.pdf.

  17. Hirzebruch F. (1973): Hilbert modular surfaces. Enseign. Math. 19, 183–281

    MathSciNet  Google Scholar 

  18. Jannsen, U.: Mixed Motives, Motivic Cohomology and Ext-Groups. In: Proceedings of the ICM, vol. 1, Zürich, 1994, pp. 667–679. Birkhäuser, Basel (1995)

  19. Kleinjung, Th.: Gerahmte gemischte Tate-Motive und die Werte von Zetafunktionen zu Zahlkörpern an den Stellen 2 und 3. (German) [Framed mixed Tate motives and the values of zeta functions for number fields at the places 2 and 3] Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 2000. Bonner Mathematische Schriften [Bonn Mathematical Publications], 340. Universität Bonn, Mathematisches Institut, Bonn, 2001

  20. Ramachandran N. (2004): One-motives and a conjecture of deligne. J. Algebr. Geom. 13(1): 29–80

    MathSciNet  Google Scholar 

  21. Rapoport M. (1978): Compactifications de l’espace de modules de Hilbert - Blumenthal. Compositio Mathematica 36(3): 255–335

    MathSciNet  Google Scholar 

  22. Scholl, A.J.: Height pairings and special values of L-functions motives. In: Proceedings Symposium on Pure Mathematics, vol. 55, (Seattle, WA, 1991), Part 1, pp. 571–598. American Mathematical Society, Providence, RI, 1994

  23. Serre, J.P.: Le problème des groupes de congruence pour SL 2. Ann. Math. (2) 92, 489–527 (1970)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland

    Alexander Caspar

Authors
  1. Alexander Caspar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Caspar.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Caspar, A. Realisations of Kummer–Chern–Eisenstein-motives. manuscripta math. 122, 23–57 (2007). https://doi.org/10.1007/s00229-006-0052-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-006-0052-4

Mathematics Subject Classification

Search

Navigation