Skip to main content
SpringerLink
Log in

On p-adic Interpolation of Motivic Eisenstein Classes

  • Conference paper
  • First Online:
Elliptic Curves, Modular Forms and Iwasawa Theory (JHC70 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 188))

Abstract

In this paper we prove that the motivic Eisenstein classes associated to polylogarithms of commutative group schemes can be p-adically interpolated in étale cohomology. This connects them to Iwasawa theory and generalizes and strengthens the results for elliptic curves obtained in our former work. In particular, degeneration questions can be treated easily.

To John Coates, on the occasion of his 70th birthday

This research was supported by the DFG grant: SFB 1085 “Higher invariants”.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Berthelot, P., Ogus, A.: Notes on crystalline cohomology. University of Tokyo Press, Tokyo, Princeton University Press, Princeton (1978)

    MATH  Google Scholar 

  2. Brion, M., Szamuely, T.: Prime-to-\(p\) étale covers of algebraic groups and homogeneous spaces. Bull. Lond. Math. Soc. 45(3), 602–612 (2013). doi:10.1112/blms/bds110

    Article  MathSciNet  MATH  Google Scholar 

  3. Colmez, P.: Fonctions \(L\) \(p\)-adiques. Astérisque 266, Exp. No. 851, 3, pp. 21–58 (2000). Séminaire Bourbaki, vol. 1998/99. http://www.numdam.org/item?id=SB_1998-1999__41__21_0

  4. Deligne, P.: Cohomologie étale. In: Lecture Notes in Mathematics, vol. 569. Springer, Berlin-New York (1977)

    Google Scholar 

  5. Ekedahl, T.: On the adic formalism. In: The Grothendieck Festschrift, vol. II, Progr. Math., vol. 87, pp. 197–218. Birkhäuser Boston, Boston, MA (1990)

    Google Scholar 

  6. Huber, A., Kings, G.: Degeneration of \(l\)-adic Eisenstein classes and of the elliptic polylog. Invent. Math. 135(3), 545–594 (1999). doi:10.1007/s002220050295

    Article  MathSciNet  MATH  Google Scholar 

  7. Huber, A., Kings, G.: Polylogarithm for Families of Commutative Group Schemes (2015) (Preprint)

    Google Scholar 

  8. Illusie, L.: Complexe Cotangent et Déformations I. In: Lecture Notes in Mathematics, 239, vol. 239. Springer (1971)

    Google Scholar 

  9. Jannsen, U.: Continuous étale cohomology. Math. Ann. 280(2), 207–245 (1988). doi:10.1007/BF01456052

    Article  MathSciNet  MATH  Google Scholar 

  10. Katz, N.M.: \(P\)-adic interpolation of real analytic Eisenstein series. Ann. Math. (2)104(3), 459–571 (1976). doi:10.2307/1970966

    Google Scholar 

  11. Kings, G.: The Tamagawa number conjecture for CM elliptic curves. Invent. Math. 143(3), 571–627 (2001). doi:10.1007/s002220000115

    Article  MathSciNet  MATH  Google Scholar 

  12. Kings, G.: Eisenstein classes, elliptic Soulé elements and the \(\ell \)-adic elliptic polylogarithm. In: The Bloch–Kato conjecture for the Riemann zeta function, London Math. Soc. Lecture Note Ser., vol. 418, pp. 237–296. Cambridge University Press (2015)

    Google Scholar 

  13. Kings, G., Loeffler, D., Zerbes, S.: Rankin-Eisenstein classes and explicit reciprocity laws (2015) (Preprint)

    Google Scholar 

  14. Roos, J.E.: Sur les foncteurs dérivés de \(\underleftarrow{\lim }\) applications. C. R. Acad. Sci. Paris 252, 3702–3704 (1961)

    MATH  Google Scholar 

  15. Cohomologie \(l\)-adique et fonctions \(L\). Lecture Notes in Mathematics, vol. 589. Springer, Berlin-New York. Séminaire de Géometrie Algébrique du Bois-Marie 1965–1966 (SGA 5), Edité par Luc Illusie (1977)

    Google Scholar 

Download references

Acknowledgements

This paper is a sequel to [12], which was written as a response to John Coates’ wish to have an exposition of the results in [6] at a conference in Pune, India. This triggered a renewed interest of mine in the p-adic interpolation of motivic Eisenstein classes which goes back to [11]. In this sense the paper would not exist without the persistence of John Coates. The paper [7] with Annette Huber created the right framework to treat these questions. It is a pleasure to thank them both and to dedicate this paper to John Coates on the occasion of his seventieth birthday.

Author information

Authors and Affiliations

  1. Fakultät Für Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Guido Kings

Authors
  1. Guido Kings
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guido Kings .

Editor information

Editors and Affiliations

  1. Mathematics Institute, University of Warwick, Coventry, United Kingdom

    David Loeffler

  2. Department of Mathematics, University College London, London, United Kingdom

    Sarah Livia Zerbes

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Kings, G. (2016). On p-adic Interpolation of Motivic Eisenstein Classes. In: Loeffler, D., Zerbes, S. (eds) Elliptic Curves, Modular Forms and Iwasawa Theory. JHC70 2015. Springer Proceedings in Mathematics & Statistics, vol 188. Springer, Cham. https://doi.org/10.1007/978-3-319-45032-2_10

Download citation

Publish with us

Policies and ethics

Search

Navigation