Skip to main content
SpringerLink
Log in

K(1)-local topological modular forms

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We construct the Witten orientation of the topological modular forms spectrum tmf in the K(1)-local setting by attaching E∞ cells to the bordism theory MO<8>. We also identify the KO-homology of tmf with the ring of divided congruences.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, J.F.: Lectures on generalised cohomology. Category Theory Homology Theory Appl., Proc. Conf. Seattle Res. Center Battelle Mem. Inst. 1968, 3, 1–138 (1969)

    Google Scholar 

  2. Adams, J.F.: Stable homotopy and generalised homology. Chicago Lectures in Mathematics. Chicago, London: The University of Chicago Press 1974

  3. Anderson, D.F.: PhD thesis. Berkely 1964

  4. Anderson, D.W., Brown, E.H., Peterson, F.P.: Spin cobordism. Bull. Am. Math. Soc. 72, 256–260 (1966)

    Article  Google Scholar 

  5. Ando, M., Hopkins, M.J., Strickland, N.P.: Elliptic homology of BO<8> in positive charachteristic. Preprint 1999

  6. Ando, M., Hopkins, M.J., Strickland, N.P.: Elliptic spectra, the Witten genus and the theorem of the cube. Invent. Math. 146, 595–687 (2001)

    Article  MathSciNet  Google Scholar 

  7. Ando, M.: Isogenies of formal group laws and power operations in the cohomology theories E n . Duke Math. J. 79, 423–485 (1995)

    Article  MathSciNet  Google Scholar 

  8. Ando, M.: Power operations in elliptic cohomology and representations of loop groups. Trans. Am. Math. Soc. 352, 5619–5666 (2000)

    Article  Google Scholar 

  9. Atiyah, M.F.: Power operations in K-theory. Q. J. Math., Oxf. II. Ser. 17, 165–193 (1966)

    Article  MathSciNet  Google Scholar 

  10. Atiyah, M.F., Bott, R., Shapiro, A.: Clifford modules. Topology 3, Suppl. 1, 3–38 (1964)

  11. Baker, A.: Operations and cooperations in elliptic cohomology. I: Generalized modular forms and the cooperation algebra. New York J. Math. 1, 39–74 (1994)

    Google Scholar 

  12. Birch, B.J., Kuyk, W. (eds.): Modular functions of one variable. IV. Proceedings of the international summer school, University of Antwerp, RUCA, July 17–August 3, 1972. Lecture Notes in Mathematics 476. Berlin, Heidelberg, New York: Springer 1975

  13. Bousfield, A.K.: The localization of spectra with respect to homology. Topology 18, 257–281 (1979)

    Article  MathSciNet  Google Scholar 

  14. Bousfield, A.K.: On λ-rings and the K-theory of infinite loop spaces. K-Theory 10, 1–30 (1996)

    Article  MathSciNet  Google Scholar 

  15. Bousfield, A.K.: On p-adic λ-rings and the K-theory of H-spaces. Math. Z. 223, 483–519 (1996)

    Article  MathSciNet  Google Scholar 

  16. Bousfield, A.K.: The K-theory localizations and v1-periodic homotopy groups of H-spaces. Topology 38, 1239–1264 (1999)

    Article  MathSciNet  Google Scholar 

  17. Bousfield, A.K., Kan, D.M.: Homotopy limits, completions and localizations. Lecture Notes in Mathematics 304. Berlin, Heidelberg, New York: Springer 1972

  18. Bruner, R.R., May, J.P., McClure, J.E., Steinberger, M.: H∞ ring spectra and their applications. Lecture Notes in Mathematics 1176. Berlin, Heidelberg, New York: Springer 1986

  19. Bruner, R.R., May, J.P., McClure, J.E., Steinberger, M.: H∞ ring spectra and their applications. Lecture Notes in Mathematics 1176. Berlin, Heidelberg, New York: Springer 1986

  20. Brylinski, J.-L.: Representations of loop groups, Dirac operators on loop space, and modular forms. Topology 29, 461–480 (1990)

    Article  MathSciNet  Google Scholar 

  21. Clarke, F., Johnson, K.: Cooperations in elliptic homology. In: Adams memorial symposium on algebraic topology, vol. 2, Proc. Symp., Manchester/UK 1990. Lond. Math. Soc. Lect. Note Ser. 176, 131–143 (1992)

  22. Deligne, P.: Courbes elliptiques: formulaire d’après J. Tate. In: Modular Functions one Variable IV, Proc. Internat. Summer School, Univ. Antwerp 1972. Lecture Notes Math. 476, 53–73 (1975)

  23. Dwork, B.: Norm residue symbol in local number fields. Abh. Math. Semin. Univ. Hamb. 22, 180–190 (1958)

    Article  MathSciNet  Google Scholar 

  24. Elmendorf, A.D., Kriz, I., Mandell, M.A., May, J.P.: Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole. Mathematical Surveys and Monographs 47. Providence, RI: American Mathematical Society 1997

  25. Franke, J.: On the construction of elliptic cohomology. Math. Nachr. 158, 43–65 (1992)

    Article  MathSciNet  Google Scholar 

  26. Hazewinkel, M.: A universal formal group and complex cobordism. Bull. Am. Math. Soc. 81, 930–933 (1975)

    Article  MathSciNet  Google Scholar 

  27. Hazewinkel, M.: Formal groups and applications. Pure and Applied Mathematics, Vol. 78. New York, San Francisco, London: Academic Press 1978

  28. Hirzebruch, F., Berger, T., Jung, R.: Manifolds and modular forms. Transl. by P.S. Landweber. Aspects of Mathematics. E 20. Wiesbaden: Friedr. Vieweg 1992

  29. Hopkins, M.J.: Topological modular forms, the Witten genus, and the theorem of the cube. In: Chatterji, S.D. (ed.), Proceedings of the international congress of mathematicians, ICM ’94, August 3–11, 1994, Zuerich, Switzerland. Vol. I, p. 554–565. Basel: Birkhäuser 1995

  30. Hopkins, M.J.: K(1)-local E∞ ring spectra. Unpublished notes 1998

  31. Hopkins, M.J., Miller, H.R.: A∞-structures on elliptic spectra. Unpublished notes. 1995

  32. Johnson, D.C., Wilson, W.S.: The Brown-Peterson homology of elementary p-groups. Am. J. Math. 107, 427–453 (1985)

    Article  Google Scholar 

  33. Katz, N.M.: P-adic properties of modular schemes and modular forms. In: Modular Functions one Variable III, Proc. Internat. Summer School, Univ. Antwerp 1972. Lect. Notes Math. 350, 69–190 (1973)

  34. Katz, N.M.: Higher congruences between modular forms. Ann. Math. (2) 101, 332–367 (1975)

  35. Katz, N.M.: p-adic L-functions via moduli of elliptic curves. In: Algebraic Geom., Proc. Symp. Pure Math. 29, Arcata 1974, 479–506 (1975)

  36. Katz, N.M.: The Eisenstein measure and p-adic interpolation. Am. J. Math. 99, 238–311 (1977)

    Article  Google Scholar 

  37. Kervaire, M.A., Milnor, J.W.: Groups of homotopy spheres. I. Ann. Math. (2) 77, 504–537 (1963)

  38. Kreck, M., Stolz, S.: HP2-bundles and elliptic homology. Acta Math. 171, 231–261 (1993)

    Article  MathSciNet  Google Scholar 

  39. Landweber, P.S.: Homological properties of comodules over MU*(MU) and BP*(BP). Am. J. Math. 98, 591–610 (1976)

    Article  MathSciNet  Google Scholar 

  40. Landweber, P.S., Ravenel, D.C., Stong, R.E.: Periodic cohomology theories defined by elliptic curves. In: Cenkl, M. et al. (eds.), The Cech centennial. A conference on homotopy theory dedicated to Eduard Cech on the occasion of his 100th birthday, June 22–26, 1993, Northeastern University, Boston, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 181, 317–337 (1995)

    Article  MathSciNet  Google Scholar 

  41. Laures, G.: The topological q-expansion principle. Topology 38, 387–425 (1999)

    Article  MathSciNet  Google Scholar 

  42. Laures, G.: An E∞-splitting of spin bordism. Am. J. Math. 125 (2003)

  43. Lawson jun., H.B., Michelsohn, M.-L.: Spin geometry. Princeton Mathematical Series 38. Princeton: Princeton University Press 1989

  44. Lichnerowicz, A.: Laplacien sur une variete riemannienne et spineurs. Atti Accad. Naz. Lincei, Rend., Cl. Sci. Fis. Mat. Nat., VIII. Ser. 33, 187–191 (1963)

    MathSciNet  Google Scholar 

  45. Meier, W.: Complex and real K-theory and localization. J. Pure Appl. Algebra 14, 59–71 (1979)

    Article  MathSciNet  Google Scholar 

  46. Quillen, D.: Elementary proofs of some results of cobordism theory using Steenrod operations. Adv. Math. 7, 29–56 (1971)

    Article  MathSciNet  Google Scholar 

  47. Ravenel, D.C.: Localization with respect to certain periodic homology theories. Am. J. Math. 106, 351–414 (1984)

    Article  MathSciNet  Google Scholar 

  48. Ravenel, D.C., Wilson, W.S.: The Morava K-theories of Eilenberg-MacLane spaces and the Conner-Floyd conjecture. Am. J. Math. 102, 691–748 (1980)

    Article  MathSciNet  Google Scholar 

  49. Segal, G.: Elliptic cohomology [after Landweber-Stong, Ochanine, Witten, and others]. Semin. Bourbaki, 40eme Annee, Vol. 1987/88, Exp. No. 695, Asterisque 161–162, 187–201 (1988)

  50. Serre, J.P.: Formes modulaires et fonctions zeta p-adiques. In: Modular Functions one Variable III, Proc. Internat. Summer School, Univ. Antwerp 1972. Lecture Notes Math. 350, 191–268 (1973)

  51. Silverman, J.H.: The arithmetic of elliptic curves. Graduate Texts in Mathematics 106. New York etc.: Springer 1986

  52. Silverman, J.H.: Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics 151. New York: Springer 1998

  53. Witten, E.: The index of the Dirac operator in loop space. Elliptic curves and modular forms in algebraic topology. Proc. Conf., Princeton/NJ 1986, Lect. Notes Math. 1326, 161–181 (1988)

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, 69120, Heidelberg, Germany

    Gerd Laures

Authors
  1. Gerd Laures
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gerd Laures.

Additional information

Mathematics Subject Classification (2000)

55P43, 55N22, 55P43, 55P48, 55P60, 55S12, 55S25, 11G07

Rights and permissions

Reprints and permissions

About this article

Cite this article

Laures, G. K(1)-local topological modular forms. Invent. math. 157, 371–403 (2004). https://doi.org/10.1007/s00222-003-0355-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-003-0355-y

Keywords

Search

Navigation