Skip to main content
Log in

Denominators of Eisenstein cohomology classes for GL2 over imaginary quadratic fields

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We study the arithmetic of Eisenstein cohomology classes for symmetric spaces associated to GL2 over imaginary quadratic fields. We prove in many cases a lower bound on their denominator in terms of an L-value of a Hecke character providing evidence for a conjecture of Harder that the denominator is given by this L-value. Furthermore, we exibit conditions under which the restriction of the classes to the boundary is integral.

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

Access this article

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. Arnold T. (2007). Anticyclotomic main conjectures for CM modular forms. J. Reine Angew. Math. 606: 41–78

    MATH  MathSciNet  Google Scholar 

  2. Berger, T.: An Eisenstein ideal for imaginary quadratic fields, Ph.D. thesis, University of Michigan, Ann Arbor (2005)

  3. Berger, T.: An Eisenstein ideal for imaginary quadratic fields and the Bloch–Kato conjecture for Hecke characters, MPI Preprint No. 2006-127 (2006)

  4. Borel, A.: Stable real cohomology of arithmetic groups. II Manifolds and Lie groups (Notre Dame, Ind., 1980), Progr. Math., vol. 14, pp. 21–55. Birkhäuser, Boston (1981)

  5. Borel, A., Serre, J.-P.: Corners and arithmetic groups. Comment. Math. Helv. 48, 436–491 (1973), Avec un appendice: Arrondissement des variétés à coins, par A. Douady et L. Hérault

  6. Borel, A., Wallach, N.R.: Continuous cohomology, discrete subgroups, and representations of reductive groups. Annals of Mathematics Studies, vol. 94. Princeton University Press, Princeton (1980)

  7. Bredon, G.E.: Sheaf theory, 2nd edn. Graduate Texts in Mathematics, vol. 170. Springer, New York (1997)

  8. Casselman W. (1973). On some results of Atkin and Lehner. Math. Ann. 201: 301–314

    Article  MATH  MathSciNet  Google Scholar 

  9. Chellali M. (1990). Congruence entre nombres de Bernoulli–Hurwitz dans le cas supersingulier. J Number Theory 35: 157–179

    Article  MATH  MathSciNet  Google Scholar 

  10. Conrad, B.: Modular forms, cohomology and the Ramanujan conjecture. Manuscript

  11. de Shalit, E.: Iwasawa theory of elliptic curves with complex multiplication. Perspectives in Mathematics, vol. 3. Academic, Boston (1987)

  12. Dee, J.: Selmer groups of Hecke characters and Chow groups of self products of CM elliptic curves (1999, preprint). arXiv:math.NT/9901155

  13. Feldhusen, D.: Nenner der Eisensteinkohomologie der GL(2) über imaginär quadratischen Zahlkörpern, Bonner Mathematische Schriften [Bonn Mathematical Publications], 330, Universität Bonn Mathematisches Institut, Bonn, 2000, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn (2000)

  14. Finis T. (2006). Divisibility of anticyclotomic L-functions and theta functions with complex multiplication. Ann. Math. (2) 163(3): 767–807

    Article  MATH  MathSciNet  Google Scholar 

  15. Franke J. and Schwermer J. (1998). A decomposition of spaces of automorphic forms and the Eisenstein cohomology of arithmetic groups. Math. Ann. 311: 765–790

    Article  MATH  MathSciNet  Google Scholar 

  16. Fröhlich A. and Queyrut J. (1973). On the functional equation of the Artin L-function for characters of real representations. Invent. Math. 20: 125–138

    Article  MATH  MathSciNet  Google Scholar 

  17. Fujiwara Y. (1988). On divisibilities of special values of real analytic Eisenstein series. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35: 393–410

    MATH  MathSciNet  Google Scholar 

  18. Greenberg M. (1967). Lectures on Algebraic Topology. W. A. Benjamin, New York

    MATH  Google Scholar 

  19. Greenberg R. (1983). On the Birch and Swinnerton-Dyer conjecture. Invent. Math. 72(2): 241–265

    Article  MATH  MathSciNet  Google Scholar 

  20. Greenberg R. (1985). On the critical values of Hecke L-functions for imaginary quadratic fields. Invent. Math. 79(1): 79–94

    Article  MATH  MathSciNet  Google Scholar 

  21. Harder, G.: Algebraic Geometry. Vieweg. Manuscript (to appear)

  22. Harder, G.: The arithmetic properties of Eisenstein classes. Unpublished manuscript

  23. Harder, G.: Cohomology of arithmetic groups. Manuscript

  24. Harder, G.: Period integrals of cohomology classes which are represented by Eisenstein series, automorphic forms, representation theory and arithmetic (Bombay, 1979). Tata Inst. Fund. Res. Studies in Math., vol. 10, Tata Inst. Fundamental Res., pp. 41–115. Bombay (1981)

  25. Harder, G.: Period integrals of Eisenstein cohomology classes and special values of some L-functions. Number theory related to Fermat’s last theorem (Cambridge, Mass., 1981), Progr. Math., vol. 26, pp. 103–142. Birkhäuser, Boston (1982)

  26. Harder G. (1987). Eisenstein cohomology of arithmetic groups. The case GL2. Invent. Math. 89(1): 37–118

    Article  MATH  MathSciNet  Google Scholar 

  27. Harder, G.: Eisensteinkohomologie und die Konstruktion gemischter Motive. Lecture Notes in Mathematics, vol. 1562. Springer, Berlin (1993)

  28. Harder, G.: Cohomology in the language of Adeles (2006), Chap. III of [23]. Downloadable at http://www.math.uni-bonn.de/people/harder/Manuscripts/

  29. Harder G. and Pink R. (1992). Modular konstruierte unverzweigte abelsche p-Erweiterungen von \({\bf Q}(\zeta_p)\) und die Struktur ihrer Galoisgruppen. Math. Nachr. 159: 83–99

    Article  MATH  MathSciNet  Google Scholar 

  30. Harish-Chandra: Automorphic forms on semisimple Lie groups. Notes by J. G. M. Mars. Lecture Notes in Mathematics, No. 62. Springer, Berlin (1968)

  31. Hida, H.: Elementary theory of L-functions and Eisenstein series. London Mathematical Society Student Texts, vol. 26. Cambridge University Press, Cambridge (1993)

  32. Hida, H.: Non-vanishing modulo p of Hecke L-values. Geometric aspects of Dwork theory, vols. I, II, pp. 735–784. Walter de Gruyter GmbH & Co. KG, Berlin (2004)

  33. Hida, H.: Non-vanishing modulo p of Hecke L-values and application. Durham symposium proceedings (downloadable at http://www.math.ucla.edu/~hida) (2005, in press)

  34. Hida H. and Tilouine J. (1994). On the anticyclotomic main conjecture for CM fields. Invent. Math. 117(1): 89–147

    Article  MATH  MathSciNet  Google Scholar 

  35. Kaiser C. (1990). Die Nenner von Eisensteinklassen für gewisse Kongruenzuntergruppen. Diplomarbeit Universität Bonn, Bonn

    Google Scholar 

  36. Katz N.M. (1976). p-adic interpolation of real analytic Eisenstein series. Ann. Math. (2) 104(3): 459–571

    Article  Google Scholar 

  37. Katz N.M. (1977). Formal groups and p-adic interpolation. Astèrisque 41–42: 55–65

    Google Scholar 

  38. Katz N.M. (1978). p-adic L-functions for CM fields. Invent. Math. 49(3): 199–297

    Article  MATH  MathSciNet  Google Scholar 

  39. Katz N.M. (1982). Divisibilities, congruences and Cartier duality. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28: 667–678

    Google Scholar 

  40. König, H.: Eisenstein–Kohomologie von Sl2(Z[i]), Bonner Mathematische Schriften [Bonn Mathematical Publications], 222, Universität Bonn Mathematisches Institut, Bonn, 1991, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn (1991)

  41. Lang, S.: Algebraic number theory, 2nd edn. Graduate Texts in Mathematics, vol. 110. Springer, New York

  42. Langlands, R.P.: On the functional equations satisfied by Eisenstein series. Lecture Notes in Mathematics, vol. 544. Springer, Berlin (1976)

  43. Maennel, H.: Nenner von Eisensteinklassen auf Hilbertschen Modulvarietäten und die p-adische Klassenzahlformel, Bonner Mathematische Schriften [Bonn Mathematical Publications], 247, Universität Bonn Mathematisches Institut, Bonn, 1993, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn (1992)

  44. Mahnkopf J. (1998). Modular symbols and values of L-functions on GL3. J. Reine Angew. Math. 497: 91–112

    MATH  MathSciNet  Google Scholar 

  45. Mahnkopf J. (2000). Eisenstein cohomology and the construction of p-adic analytic L-functions. Compositio Math. 124(3): 253–304

    Article  MATH  MathSciNet  Google Scholar 

  46. Ribet K.A. (1976). A modular construction of unramified p-extensions of \(Q(\mu_{p})\). Invent. Math. 34(3): 151–162

    Article  MATH  MathSciNet  Google Scholar 

  47. Rohrlich D.E. (1982). Root numbers of Hecke L-functions of CM fields. Am. J. Math. 104(3): 517–543

    Article  MATH  MathSciNet  Google Scholar 

  48. Rubin K. (1983). Congruences for special values of L-functions of elliptic curves with complex multiplication. Invent. Math. 71: 339–364

    Article  MATH  MathSciNet  Google Scholar 

  49. Rubin, K.: Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer. Arithmetic theory of elliptic curves (Cetraro, Italy 1997). Lecture Notes in Mathematics, vol. 1716, pp. 167–234. Springer, New York (1999)

  50. Schwermer, J.: Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen. Lecture Notes in Mathematics, vol. 988. Springer, Berlin (1983)

  51. Skinner, C.: The Eisenstein ideal again (2002, preprint)

  52. Taylor R. (1994). l-adic representations associated to modular forms over imaginary quadratic fields II. Invent. Math. 116(1–3): 619–643

    Article  MATH  MathSciNet  Google Scholar 

  53. Tilouine J. (1989). Sur la conjecture principale anticyclotomique. Duke Math. J. 59(3): 629–673

    Article  MATH  MathSciNet  Google Scholar 

  54. Urban E. (1995). Formes automorphes cuspidales pour GL2 sur un corps quadratique imaginaire. Valeurs spéciales de fonctions L et congruences. Compositio Math. 99(3): 283–324

    MATH  MathSciNet  Google Scholar 

  55. Urban E. (1988). Module de congruences pour GL(2) d’un corps imaginaire quadratique et théorie d’Iwasawa d’un corps CM biquadratique. Duke Math J. 92(1): 179–220

    Article  MathSciNet  Google Scholar 

  56. Waldspurger J.-L. (1985). Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie. Compositio Math. 54(2): 173–242

    MATH  MathSciNet  Google Scholar 

  57. Wang, X.D.: Die Eisensteinklasse in H 1(SL2(Z),M n (Z)) und die Arithmetik spezieller Werte von L-Funktionen, Bonner Mathematische Schriften [Bonn Mathematical Publications], 202, Universität Bonn Mathematisches Institut, Bonn, 1989, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn (1989)

  58. Weselmann U. (1988). Eisensteinkohomologie und Dedekindsummen für GL2 über imaginär-quadratischen Zahlkörpern. J. Reine Angew. Math. 389: 90–121

    MATH  MathSciNet  Google Scholar 

  59. Yang, T.: On CM abelian varieties over imaginary quadratic fields. Math. Ann. 329(1), 87–117 (2004). arXiv:math.NT/0301306

    Google Scholar 

  60. Zucker S. (1997). On the boundary cohomology of locally symmetric varieties. Vietnam J. Math. 25(4): 279–318

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tobias Berger.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berger, T. Denominators of Eisenstein cohomology classes for GL2 over imaginary quadratic fields. manuscripta math. 125, 427–470 (2008). https://doi.org/10.1007/s00229-007-0139-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-007-0139-6

Mathematics Subject Classification (2000)

Navigation