Annales mathématiques du Québec

, Volume 40, Issue 2, pp 377–395

Modular symbols and the integrality of zeta elements

Article
  • 77 Downloads

Abstract

We consider modifications of Manin symbols in first homology groups of modular curves with \(\mathbb {Z}_p\)-coefficients for an odd prime p. We show that these symbols generate homology in primitive eigenspaces for the action of diamond operators, with a certain condition on the eigenspace that can be removed on Eisenstein parts. We apply this to prove the integrality of maps taking compatible systems of Manin symbols to compatible systems of zeta elements. In the work of the first two authors on an Iwasawa-theoretic conjecture of the third author, these maps are constructed with certain bounded denominators. As a consequence, their main result on the conjecture was proven after inverting p, and the results of this paper allow one to remove this condition.

Keywords

Iwasawa theory Modular symbols Hecke algebras Eisenstein ideals 

Résumé

Cet article est consacré à l’étude de certaines modifications des symboles de Manin dans le premier groupe d’homologie d’une courbe modulaire à coefficients dans \(\mathbb {Z}_p\) pour un nombre premier p impair. On démontre que ces symboles de Manin engendrent les sous-espaces propres de l’homologie associés aux caractères primitifs pour l’action des opérateurs diamants, modulo une condition qui peut être enlevée pour les parties Eisenstein. Ces résultats servent à démontrer l’intégralité de certaines fonctions allant des systèmes compatibles d’éléments de Manin vers les systèmes compatibles d’éléments zêta. Dans des travaux antérieurs des deux premiers auteurs autour d’une conjecture du troisième auteur, ces applications furent construites, mais seulement à dénominateurs bornés, ce qui a permis seulement de démontrer la conjecture après inversion de p. Les résultats plus fins de cet article permettent d’enlever cette restriction.

Mathematics Subject Classification

11F67 11G16 11R23 11R34 

References

  1. 1.
    Brunault, F.: Beilinson–Kato elements in \(K_2\) of modular curves. Acta Arith. 134, 283–298 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Busuioc, C.: The Steinberg symbol and special values of \(L\)-functions. Trans. Am. Math. Soc. 360, 5999–6015 (2008)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Fukaya, T., Kato, K.: On conjectures of Sharifi (preprint)Google Scholar
  4. 4.
    Fukaya, T., Kato, K., Sharifi, R., Modular symbols in Iwasawa theory: state of the art and recent advances. Contrib. Math. Comput. Sci. (7) (Springer) 2014, 177–219 (2012)Google Scholar
  5. 5.
    Hida, H.: Galois representations into \({\rm GL}_2({\mathbb{Z}}_p\!\!{\llbracket }{X}{\rrbracket}\!\!)\) attached to ordinary cusp forms. Invent. Math. 85, 545–613 (1986)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Manin, J.: Parabolic points and zeta-functions of modular curves. Math. USSR Izvsetija 6, 19–64 (1972)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Merel, L.: Universal Fourier expansions of modular forms. In: On Artin’s conjecture for odd \(2\)-dimensional representations. Lecture Notes in Mathematics, vol. 1585, pp. 59–94. Springer, Berlin (1994)Google Scholar
  8. 8.
    Ohta, M.: On the \(p\)-adic Eichler–Shimura isomorphism for \(\Lambda \)-adic cusp forms. J. Reine Angew. Math. 463, 49–98 (1995)MathSciNetMATHGoogle Scholar
  9. 9.
    Ohta, M.: Ordinary étale cohomology groups attached to towers of elliptic modular curves. II. Math. Ann. 318, 557–583 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Sharifi, R.: A reciprocity map and the two variable \(p\)-adic \(L\)-function. Ann. Math. 173, 251–300 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Sharifi, R.: An extension of the method of Fukaya and Kato (in preparation)Google Scholar
  12. 12.
    Wiles, A.: The Iwasawa conjecture for totally real fields. Ann. Math. 131, 493–540 (1990)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Fondation Carl-Herz and Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsUniversity of ArizonaTucsonUSA

Personalised recommendations