Modular symbols and the integrality of zeta elements

Takako Fukaya, Kazuya Kato, Romyar Sharifi

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


We consider modifications of Manin symbols in first homology groups of modular curves with Zp-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.

Original languageEnglish (US)
Pages (from-to)377-395
Number of pages19
JournalAnnales Mathematiques du Quebec
Issue number2
StatePublished - Aug 1 2016


  • Eisenstein ideals
  • Hecke algebras
  • Iwasawa theory
  • Modular symbols

ASJC Scopus subject areas

  • Mathematics(all)


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