Modular symbols and the integrality of zeta elements

Takako Fukaya, Kazuya Kato, Romyar T Sharifi

Research output: Contribution to journalArticle

Abstract

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
Volume40
Issue number2
DOIs
StatePublished - Aug 1 2016

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Integrality
Eigenspace
Diamond Operator
Modular Curves
Homology Groups
Denominator
Homology
Odd
Coefficient

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Modular symbols and the integrality of zeta elements. / Fukaya, Takako; Kato, Kazuya; Sharifi, Romyar T.

In: Annales Mathematiques du Quebec, Vol. 40, No. 2, 01.08.2016, p. 377-395.

Research output: Contribution to journalArticle

Fukaya, Takako ; Kato, Kazuya ; Sharifi, Romyar T. / Modular symbols and the integrality of zeta elements. In: Annales Mathematiques du Quebec. 2016 ; Vol. 40, No. 2. pp. 377-395.
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