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Modular symbols and the integrality of zeta elements

Content Provider	Springer Nature Link
Author	Kato, Kazuya Fukaya, Takako Sharifi, Romyar
Copyright Year	2016
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.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.
Starting Page	377
Ending Page	395
Page Count	19
File Format	PDF
ISSN	21954755
Journal	Annales mathématiques du Québec
Volume Number	40
Issue Number	2
e-ISSN	21954763
Language	English
Publisher	Springer International Publishing
Publisher Date	2016-03-02
Publisher Place	Cham
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Iwasawa theory Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols Algebra Analysis Eisenstein ideals Modular symbols Mathematics Number Theory Hecke algebras Galois cohomology Elliptic and modular units
Content Type	Text
Resource Type	Article
Subject	Mathematics

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