Modular symbols and the integrality of zeta elements
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Modular symbols and the integrality of zeta elements Takako Fukaya1 · Kazuya Kato1 · Romyar Sharifi2 Dedicated to Professor Glenn Stevens on the occasion of his 60th birthday
Received: 3 May 2015 / Accepted: 2 February 2016 © Fondation Carl-Herz and Springer International Publishing Switzerland 2016
Abstract We consider modifications of Manin symbols in first homology groups of modular curves with 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 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
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Romyar Sharifi [email protected] Takako Fukaya [email protected] Kazuya Kato [email protected]
1
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA
2
Department of Mathematics, University of Arizona, 617 N. Santa Rita Ave., Tucson, AZ 85711, USA
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T. Fukaya et al.
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
1 Introduction 1.1 Homology and (c, d)-symbols Let p ≥ 5 be a prime, and let N be a positive integer. We let X 1 (N ) denote the compact modular curve of level N over C and let C1 (N ) denote its set of cusps. Consider the relative homology group H˜ = H1 (X 1 (N ), C1 (N ), Z p ). Manin showed that H˜ is generated by elements [u : v] attached to pairs (u, v) with u, v ∈
Z/N Z and (u, v) = (1) (see Sect. 3.1). For integers c and d greater than 1 and prime to 6N ,
we define the (c, d)-symbol for the pair (u, v) by c,d [u
: v] = c2 d 2 [u : v] − c2 [u : dv] − d 2 [
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