Modular symbols and the integrality of zeta elements

Fukaya, Takako; Katô, Kazuya; Sharifi, Romyar T.

doi:10.1007/s40316-016-0059-5

Cited by 3 publications

(1 citation statement)

References 10 publications

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“…In [FKS2], it is shown that if p ∤ ϕ(N), then there exists an integral version of z N to the primitive part of H 2 ét (Y 1 (N), Z p (2)) ord for the action of (Z/NZ) × by diamond operators, after excluding the ω −2 -eigenspace for (Z/pZ) × . Let us describe a motivic version of this, without some of these assumptions.…”

Section: Comparison Of Hecke Algebras For γ 0 and γmentioning

confidence: 99%

Eisenstein cocycles in motivic cohomology

Sharifi¹,

Venkatesh²

2020

Preprint

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Several authors have studied homomorphisms from first homology groups of modular curves to K 2 (X), with X either a cyclotomic ring or a modular curve. These maps send Manin symbols in the homology groups to Steinberg symbols of cyclotomic or Siegel units. We give a new construction of these maps and a direct proof of their Hecke equivariance, analogous to the construction of Siegel units using the universal elliptic curve. Our main tool is a 1-cocycle from GL 2 (Z) to the second K-group of the function field of a suitable group scheme over X, from which the maps of interest arise by specialization.

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Section: Comparison Of Hecke Algebras For γ 0 and γmentioning

confidence: 99%

Eisenstein cocycles in motivic cohomology

Sharifi¹,

Venkatesh²

2020

Preprint

Self Cite

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On Sharifi’s conjecture: Exceptional case

Shih¹,

Wang²

2021

Trans. Amer. Math. Soc.

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In the present article, we study the conjecture of Sharifi on the surjectivity of the map ϖ θ \varpi _{\theta } . Here θ \theta is a primitive even Dirichlet character of conductor N p Np , which is exceptional in the sense of Ohta. After localizing at the prime ideal p \mathfrak {p} of the Iwasawa algebra related to the trivial zero of the Kubota–Leopoldt p p -adic L L -function L p ( s , θ − 1 ω 2 ) L_p(s,\theta ^{-1}\omega ^2) , we compute the image of ϖ θ , p \varpi _{\theta ,\mathfrak {p}} in a local Galois cohomology group and prove that it is an isomorphism. Also, we prove that the residual Galois representations associated to the cohomology of modular curves are decomposable after taking the same localization.

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On Sharifi's conjecture: exceptional case

Shih¹,

Wang²

2020

Preprint

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In the present article, we study the conjecture of Sharifi on the surjectivity of the map ̟ θ . Here θ is a primitive even Dirichlet character of conductor N p, which is exceptional in the sense of Ohta. After localizing at the prime ideal p of the Iwasawa algebra related to the trivial zero of the Kupota-Leopoldt p-adic L-function Lp(s, θ −1 ω 2 ), we compute the image of ̟ θ,p in local Galois cohomology groups and prove that it is an isomorphism. Also, we prove that the residual Galois representations associated to the cohomology of modular curves are decomposable after taking the same localization. Sharifi's ConjectureThroughout this paper, we will denote by N a positive integer, denote by p ≥ 5 a prime number not dividing N φ(N ). The goal of this section is to review Sharifi's conjecture on the map ̟ following [13] and to state Theorem 1.1. We refer the reader to loc. cit. for more details on Sharifi's conjecture.

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Modular symbols and the integrality of zeta elements

Cited by 3 publications

References 10 publications

Eisenstein cocycles in motivic cohomology

Eisenstein cocycles in motivic cohomology

On Sharifi’s conjecture: Exceptional case

On Sharifi's conjecture: exceptional case

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