Modular Symbols in Iwasawa Theory

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

doi:10.1007/978-3-642-55245-8_5

Cited by 13 publications

(28 citation statements)

References 25 publications

(7 reference statements)

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“…As constructed in [FK,Section 3.3], the map z ♯ is only defined rationally, which is to say that it takes values ], (1)) injects into its tensor product with Q(Λ), so we show along the way that this cohomology group is Λ-torsion free.…”

Section: Application To Iwasawa Theorymentioning

confidence: 98%

“…Set P = (θ ) /I (θ ) and Y = X (1) (θ ) As defined in [Sh1,FK], there are canonical Λ-module homomorphisms ̟: P → Y and Υ: Y → P, the first being explicitly defined to take a sequence of Manin symbols to a certain sequence of cup products of cyclotomic N -units (see also [Bu] at level p ), and the second being defined via the action of Galois on étale cohomology groups of modular curves (and constructed out of maps occurring in Hida-theoretic proofs of the main conjecture [Wi,Oh2]). In [Sh1], it was conjectured that these maps are inverse to each other, up to a unit suspected to be 1.…”

Section: Application To Iwasawa Theorymentioning

confidence: 99%

“…As aΛ-module,˜ is generated by the elements (u : v ) for u , v ∈ as above [FK,Lemma 3.2.5]. The proof is simple: the image of (u :…”

Section: Homology Of Modular Curvesmentioning

confidence: 99%

“…], (1)) has no nonzero Λ θ -torsion as zeroth Iwasawa cohomology groups are trivial. We modify the argument of [FK,Prop. 3.1.3(1)] in order to see that…”

Section: Zeta Elementsmentioning

confidence: 99%

“…where κ: × p → Λ sends a unit to the group element of its projection to 1+p p (see [FK,2.4.6] ],˜ m (1)) is Λ θ -torsion free.…”

Section: Zeta Elementsmentioning

confidence: 99%

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Modular symbols and the integrality of zeta elements

2016

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

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Section: Application To Iwasawa Theorymentioning

confidence: 98%

Section: Application To Iwasawa Theorymentioning

confidence: 99%

“…As aΛ-module,˜ is generated by the elements (u : v ) for u , v ∈ as above [FK,Lemma 3.2.5]. The proof is simple: the image of (u :…”

Section: Homology Of Modular Curvesmentioning

confidence: 99%

“…], (1)) has no nonzero Λ θ -torsion as zeroth Iwasawa cohomology groups are trivial. We modify the argument of [FK,Prop. 3.1.3(1)] in order to see that…”

Section: Zeta Elementsmentioning

confidence: 99%

“…where κ: × p → Λ sends a unit to the group element of its projection to 1+p p (see [FK,2.4.6] ],˜ m (1)) is Λ θ -torsion free.…”

Section: Zeta Elementsmentioning

confidence: 99%

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Modular symbols and the integrality of zeta elements

2016

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Ordinary pseudorepresentations and modular forms

Wake

Wang-Erickson

2017

Proc. Amer. Math. Soc. Ser. B

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Abstract. In this note, we observe that the techniques of our paper "Pseudomodularity and Iwasawa theory" can be used to provide a new proof of some of the residually reducible modularity lifting results of Skinner and Wiles. In these cases, we have found that a deformation ring of ordinary pseudorepresentations is equal to the Eisenstein local component of a Hida Hecke algebra. We also show that Vandiver's conjecture implies Sharifi's conjecture.

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The geometry of Hida families II: -adic -modules and -adic Hodge theory

Cais¹

2018

Compositio Math.

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To Haruzo Hida, on the occasion of his 60 th birthday.Abstract. We construct the Λ-adic crystalline and Dieudonné analogues of Hida's ordinary Λ-adić etale cohomology, and employ integral p-adic Hodge theory to prove Λ-adic comparison isomorphisms between these cohomologies and the Λ-adic de Rham cohomology studied in [Cai14] as well as Hida's Λ-adicétale cohomology. As applications of our work, we provide a "cohomological" construction of the family of (ϕ, Γ)-modules attached to Hida's ordinary Λ-adicétale cohomology by [Dee01], and we give a new and purely geometric proof of Hida's finitenes and control theorems. We also prove suitable Λ-adic duality theorems for each of the cohomologies we construct. N are the adjoint diamond operators; see [Cai14, §2.2]. 2 This convention is unfortunately somewhat at odds with our notation ΛA, which (as an A-module) is in general neither the tensor product Λ ⊗ Zp A nor (unless A is a complete Zp-algebra) the completed tensor product Λ ⊗ Zp A; we hope that this small abuse causes no confusion.3 That is, ψ(στ ) = ψ(σ) · σψ(τ ) for all σ, τ ∈ Γ,

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Modular Symbols in Iwasawa Theory

Cited by 13 publications

References 25 publications

Modular symbols and the integrality of zeta elements

Modular symbols and the integrality of zeta elements

Ordinary pseudorepresentations and modular forms

The geometry of Hida families II: -adic -modules and -adic Hodge theory

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