A reciprocity map and the two-variable $p$-adic $L$-function

Sharifi, Romyar T.

doi:10.4007/annals.2011.173.1.7

Cited by 40 publications

(62 citation statements)

References 19 publications

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“…As applications of his theory, Ohta provides a construction of two-variable p-adic L-functions attached to families of ordinary cuspforms differing from that of Kitagawa [Kit94], and, in a subsequent paper [Oht00], provides a new and streamlined proof of the theorem of Mazur-Wiles [MW84] (Iwasawa's Main Conjecture for Q; see also [Wil90]). We remark that Ohta's Λ-adic Hodge-Tate isomorphism is a crucial ingredient in the forthcoming partial proof of Sharifi's conjectures [Sha11], [Sha07] due to Fukaya and Kato [FK12].…”

mentioning

confidence: 99%

The geometry of Hida families I: $$\Lambda $$-adic de Rham cohomology

Cais

2017

Math. Ann.

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We construct the Λ-adic de Rham analogue of Hida's ordinary Λ-adicétale cohomology and of Ohta's Λ-adic Hodge cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of Qp, we give a purely geometric proof of the expected finiteness, control, and Λ-adic duality theorems. Following Ohta, we then prove that our Λ-adic module of differentials is canonically isomorphic to the space of ordinary Λ-adic cuspforms. In the sequel [Cai14] to this paper, we construct the crystalline counterpart to Hida's ordinary Λ-adicétale cohomology, and employ integral p-adic Hodge theory to prove Λ-adic comparison isomorphisms between all of these cohomologies. As applications of our work in this paper and [Cai14], we will be able to provide a "cohomological" construction of the family of (ϕ, Γ)-modules attached to Hida's ordinary Λ-adicétale cohomology by [Dee01], as well as a new and purely geometric proof of Hida's finitenes and control theorems. We are also able to prove refinements of the main theorems in [MW86] and [Oht95].

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confidence: 99%

The geometry of Hida families I: $$\Lambda $$-adic de Rham cohomology

Cais

2017

Math. Ann.

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“…One consequence of Sharifi's conjecture is that X χ (1) ∼ = H − /IH − as Λ-modules. Since X χ has no p-torsion, this would imply that H − /IH − has no p-torsion, which Sharifi explicitly conjectures in [S1,Remark,pg. 51].…”

Section: Condition (1) Holds If and Only If Conditions (2) And (3) Bomentioning

confidence: 94%

Eisenstein Hecke algebras and conjectures in Iwasawa theory

Wake

2015

Algebra Number Theory

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“…Recall that Sharifi's conjecture states that two maps Υ and are isomorphisms [Sha11]. In [FK12,FKS14] …”

Section: Lemma 422 Let a Be A Ring Let M Be A Finitely Presented mentioning

confidence: 99%

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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A reciprocity map and the two-variable $p$-adic $L$-function

Cited by 40 publications

References 19 publications

The geometry of Hida families I: $$\Lambda $$-adic de Rham cohomology

The geometry of Hida families I: $$\Lambda $$-adic de Rham cohomology

Eisenstein Hecke algebras and conjectures in Iwasawa theory

Ordinary pseudorepresentations and modular forms

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