Companion forms and the structure of<i>p</i>-adic Hecke algebras

Ohta, Masami

doi:10.1515/crll.2005.2005.585.141

Cited by 13 publications

(1 citation statement)

References 18 publications

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“…Remark 4.7. We thank one of the referees for informing us that this Corollary was also proven in [Oht05] (3.4.2) and [Laf15] Section 3.3.2.…”

Section: R = T Theorem In the Ordinary Casementioning

confidence: 90%

𝑅=𝑇 theorems for weight one modular forms

Berger¹,

Klosin²

2023

Trans. Amer. Math. Soc.

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We prove modularity of certain residually reducible ordinary 2-dimensional p p -adic Galois representations with determinant a finite order odd character χ \chi . For certain non-quadratic χ \chi we prove an R = T R=T result for T T the weight 1 specialisation of the cuspidal Hida Hecke algebra acting on non-classical weight 1 forms. Under the additional assumption that no two cuspidal Hida families congruent to an Eisenstein series cross in weight 1 we show that T T is reduced. For quadratic χ \chi we prove that the quotient of R R corresponding to deformations split at p p is isomorphic to the Hecke algebra acting on classical CM weight 1 modular forms.

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“…Remark 4.7. We thank one of the referees for informing us that this Corollary was also proven in [Oht05] (3.4.2) and [Laf15] Section 3.3.2.…”

Section: R = T Theorem In the Ordinary Casementioning

confidence: 90%

𝑅=𝑇 theorems for weight one modular forms

Berger¹,

Klosin²

2023

Trans. Amer. Math. Soc.

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Hida duality and the Iwasawa main conjecture

Lafferty

2019

Math. Ann.

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The central result of this paper is a refinement of Hida's duality theorem between ordinary Λ-adic modular forms and the universal ordinary Hecke algebra. Specifically, we give a necessary condition for this duality to be integral with respect to particular submodules of the space ordinary Λ-adic modular forms. This refinement allows us to give a simple proof that the universal ordinary cuspidal Hecke algebra modulo Eisenstein ideal is isomorphic to the Iwasawa algebra modulo an ideal related to the Kubota-Leopoldt p-adic L-function. The motivation behind these results stems from Ohta's proof of the Iwasawa main conjecture over Q. Specifically, the most general application of this argument, which employs results on congruence modules and requires one to make some restrictive hypotheses. Using our results we are able to extend Ohta's argument and remove these hypotheses.

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The Eisenstein ideal with squarefree level

Wake

Wang-Erickson²

2021

Advances in Mathematics

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Companion forms and the structure ofp-adic Hecke algebras

Cited by 13 publications

References 18 publications

𝑅=𝑇 theorems for weight one modular forms

𝑅=𝑇 theorems for weight one modular forms

Hida duality and the Iwasawa main conjecture

The Eisenstein ideal with squarefree level

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