On the Brumer-Stark Conjecture

Dasgupta, Samit; Kakde, Mahesh

doi:10.48550/arxiv.2010.00657

Cited by 6 publications

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“…On the other hand, in a recent preprint [11], Dasgupta and Kakde unconditionally proved the Brumer-Stark conjecture (we do not review the statement), except for the 2-components. A promising strategy to prove the Brumer-Stark conjecture had been again to descend from the (equivariant) Iwasawa main conjecture (e.g., Greither and Popescu [19]).…”

Section: Introductionmentioning

confidence: 86%

“…In more recent work, using the work [11], Nickel [25] proved the eTNC − p as long as all the p-adic primes are almost tamely ramified, without assuming the vanishing of the µinvariant. Also, using the work [11], Johnston and Nickel [21] proved the relevant Iwasawa main conjecture without assuming the vanishing of the µ-invariant.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, by adapting the method of Dasgupta and Kakde [11] on the Brumer-Stark conjecture, we prove the eTNC − p in much more cases. The main theorem is the following.…”

Section: Introductionmentioning

confidence: 99%

“…After the authors had completed this project, very recently Bullach, Burns, Daoud, and Seo [3] announced an unconditional proof of the eTNC − . This is achieved by combining the result of [11] with a newly developed general theory of Euler system. On the other hand, this present paper does not rely on the theory of Euler systems at all and instead directly adapt the strategy of [11] (see §1.3).…”

Section: Introductionmentioning

confidence: 99%

“…This is achieved by combining the result of [11] with a newly developed general theory of Euler system. On the other hand, this present paper does not rely on the theory of Euler systems at all and instead directly adapt the strategy of [11] (see §1.3).…”

Section: Introductionmentioning

confidence: 99%

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On the minus component of the equivariant Tamagawa number conjecture for $\mathbb{G}_m$

Atsuta¹,

Kataoka²

2021

Preprint

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The equivariant Tamagawa number conjecture (hereinafter called the eTNC) predicts close relationships between algebraic and analytic aspects of motives. In this paper, we prove a lot of new cases of the minus component of the eTNC for G m and for CM abelian extensions. One of the main results states that the p-component of the eTNC is true when there exists at least one p-adic prime that is tamely ramified. The fundamental strategy is inspired by the work of Dasgupta and Kakde on the Brumer-Stark conjecture. Contents 2.1. Gruenberg's translation functor 2.2. Local consideration 2.3. Global consideration 2.4. Compatibility between the local and global diagrams 2.5. Construction of the Ritter-Weiss type module 2.6. Cohomological triviality 2.7. The extension class 3. Compatibility of the eTNC 3.1. Statements 3.2. Preliminaries on Fitting ideals 3.3. A minor variant of the Ritter-Weiss type module 3.4. The proof of Propositions 3.1 and 3.2 4. Reduction of the main theorems 4.1. The integrality of the Stickelberger element 4.2. An application of the analytic class number formula 4.3. Deducing Theorems 1.1 and 1.8 from Theorem 1.10 4.4. Other formulations of the eTNC 5. Modifications of the Eisenstein series 5.1. Notation on Hilbert modular forms 5.2. The Eisenstein series

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Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

“…In this paper, by adapting the method of Dasgupta and Kakde [11] on the Brumer-Stark conjecture, we prove the eTNC − p in much more cases. The main theorem is the following.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the minus component of the equivariant Tamagawa number conjecture for $\mathbb{G}_m$

Atsuta¹,

Kataoka²

2021

Preprint

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An unconditional proof of the abelian equivariant Iwasawa main conjecture and applications

Johnston¹,

Nickel²

2020

Preprint

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Let p be an odd prime. We give an unconditional proof of the equivariant Iwasawa main conjecture for totally real fields for every admissible one-dimensional padic Lie extension whose Galois group has an abelian Sylow p-subgroup. Crucially, this result does not depend on the vanishing of any µ-invariant. As applications, we deduce the Coates-Sinnott conjecture away from its 2-primary part and new cases of the equivariant Tamagawa number conjecture for Tate motives.

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Fitting ideals of class groups for CM abelian extensions

Atsuta¹,

Kataoka²

2021

Preprint

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Let K/k be a finite abelian CM-extension and T a suitable finite set of finite primes of k. In this paper, we determine the Fitting ideal of the minus component of the T -ray class group of K, except for the 2-component, assuming the validity of the equivariant Tamagawa number conjecture. As an application, we give a necessary and sufficient condition for the Stickelberger element to lie in that Fitting ideal.

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On the Brumer-Stark Conjecture

Cited by 6 publications

References 0 publications

On the minus component of the equivariant Tamagawa number conjecture for $\mathbb{G}_m$

On the minus component of the equivariant Tamagawa number conjecture for $\mathbb{G}_m$

An unconditional proof of the abelian equivariant Iwasawa main conjecture and applications

Fitting ideals of class groups for CM abelian extensions

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