Brumer-Stark Units and Hilbert's 12th Problem

Dasgupta, Samit; Kakde, Mahesh

doi:10.48550/arxiv.2103.02516

Cited by 4 publications

(5 citation statements)

References 24 publications

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“…They construct a suitable cuspform and study the associated Galois representation. A key point we need to care about is that we have to work over a group ring like Z p OEG, not over a domain like C. Let us also mention that subsequently Dasgupta and Kakde [11] succeeded in applying the strategy to establish explicit class field theory over totally real fields. Now let us explain the organization of this paper and at the same time the outline the proof of Theorem 1.10.…”

Section: An Outline Of the Proof And The Organization Of This Papermentioning

confidence: 99%

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

Atsuta

Kataoka

2023

Doc. Math.

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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.

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Section: An Outline Of the Proof And The Organization Of This Papermentioning

confidence: 99%

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

Atsuta

Kataoka

2023

Doc. Math.

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“…There is no difficulty in defining the ratios (39) and (40), since the quantities live in a p-adic field and the denominators are non-zero. The analogue of this situation for Gross's For this reason, we introduce in [17] an R-algebra R L that is generated by an element L that plays the role of the analytic L -invariant, i.e. the "ratio" between Θ L and Θ H .…”

Section: The Greenberg-stevens L -Invariantmentioning

confidence: 99%

“…To prove (42), we define a generalized Ritter-Weiss module ∇ L over the ring R L that can be viewed as a gluing of the modules ∇ T S (H) and ∇ T S ′ (L). We show in [17,Theorem 4.6] that the Fitting ideal Fitt R L (∇ L ) is generated by the element rec G (u p ) − L ord G (u p ) ∈ I/I 2 , and hence that (42) is equivalent to…”

Section: The Greenberg-stevens L -Invariantmentioning

confidence: 99%

On the Brumer-Stark Conjecture and Refinements

Dasgupta¹,

Kakde²

2022

Preprint

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We state the Brumer-Stark conjecture and motivate it from two perspectives. Stark's perspective arose in his attempts to generalize the classical Dirichlet class number formula for the leading term of the Dedekind zeta function at s = 1 (equivalently, s = 0). Brumer's perspective arose by generalizing Stickelberger's work regarding the factorization of Gauss sums and the annihilation of class groups of cyclotomic fields. These viewpoints were synthesized by Tate, who stated the Brumer-Stark conjecture in its current form.The conjecture considers a totally real field F and a finite abelian CM extension H/F . It states the existence of p-units in H whose valuations at places above p are related to the special values of the L-functions of the extension H/F at s = 0. Essentially equivalently, the conjecture states that a Stickelberger element associated to H/F annihilates the (appropriately smoothed) class group of H.This conjecture has been refined by many authors in multiple directions. Notably, Kurihara conjectured that the Stickelberger element lies in the Fitting ideal of the Pontryagin dual of the class group, and furthermore conjectured an exact formula for this Fitting ideal. Burns constructed a Selmer group whose Fitting ideal he conjectured to be generated by the Stickelberger element. Atsuta and Kataoka conjectured a formula for the Fitting ideal of the class group, rather than its dual. Rubin stated a higher rank generalization of the Brumer-Stark conjecture. The first author and his collaborators stated an exact p-adic analytic formula for Brumer-Stark units, generalizing conjectures of Gross that give the image of the units under the Artin reciprocity map.We conclude by stating our results toward these various conjectures and summarizing the proofs. In particular, we prove the Brumer-Stark conjecture, Rubin's higher rank version, and Kurihara's conjecture, all "away from 2." We also prove strong partial results toward Gross's conjecture and the exact p-adic analytic formula for Brumer-Stark units. The key technique involved in the proofs is Ribet's method. We demonstrate congruences between Hilbert modular Eisenstein series and cusp forms, and use the associated Galois representations to construct Galois cohomology classes. These cohomology classes are interpreted in terms of Ritter-Weiss modules, from which results on class groups may be deduced.

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“…Recent work of Dasgupta and Kakde [DK23] on the Brumer-Stark conjecture refines this by removing the norm. The computation of Gross-Stark units over quadratic fields has been studied in [TY13] when p splits in F , and [Das07], [DK21], and [FL22] for p inert.…”

Section: Introductionmentioning

confidence: 99%

Modular algorithms for Gross-Stark units and Stark-Heegner points

Damm-Johnsen¹

2023

Preprint

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In recent work, Darmon, Pozzi and Vonk explicitly construct a modular form whose spectral coefficients are p-adic logarithms of Gross-Stark units and Stark-Heegner points. Here we describe how this construction gives rise to a practical algorithm for explicitly computing these logarithms to specified precision, and how to recover the exact values of the Gross-Stark units and Stark-Heegner points from them.Key tools are overconvergent modular forms, reduction theory of quadratic forms and Newton polygons. As an application, we tabulate Brumer-Stark units in narrow Hilbert class fields of real quadratic fields with discriminants up to 10000, for primes less than 20, as well as Stark-Heegner points on elliptic curves.

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Brumer-Stark Units and Hilbert's 12th Problem

Cited by 4 publications

References 24 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 $\mathbb{G}_m$

On the Brumer-Stark Conjecture and Refinements

Modular algorithms for Gross-Stark units and Stark-Heegner points

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