Displaying similar documents to “Abelian fields and the Brumer-Stark conjecture”

Class groups of abelian fields, and the main conjecture

Cornelius Greither (1992)

Annales de l'institut Fourier

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This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case p = 2 , by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of χ -parts of p -class groups of abelian number fields: first for relative class groups of real fields (again including the case p = 2 ). As a consequence, a generalization of the Gras conjecture...

The Bloch-Kato conjecture on special values of L -functions. A survey of known results

Guido Kings (2003)

Journal de théorie des nombres de Bordeaux

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This paper contains an overview of the known cases of the Bloch-Kato conjecture. It does not attempt to overview the known cases of the Beilinson conjecture and also excludes the Birch and Swinnerton-Dyer point. The paper starts with a brief review of the formulation of the general conjecture. The final part gives a brief sketch of the proofs in the known cases.

On Tate’s refinement for a conjecture of Gross and its generalization

Noboru Aoki (2004)

Journal de Théorie des Nombres de Bordeaux

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We study Tate’s refinement for a conjecture of Gross on the values of abelian L -function at s = 0 and formulate its generalization to arbitrary cyclic extensions. We prove that our generalized conjecture is true in the case of number fields. This in particular implies that Tate’s refinement is true for any number field.

A Stark conjecture “over 𝐙 ” for abelian L -functions with multiple zeros

Karl Rubin (1996)

Annales de l'institut Fourier

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Suppose K / k is an abelian extension of number fields. Stark’s conjecture predicts, under suitable hypotheses, the existence of a global unit ϵ of K such that the special values L ' ( χ , 0 ) for all characters χ of Gal / ( K / k ) can be expressed as simple linear combinations of the logarithms of the different absolute values of ϵ . In this paper we formulate an extension of this conjecture, to attempt to understand the values L ( r ) ( χ , 0 ) when the order of vanishing r may be greater than one. This conjecture...

p -adic Abelian Stark conjectures at s = 1

David Solomon (2002)

Annales de l’institut Fourier

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A p -adic version of Stark’s Conjecture at s = 1 is attributed to J.-P. Serre and stated (faultily) in Tate’s book on the Conjecture. Building instead on our previous paper (and work of Rubin) on the complex abelian case, we give a new approach to such a conjecture for real ray-class extensions of totally real number fields. We study the coherence of our p -adic conjecture and then formulate some integral refinements, both alone and in combination with its complex analogue. A ‘Weak Combined...

Refined theorems of the Birch and Swinnerton-Dyer type

Ki-Seng Tan (1995)

Annales de l'institut Fourier

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In this paper, we generalize the context of the Mazur-Tate conjecture and sharpen, in a certain way, the statement of the conjecture. Our main result will be to establish the truth of a part of these new sharpened conjectures, provided that one assume the truth of the classical Birch and Swinnerton-Dyer conjectures. This is particularly striking in the function field case, where these results can be viewed as being a refinement of the earlier work of Tate and Milne.