Refined theorems of the Birch and Swinnerton-Dyer type

Ki-Seng Tan

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Displaying similar documents to “Refined theorems of the Birch and Swinnerton-Dyer type”

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.

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 the conjectures of Birch and Swinnerton-Dyer and a geometric analog

John Tate (1964-1966)

Séminaire Bourbaki

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On the Birch and Swinnerton-Dyer conjecture

Cristian Virdol (2011)

Acta Arithmetica

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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...

On Birch and Swinnerton-Dyer's conjecture for elliptic curves with complex multiplication. I

Nicole Arthaud (1978)

Compositio Mathematica

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Numerical verification of the Stark-Chinburg conjecture for some icosahedral representations.

Jehanne, Arnaud, Roblot, Xavier-Francois, Sands, Jonathan (2003)

Experimental Mathematics

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Some remarks on conjectures about cyclotomic fields and $K$ -groups of $𝐙$

Masato Kurihara (1992)

Compositio Mathematica

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