Formules de classes pour les corps abéliens réels

Belliard, Jean-Robert; Quang, Thong Nguyen

doi:10.5802/aif.1840

Cited by 15 publications

(16 citation statements)

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“…Our Theorem 8.1 is closely related to one of the principal results of L. V. Kuzmin in [8], which was reproved in a more direct way by J.-R. Belliard and T. Nguyen Quang Do in [1]. If we fix a prime p (which is supposed to be odd in [1]), any real abelian field F can be written as the compositum F = KL, where the degree of K/Q is a power of p and the degree of L/Q is relatively prime to p. Taking any Z p -valued Q p -irreducible character χ of Gal(F/K), the mentioned result describes the fudge factor c χ in the following formula…”

Section: Annales De L'institut Fouriersupporting

confidence: 57%

“…This follows from the fact that each factor is a positive integer since [11,Lemma 5.1] holds true for Q p [G] even though it is formulated for Q [G] only. The authors of [1] probably had exactly this reasoning in mind in their remark a)(i) on page 921.…”

Section: Annales De L'institut Fouriermentioning

confidence: 85%

“…If we fix a prime p (which is supposed to be odd in [1]), any real abelian field F can be written as the compositum F = KL, where the degree of K/Q is a power of p and the degree of L/Q is relatively prime to p. Taking any Z p -valued Q p -irreducible character χ of Gal(F/K), the mentioned result describes the fudge factor c χ in the following formula…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

See 2 more Smart Citations

Eigenspaces of the ideal class group

Greither¹,

Kučera²

2014

Ann. inst. Fourier

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Section: Annales De L'institut Fouriersupporting

confidence: 57%

Section: Annales De L'institut Fouriermentioning

confidence: 85%

Section: Annales De L'institut Fouriermentioning

confidence: 99%

See 1 more Smart Citation

Eigenspaces of the ideal class group

Greither¹,

Kučera²

2014

Ann. inst. Fourier

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“…), which has been proved in [20] and later with a different method in [18] (2) A proof of the equivariant conjecture for Tate motives Q(r) and r G 0 over abelian number fields and p # 2 is given by Burns and Greither in [6]. (3) [3] and [29] Remark.…”

Section: Introductionmentioning

confidence: 99%

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

Kings¹

2003

Journal De Théorie Des Nombres De Bordeaux

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“…As for the general case of the Lichtenbaum conjecture, it is now proven with the same caveat (Bloch-Kato conjecture) for K abelian over Q, by the work of Fleckinger-Kolster-Nguyen Quang Do [47] (see also [15] and [16, appendix]). For nonabelian K we are still far from a proof.…”

mentioning

confidence: 99%

Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry

Kahn¹

Handbook of K-Theory

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IntroductionWarning: This paper is full of conjectures. If you are allergic to them it may be harmful to your health. Parts of them are proven, though.In algebraic geometry, one encounters two important kinds of objects: vector bundles and algebraic cycles. The first lead to algebraic K-theory while the second lead to motivic cohomology. They are related via the Chern character and Atiyah-Hirzebruch-like spectral sequences.Vector bundles and algebraic cycles offer very different features. On the one hand, it is often more powerful and easier to work with vector bundles: for example Quillen's localisation theorem in algebraic K-theory is considerably easier to prove than the corresponding localisation theorem of Bloch for higher Chow groups. In short, no moving lemmas are needed to handle vector bundles. In appropriate cases they can be classified by moduli spaces, which underlies the proof of finiteness theorems like Tate's theorem [189] and Faltings' proof of the Mordell conjecture, or Quillen's finite generation theorem for K-groups of curves over a finite field [66]. They also have a better functoriality than algebraic cycles, and this has been used for example by Takeshi Saito in [163, Proof of Lemma 2.4.2] to establish functoriality properties of the weight spectral sequences for smooth projective varieties over Q p .On the other hand, it is fundamental to work with motivic cohomology: as E 2 -terms of a spectral sequence converging to K-theory, the groups involved contain finer torsion information than algebraic K-groups (see Remark 2.2.2), they appear naturally as Hom groups in triangulated categories of motives and they appear naturally, rather than K-groups, in the arithmetic conjectures of Lichtenbaum on special values of zeta functions.In this survey we shall try and clarify for the reader the interaction between these two mathematical objects and give a state of the art of the (many) 2 Bruno Kahn conjectures involving them, and especially the various implications between these conjectures. We shall also explain some unconditional results.Sections 1 to 4 are included for the reader's convenience and are much more developed in other chapters of this Handbook: the reader is invited to refer to those for more details. These sections are also used for reference purposes. The heart of the chapter is in Sections 6 and 7: in the first we try and explain in much detail the conjectures of Soulé and Lichtenbaum on the order of zeroes and special values of zeta functions of schemes of finite type over Spec Z, and an approach to prove them, in characteristic p (we don't touch the much more delicate Beilinson conjectures on special values of L-functions of Q-varieties and their refinements by Bloch-Kato and Fontaine-Perrin-Riou; these have been excellently exposed in many places of the literature anyway). In the second, we indicate some cases where they can be proven, following [95].There are two sources for the formulation of (ii) in Theorem 6.7

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Formules de classes pour les corps abéliens réels

Cited by 15 publications

References 2 publications

Eigenspaces of the ideal class group

Eigenspaces of the ideal class group

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

Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry

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