Refined class number formulas for $\mathbb{G}_m$

Barry Mazur; Karl Rubin

doi:10.5802/jtnb.934

Refined class number formulas for

𝔾_{m}

Barry Mazur ; Karl Rubin

Journal de Théorie des Nombres de Bordeaux, Tome 28 (2016) no. 1, pp. 185-211.

Résumé
Abstract

Nous formulons une généralisation d’une “formule du nombre de classes raffinée” de Darmon. Notre conjecture concerne des éléments de type Stickelberger formés à partir d’unités de Stark généralisées. En utilisant la théorie des systèmes de Kolyvagin, nous démontrons une grande partie de cette conjecture lorsque l’ordre d’annulation de la fonction $L$ complexe correspondante est $1$ .

We formulate a generalization of a “refined class number formula” of Darmon. Our conjecture deals with Stickelberger-type elements formed from generalized Stark units, and has two parts: the “order of vanishing” and the “leading term”. Using the theory of Kolyvagin systems we prove a large part of this conjecture when the order of vanishing of the corresponding complex $L$ -function is $1$ .

Reçu le : 2013-12-16
Accepté le : 2015-05-28
Publié le : 2017-01-02

Zbl 1414.11155

DOI : https://doi.org/10.5802/jtnb.934
Classification : 11R42, 11R27, 11R23, 11R29
Mots clés : Class number formulas, Euler systems, Kolyvagin systems, Stark conjectures, L-functions.

@article{JTNB_2016__28_1_185_0,
     author = {Barry Mazur and Karl Rubin},
     title = {Refined class number formulas for $\mathbb{G}_m$},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {185--211},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {28},
     number = {1},
     year = {2016},
     doi = {10.5802/jtnb.934},
     zbl = {1414.11155},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.934/}
}

Barry Mazur; Karl Rubin. Refined class number formulas for $\mathbb{G}_m$. Journal de Théorie des Nombres de Bordeaux, Tome 28 (2016) no. 1, pp. 185-211. doi : 10.5802/jtnb.934. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.934/

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