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 ¹ ; Karl Rubin ²

¹ Department of Mathematics Harvard University Cambridge, MA 02138, USA
² Department of Mathematics UC Irvine Irvine, CA 92697, USA

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

Abstract (VO)
Résumé

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 $1$ .

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 $1$ .

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

Zbl

DOI : 10.5802/jtnb.934

Classification : 11R42, 11R27, 11R23, 11R29
Mots-clés : Class number formulas, Euler systems, Kolyvagin systems, Stark conjectures, L-functions.

Affiliations des auteurs :

Barry Mazur ¹ ; Karl Rubin ²

¹ Department of Mathematics Harvard University Cambridge, MA 02138, USA
² Department of Mathematics UC Irvine Irvine, CA 92697, USA

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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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Henri Darmon The work of Barry Mazur, International congress of mathematicians 2022, ICM 2022, Helsinki, Finland, virtual, July 6–14, 2022. Volume 1. Prize lectures, Berlin: European Mathematical Society (EMS), 2023, pp. 118-141 | DOI:10.4171/icm2022/217 | Zbl:1536.01017
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Barry Mazur; Karl Rubin Controlling Selmer groups in the higher core rank case, Journal de Théorie des Nombres de Bordeaux, Volume 28 (2016) no. 1, pp. 145-183 | DOI:10.5802/jtnb.933 | Zbl:1411.11065
Takamichi Sano On a conjecture for Rubin-Stark elements in a special case, Tokyo Journal of Mathematics, Volume 38 (2015) no. 2, pp. 459-476 | DOI:10.3836/tjm/1452806050 | Zbl:1377.11117
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Cité par 15 documents. Sources : Crossref, zbMATH