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 -function is .
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 complexe correspondante est .
Accepted:
Published online:
DOI: 10.5802/jtnb.934
Classification: 11R42, 11R27, 11R23, 11R29
Keywords: Class number formulas, Euler systems, Kolyvagin systems, Stark conjectures, L-functions.
Author's affiliations:
@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/} }
TY - JOUR TI - Refined class number formulas for $\mathbb{G}_m$ JO - Journal de Théorie des Nombres de Bordeaux PY - 2016 DA - 2016/// SP - 185 EP - 211 VL - 28 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.934/ UR - https://zbmath.org/?q=an%3A1414.11155 UR - https://doi.org/10.5802/jtnb.934 DO - 10.5802/jtnb.934 LA - en ID - JTNB_2016__28_1_185_0 ER -
Barry Mazur; Karl Rubin. Refined class number formulas for $\mathbb{G}_m$. Journal de Théorie des Nombres de Bordeaux, Volume 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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