In this article we obtain class invariants and cyclotomic unit groups by considering specializations of modular units. We construct these modular units from functional solutions to higher order -recurrence equations given by Selberg in his work generalizing the Rogers-Ramanujan identities. As a corollary, we provide a new proof of a result of Zagier and Gupta, originally considered by Gauss, regarding the Gauss periods. These results comprise part of the author’s 2006 Ph.D. thesis [6] in which the structure of these modular unit groups and their associated cuspidal divisor class groups are also characterized, and a cuspidal divisor class number formula is given in terms of products of -functions and compared to the classical relative class number formula within the cyclotomic fields [6, 7].
Dans cet article, nous obtenons des invariants de classe et des groupes d’unités cyclotomiques en considérant des spécialisations d’unités modulaires. Nous construisons ces unités modulaires à partir de solutions d’équations fonctionnelles de -récurrence données par Selberg dans son travail généralisant les identités de Rogers-Ramanujan. Commme corollaire, nous donnons une nouvelle preuve d’un résultat de Zagier et Gupta, originellement considéré par Gauss, à propos des périodes de Gauss. Ces résultats proviennent pour partie de la thèse de l’auteur en 2006 [6] dans laquelle la structure de ces groupes d’unités modulaires et de leur groupe de classes de diviseurs cuspidaux associé est donnée en termes de produits de fonctions et comparée à la formule classique du nombre de classes relatives pour les corps cyclotomiques [6, 7].
@article{JTNB_2008__20_2_289_0,
author = {Amanda Folsom},
title = {Class invariants and cyclotomic unit groups from special values of modular units},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {289--325},
publisher = {Universit\'e Bordeaux 1},
volume = {20},
number = {2},
year = {2008},
doi = {10.5802/jtnb.628},
mrnumber = {2477505},
zbl = {1172.11019},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.628/}
}
TY - JOUR AU - Amanda Folsom TI - Class invariants and cyclotomic unit groups from special values of modular units JO - Journal de théorie des nombres de Bordeaux PY - 2008 SP - 289 EP - 325 VL - 20 IS - 2 PB - Université Bordeaux 1 UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.628/ DO - 10.5802/jtnb.628 LA - en ID - JTNB_2008__20_2_289_0 ER -
%0 Journal Article %A Amanda Folsom %T Class invariants and cyclotomic unit groups from special values of modular units %J Journal de théorie des nombres de Bordeaux %D 2008 %P 289-325 %V 20 %N 2 %I Université Bordeaux 1 %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.628/ %R 10.5802/jtnb.628 %G en %F JTNB_2008__20_2_289_0
Amanda Folsom. Class invariants and cyclotomic unit groups from special values of modular units. Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 2, pp. 289-325. doi : 10.5802/jtnb.628. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.628/
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