In this paper we study the relationship between non-totally real number fields and toroidal groups , as well as meromorphic periodic functions, exploiting a representation of as the generalized Jacobian of a suitable elliptic curve . We consider in detail the cubic and quartic cases.
In these cases, we write down the relations between the minimal polynomial of a suitable primitive element of and the parameters defining the generalized Jacobian corresponding to the toroidal group associated with the ring of integers. Furthermore, for such a toroidal group we explicitly show the analytic and rational representations of its ring of endomorphisms, the former giving in turn a new (complex) representation of the ring of integers of .
Moreover, for the cubic case, we give an explicit description of the -torsion of in the geometric correspondence of with , as image of a fractional ideal of .
Dans cet article, nous étudions la relation entre les corps de nombres non totalement réels et les groupes toroïdaux , ainsi que les fonctions périodiques méromorphes, en exploitant une représentation de en termes de la jacobienne généralisée d’une courbe elliptique appropriée . Nous considérons en détail les cas cubique et quartique.
Dans ces cas, nous écrivons les relations entre le polynôme minimal d’un élément primitif convenable de et les paramètres définissant la jacobienne généralisée correspondant au groupe toroïdal associé à l’anneau des entiers. En outre, pour un tel groupe toroïdal, nous décrivons explicitement les représentations analytique et rationnelle de son anneau d’endomorphismes, le premier donnant une nouvelle représentation (complexe) de l’anneau des entiers de .
De plus, dans le cas cubique, nous donnons une description explicite de la -torsion de en termes de la correspondance géométrique entre et comme l’image d’un idéal fractionnaire de .
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1281
Keywords: Toroidal groups, non-totally real number fields, generalized Jacobians
Alessandro Dioguardi Burgio 1; Giovanni Falcone 1; Mario Galici 1
CC-BY-ND 4.0
@article{JTNB_2024__36_1_339_0,
author = {Alessandro Dioguardi Burgio and Giovanni Falcone and Mario Galici},
title = {Non-totally real number fields and toroidal groups},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {339--359},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {36},
number = {1},
year = {2024},
doi = {10.5802/jtnb.1281},
mrnumber = {4788376},
zbl = {07892787},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1281/}
}
TY - JOUR AU - Alessandro Dioguardi Burgio AU - Giovanni Falcone AU - Mario Galici TI - Non-totally real number fields and toroidal groups JO - Journal de théorie des nombres de Bordeaux PY - 2024 SP - 339 EP - 359 VL - 36 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1281/ DO - 10.5802/jtnb.1281 LA - en ID - JTNB_2024__36_1_339_0 ER -
%0 Journal Article %A Alessandro Dioguardi Burgio %A Giovanni Falcone %A Mario Galici %T Non-totally real number fields and toroidal groups %J Journal de théorie des nombres de Bordeaux %D 2024 %P 339-359 %V 36 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1281/ %R 10.5802/jtnb.1281 %G en %F JTNB_2024__36_1_339_0
Alessandro Dioguardi Burgio; Giovanni Falcone; Mario Galici. Non-totally real number fields and toroidal groups. Journal de théorie des nombres de Bordeaux, Volume 36 (2024) no. 1, pp. 339-359. doi : 10.5802/jtnb.1281. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1281/
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