Connected abelian complex Lie groups and number fields
Journal de Théorie des Nombres de Bordeaux, Tome 24 (2012) no. 1, pp. 201-229.

Dans cet article, nous expliquons une façon d’associer à tout corps de nombres certains groupes de Lie complexes et connexes. Nous étudions en particulier le cas des corps de nombres de degré 3 sur qui ne sont pas totalement réels et expliquons le lien entre ceux-ci et les groupes de Cousin (“groupes toroidaux”) de dimension complexe 2 et de rang 3.

In this note we explain a way to associate to any number field some connected complex abelian Lie groups. Further, we study the case of non-totally real cubic number fields, and we see that they are intimately related with the Cousin groups (toroidal groups) of complex dimension 2 and rank 3.

Reçu le :
Révisé le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.793
@article{JTNB_2012__24_1_201_0,
     author = {Daniel Valli\`eres},
     title = {Connected abelian complex {Lie} groups and number fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {201--229},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {1},
     year = {2012},
     doi = {10.5802/jtnb.793},
     zbl = {1282.22003},
     mrnumber = {2914906},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.793/}
}
Daniel Vallières. Connected abelian complex Lie groups and number fields. Journal de Théorie des Nombres de Bordeaux, Tome 24 (2012) no. 1, pp. 201-229. doi : 10.5802/jtnb.793. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.793/

[1] F. Capocasa and F. Catanese, Periodic meromorphic functions. Acta Math. 166 (1991), 1-2, 27–68. | MR 1088982 | Zbl 0719.32005

[2] P. Cousin, Sur les fonctions triplement périodiques de deux variables. Acta. Math. 33 (1910), 1, 105–232. | JFM 41.0492.02 | MR 1555058

[3] F. Gherardelli, Varieta’ quasi abeliane a moltiplicazione complessa. Rend. Sem. Mat. Fis. Milano. 57 (1989), 8, 31–36. | MR 1017917 | Zbl 0704.14033

[4] P. de la Harpe, Introduction to complex tori. Complex analysis and its applications (Lectures, Internat. Sem., Trieste, 1975), Vol. II, pp. 101–144. Internat. Atomic Energy Agency, Vienna, 1976. | MR 480541 | Zbl 0343.14016

[5] A. Morimoto, On the classification of noncompact complex abelian Lie groups. Trans. Amer. Math. Soc. 123 (1966), 220–228. | MR 207893 | Zbl 0144.07903

[6] G. Shimura, Introduction to the arithmetic theory of automorphic functions. Princeton University Press, 1994. | MR 1291394 | Zbl 0872.11023

[7] G. Shimura and Y. Taniyama, Complex multiplication of abelian varieties and its applications to number theory. The Mathematical Society of Japan, 1961. | MR 125113 | Zbl 0112.03502

[8] G. Shimura, Abelian varieties with complex multiplication and modular functions. Princeton University Press, 1998. | MR 1492449 | Zbl 0908.11023

[9] C. L. Siegel, Topics in complex function theory I, II, III. Wiley-Interscience, 1969. | Zbl 0211.10501