Connected abelian complex Lie groups and number fields
Daniel Vallières1
1University of California, San Diego 9500 Gilman Drive # 0112 La Jolla, CA 92093-0112 USA Current address : Fakultät für Mathematik Institut für theoritische Informatik und Mathematik Universität der Bundeswehr Münschen 85577 Neubiberg Deutschland
Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 1, pp. 201-229

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.

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 ℚ 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.

Received:
Revised:
Published online:
DOI: 10.5802/jtnb.793

Daniel Vallières  1

1 University of California, San Diego 9500 Gilman Drive # 0112 La Jolla, CA 92093-0112 USA Current address : Fakultät für Mathematik Institut für theoritische Informatik und Mathematik Universität der Bundeswehr Münschen 85577 Neubiberg Deutschland 
Daniel Vallières. Connected abelian complex Lie groups and number fields. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 1, pp. 201-229. doi: 10.5802/jtnb.793
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[1] F. Capocasa and F. Catanese, Periodic meromorphic functions. Acta Math. 166 (1991), 1-2, 27–68. | MR | Zbl

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

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

[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 | Zbl

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

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

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

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

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

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