no. 2
Enumerating quartic dihedral extensions of ℚ with signatures
[Énumération des corps quartiques diédraux avec conditions de signature]
Cohen, Henri1
1 Université Bordeaux I, Laboratoire A2X, UR 5465 du CNRS, 351 cours de la Libération, 33405 Talence Cedex (France)
Annales de l'Institut Fourier, Tome 53 (2003) no. 2, pp. 339-377.

Dans un précédent article, nous avons donné des formules asymptotiques pour le nombre de classes d’isomorphismes d’extensions D 4 classées par discriminant croissant, que la signature des extensions soit ou non spécifiée. Nous avons également donné des formules exactes très efficaces pour ce nombre quand on ne spécifie pas la signature. Le but du présent article est de donner de telles formules exactes quand la signature est imposée. Le problème se complique du fait de l’apparition de caractères sur les groupes de classes de rayon qui ne sont pas des caractères de genre.

In a previous paper, we have given asymptotic formulas for the number of isomorphism classes of D 4 -extensions with discriminant up to a given bound, both when the signature of the extensions is or is not specified. We have also given very efficient exact formulas for this number when the signature is not specified. The aim of this paper is to give such exact formulas when the signature is specified. The problem is complicated by the fact that the ray class characters which appear are not all genus characters.

DOI : 10.5802/aif.1946
Classification : 11R16, 11R29, 11R45, 11Y40
Keywords: discriminant counting, genus character, quartic reciprocity
Mot clés : discriminant croissant, caractère de genre, réciprocité quartique
Cohen, Henri 1

1 Université Bordeaux I, Laboratoire A2X, UR 5465 du CNRS, 351 cours de la Libération, 33405 Talence Cedex (France) 
@article{AIF_2003__53_2_339_0,
     author = {Cohen, Henri},
     title = {Enumerating quartic dihedral extensions of ${\mathbb {Q}}$ with signatures},
     journal = {Annales de l'Institut Fourier},
     pages = {339--377},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {2},
     year = {2003},
     doi = {10.5802/aif.1946},
     mrnumber = {1990000},
     zbl = {01940698},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1946/}
}
TY  - JOUR
AU  - Cohen, Henri
TI  - Enumerating quartic dihedral extensions of ${\mathbb {Q}}$ with signatures
JO  - Annales de l'Institut Fourier
PY  - 2003
SP  - 339
EP  - 377
VL  - 53
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1946/
DO  - 10.5802/aif.1946
LA  - en
ID  - AIF_2003__53_2_339_0
ER  - 
%0 Journal Article
%A Cohen, Henri
%T Enumerating quartic dihedral extensions of ${\mathbb {Q}}$ with signatures
%J Annales de l'Institut Fourier
%D 2003
%P 339-377
%V 53
%N 2
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1946/
%R 10.5802/aif.1946
%G en
%F AIF_2003__53_2_339_0
Cohen, Henri. Enumerating quartic dihedral extensions of ${\mathbb {Q}}$ with signatures. Annales de l'Institut Fourier, Tome 53 (2003) no. 2, pp. 339-377. doi : 10.5802/aif.1946. http://www.numdam.org/articles/10.5802/aif.1946/

[1] M. Bhargava Higher Composition Laws (June 2001) (PhD Thesis, Princeton Univ.)

[2] H. Cohen A Course in Computational Algebraic Number Theory (fourth corrected printing), Graduate Texts in Math, 138, Springer-Verlag, 2000 | MR | Zbl

[3] H. Cohen Comptage exact de discriminants d'extensions abéliennes, J. Th. Nombres Bordeaux, Volume 12 (2000), pp. 379-397 | DOI | Numdam | MR | Zbl

[4] H. Cohen; F. Diaz y Diaz; M. Olivier Counting discriminants of number fields (preprint)

[5] H. Cohen; F. Diaz y Diaz; M. Olivier Construction of tables of quartic fields using Kummer theory, Proceedings ANTS IV, Leiden (2000) (Lecture Notes in Computer Science), Volume 1838 (2000), pp. 257-268 | Zbl

[6] H. Cohen; F. Diaz y Diaz; M. Olivier Counting discriminants of number fields of degree up to four, Proceedings ANTS IV, Leiden (2000) (Lecture Notes in Computer Science), Volume 1838 (2000), pp. 269-283 | Zbl

[7] H. Cohen; F. Diaz y Diaz; M. Olivier Enumerating quartic dihedral extensions of ℚ , Compositio Math, Volume 133 (2002), pp. 65-93 | DOI | MR | Zbl

[8] H. Cohen; F. Diaz y Diaz; M. Olivier Densité des discriminants des extensions cycliques de degré premier, C.R. Acad. Sci. Paris, Volume 330 (2000), pp. 61-66 | MR | Zbl

[9] H. Cohen; F. Diaz y Diaz; M. Olivier On the Density of Discriminants of Cyclic Extensions of Prime Degree, J. reine angew. Math, Volume 550 (2002), pp. 169-209 | DOI | MR | Zbl

[10] B. Datskovsky; D. J. Wright Density of discriminants of cubic extensions, J. reine angew. Math, Volume 386 (1988), pp. 116-138 | DOI | EuDML | MR | Zbl

[11] H. Davenport; H. Heilbronn On the density of discriminants of cubic fields I, Bull. London Math. Soc, Volume 1 (1969), pp. 345-348 | DOI | MR | Zbl

[12] H. Davenport; H. Heilbronn On the density of discriminants of cubic fields II, Proc. Royal. Soc. A, Volume 322 (1971), pp. 405-420 | DOI | MR | Zbl

[13] F. Lemmermeyer Reciprocity laws, Springer Monographs in Math., Springer-Verlag, 2000 | MR | Zbl

[14] S. Mäki On the density of abelian number fields (1985) (Thesis, Helsinki) | MR | Zbl

[15] S. Mäki The conductor density of abelian number fields, J. London Math. Soc, Volume 47 (1993) no. 2, pp. 18-30 | DOI | MR | Zbl

[16] D. J. Wright Distribution of discriminants of abelian extensions, Proc. London Math. Soc, Volume 58 (1989) no. 3, pp. 17-50 | DOI | MR | Zbl

[17] D. J. Wright; A. Yukie Prehomogeneous vector spaces and field extensions, Invent. Math, Volume 110 (1992), pp. 283-314 | DOI | EuDML | MR | Zbl

[18] A. Yukie Density theorems related to prehomogeneous vector spaces (preprint) | MR | Zbl

Cité par Sources :