In a previous paper, we have given asymptotic formulas for the number of isomorphism classes of -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.
Dans un précédent article, nous avons donné des formules asymptotiques pour le nombre de classes d’isomorphismes d’extensions 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.
Keywords: discriminant counting, genus character, quartic reciprocity
Mot clés : discriminant croissant, caractère de genre, réciprocité quartique
@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, Volume 53 (2003) no. 2, pp. 339-377. doi : 10.5802/aif.1946. http://www.numdam.org/articles/10.5802/aif.1946/
[1] Higher Composition Laws (June 2001) (PhD Thesis, Princeton Univ.)
[2] A Course in Computational Algebraic Number Theory (fourth corrected printing), Graduate Texts in Math, 138, Springer-Verlag, 2000 | MR | Zbl
[3] Comptage exact de discriminants d'extensions abéliennes, J. Th. Nombres Bordeaux, Volume 12 (2000), pp. 379-397 | DOI | Numdam | MR | Zbl
[4] Counting discriminants of number fields (preprint)
[5] 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] 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] Enumerating quartic dihedral extensions of , Compositio Math, Volume 133 (2002), pp. 65-93 | DOI | MR | Zbl
[8] Densité des discriminants des extensions cycliques de degré premier, C.R. Acad. Sci. Paris, Volume 330 (2000), pp. 61-66 | MR | Zbl
[9] 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] Density of discriminants of cubic extensions, J. reine angew. Math, Volume 386 (1988), pp. 116-138 | DOI | EuDML | MR | Zbl
[11] On the density of discriminants of cubic fields I, Bull. London Math. Soc, Volume 1 (1969), pp. 345-348 | DOI | MR | Zbl
[12] On the density of discriminants of cubic fields II, Proc. Royal. Soc. A, Volume 322 (1971), pp. 405-420 | DOI | MR | Zbl
[13] Reciprocity laws, Springer Monographs in Math., Springer-Verlag, 2000 | MR | Zbl
[14] On the density of abelian number fields (1985) (Thesis, Helsinki) | MR | Zbl
[15] The conductor density of abelian number fields, J. London Math. Soc, Volume 47 (1993) no. 2, pp. 18-30 | DOI | MR | Zbl
[16] Distribution of discriminants of abelian extensions, Proc. London Math. Soc, Volume 58 (1989) no. 3, pp. 17-50 | DOI | MR | Zbl
[17] Prehomogeneous vector spaces and field extensions, Invent. Math, Volume 110 (1992), pp. 283-314 | DOI | EuDML | MR | Zbl
[18] Density theorems related to prehomogeneous vector spaces (preprint) | MR | Zbl
Cited by Sources: