Enumerating quartic dihedral extensions of ℚ with signatures

Henri Cohen[1]

  • [1] Université Bordeaux I, Laboratoire A2X, UR 5465 du CNRS, 351 cours de la Libération, 33405 Talence Cedex (France)

Annales de l’institut Fourier (2003)

  • Volume: 53, Issue: 2, page 339-377
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Cohen, Henri. "Enumerating quartic dihedral extensions of ${\mathbb {Q}}$ with signatures." Annales de l’institut Fourier 53.2 (2003): 339-377. <http://eudml.org/doc/116039>.

@article{Cohen2003,
abstract = {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.},
affiliation = {Université Bordeaux I, Laboratoire A2X, UR 5465 du CNRS, 351 cours de la Libération, 33405 Talence Cedex (France)},
author = {Cohen, Henri},
journal = {Annales de l’institut Fourier},
keywords = {discriminant counting; genus character; quartic reciprocity},
language = {eng},
number = {2},
pages = {339-377},
publisher = {Association des Annales de l'Institut Fourier},
title = {Enumerating quartic dihedral extensions of $\{\mathbb \{Q\}\}$ with signatures},
url = {http://eudml.org/doc/116039},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Cohen, Henri
TI - Enumerating quartic dihedral extensions of ${\mathbb {Q}}$ with signatures
JO - Annales de l’institut Fourier
PY - 2003
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 2
SP - 339
EP - 377
AB - 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.
LA - eng
KW - discriminant counting; genus character; quartic reciprocity
UR - http://eudml.org/doc/116039
ER -

References

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  3. H. Cohen, Comptage exact de discriminants d'extensions abéliennes, J. Th. Nombres Bordeaux 12 (2000), 379-397 Zbl0976.11055MR1823191
  4. H. Cohen, F. Diaz y Diaz, M. Olivier, Counting discriminants of number fields Zbl1193.11109
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  6. H. Cohen, F. Diaz y Diaz, M. Olivier, Counting discriminants of number fields of degree up to four, Proceedings ANTS IV, Leiden (2000) 1838 (2000), 269-283, Springer-Verlag Zbl0987.11080
  7. H. Cohen, F. Diaz y Diaz, M. Olivier, Enumerating quartic dihedral extensions of ℚ , Compositio Math 133 (2002), 65-93 Zbl1050.11104MR1918290
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  9. H. Cohen, F. Diaz y Diaz, M. Olivier, On the Density of Discriminants of Cyclic Extensions of Prime Degree, J. reine angew. Math 550 (2002), 169-209 Zbl1004.11063MR1925912
  10. B. Datskovsky, D. J. Wright, Density of discriminants of cubic extensions, J. reine angew. Math 386 (1988), 116-138 Zbl0632.12007MR936994
  11. H. Davenport, H. Heilbronn, On the density of discriminants of cubic fields I, Bull. London Math. Soc 1 (1969), 345-348 Zbl0211.38602MR254010
  12. H. Davenport, H. Heilbronn, On the density of discriminants of cubic fields II, Proc. Royal. Soc. A 322 (1971), 405-420 Zbl0212.08101MR491593
  13. F. Lemmermeyer, Reciprocity laws, (2000), Springer-Verlag Zbl0949.11002MR1761696
  14. S. Mäki, On the density of abelian number fields, (1985) Zbl0566.12001MR791087
  15. S. Mäki, The conductor density of abelian number fields, J. London Math. Soc 47 (1993), 18-30 Zbl0727.11041MR1200974
  16. D. J. Wright, Distribution of discriminants of abelian extensions, Proc. London Math. Soc 58 (1989), 17-50 Zbl0628.12006MR969545
  17. D. J. Wright, A. Yukie, Prehomogeneous vector spaces and field extensions, Invent. Math 110 (1992), 283-314 Zbl0803.12004MR1185585
  18. A. Yukie, Density theorems related to prehomogeneous vector spaces Zbl0969.11538MR1840077

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