Asymptotics of number fields and the Cohen–Lenstra heuristics

Jürgen Klüners[1]

  • [1] Heinrich-Heine-Universität Düsseldorf, Mathematisches Institut Universitätsstr. 1, 40225 Düsseldorf, Germany

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 3, page 607-615
  • ISSN: 1246-7405

Abstract

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We study the asymptotics conjecture of Malle for dihedral groups D ℓ of order 2 ℓ , where ℓ is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen–Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds.

How to cite

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Klüners, Jürgen. "Asymptotics of number fields and the Cohen–Lenstra heuristics." Journal de Théorie des Nombres de Bordeaux 18.3 (2006): 607-615. <http://eudml.org/doc/249652>.

@article{Klüners2006,
abstract = {We study the asymptotics conjecture of Malle for dihedral groups $D_\ell $ of order $2\ell $, where $\ell $ is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen–Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds.},
affiliation = {Heinrich-Heine-Universität Düsseldorf, Mathematisches Institut Universitätsstr. 1, 40225 Düsseldorf, Germany},
author = {Klüners, Jürgen},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Cohen-Lenstra heuristics; Galois groups},
language = {eng},
number = {3},
pages = {607-615},
publisher = {Université Bordeaux 1},
title = {Asymptotics of number fields and the Cohen–Lenstra heuristics},
url = {http://eudml.org/doc/249652},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Klüners, Jürgen
TI - Asymptotics of number fields and the Cohen–Lenstra heuristics
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 3
SP - 607
EP - 615
AB - We study the asymptotics conjecture of Malle for dihedral groups $D_\ell $ of order $2\ell $, where $\ell $ is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen–Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds.
LA - eng
KW - Cohen-Lenstra heuristics; Galois groups
UR - http://eudml.org/doc/249652
ER -

References

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  1. H. Cohen, Advanced Topics in Computational Number Theory. Springer, Berlin, 2000. Zbl0977.11056MR1728313
  2. H. Cohen, F. Diaz y Diaz, M. Olivier, Enumerating quartic dihedral extensions of ℚ . Compositio Math. 133 (2002), 65–93. Zbl1050.11104MR1918290
  3. H. Cohen, H. W. Lenstra, Jr., Heuristics on class groups of number fields. In: Number theory, Noordwijkerhout 1983, volume 1068 of Lecture Notes in Math., pages 33–62. Springer, Berlin, 1984. Zbl0558.12002MR756082
  4. B. Datskovsky, D. Wright, Density of discriminants of cubic extensions. J. reine angew. Math 386 (1988), 116–138. Zbl0632.12007MR936994
  5. J. Ellenberg, A. Venkatesh, Reflection principles and bounds for class group torsion. To appear in Int. Math. Res. Not. Zbl1130.11060MR2181791
  6. J. Klüners, C. Fieker, Minimal discriminants for small fields with Frobenius groups as Galois groups. J. Numb. Theory 99 (2003), 318–337. Zbl1049.11139MR1968456
  7. J. Klüners, A counterexample to Malle’s conjecture on the asymptotics of discriminants. C. R. Math. Acad. Sci. Paris 340 (2005), 411–414. Zbl1083.11069
  8. J. Klüners, G. Malle, Counting nilpotent Galois extensions. J. Reine Angew. Math. 572 (2004), 1–26. Zbl1052.11075MR2076117
  9. S. Lang, Algebraic Number Theory. Springer, Berlin-Heidelberg-New York, 1986. Zbl0601.12001MR1282723
  10. G. Malle, On the distribution of Galois groups. J. Numb. Theory 92 (2002), 315–322. Zbl1022.11058MR1884706
  11. G. Malle, On the distribution of Galois groups II. Exp. Math. 13 (2004), 129–135. Zbl1099.11065MR2068887

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