Dihedral and cyclic extensions with large class numbers

Peter J. Cho[1]; Henry H. Kim[2]

  • [1] Department of Mathematics University of Toronto Toronto ON M5S 2E4 Canada
  • [2] Department of Mathematics University of Toronto Toronto ON M5S 2E4 CANADA and Korea Institute for Advanced Study

Journal de Théorie des Nombres de Bordeaux (2012)

  • Volume: 24, Issue: 3, page 583-603
  • ISSN: 1246-7405

Abstract

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This paper is a continuation of [2]. We construct unconditionally several families of number fields with large class numbers. They are number fields whose Galois closures have as the Galois groups, dihedral groups D n , n = 3 , 4 , 5 , and cyclic groups C n , n = 4 , 5 , 6 . We first construct families of number fields with small regulators, and by using the strong Artin conjecture and applying some modification of zero density result of Kowalski-Michel, we choose subfamilies such that the corresponding L -functions are zero free close to 1. For these subfamilies, the L -functions have the extremal value at s = 1 , and by the class number formula, we obtain large class numbers.

How to cite

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Cho, Peter J., and Kim, Henry H.. "Dihedral and cyclic extensions with large class numbers." Journal de Théorie des Nombres de Bordeaux 24.3 (2012): 583-603. <http://eudml.org/doc/251027>.

@article{Cho2012,
abstract = {This paper is a continuation of [2]. We construct unconditionally several families of number fields with large class numbers. They are number fields whose Galois closures have as the Galois groups, dihedral groups $D_n$, $n=3,4,5$, and cyclic groups $C_n$, $n=4,5,6$. We first construct families of number fields with small regulators, and by using the strong Artin conjecture and applying some modification of zero density result of Kowalski-Michel, we choose subfamilies such that the corresponding $L$-functions are zero free close to 1. For these subfamilies, the $L$-functions have the extremal value at $s=1$, and by the class number formula, we obtain large class numbers.},
affiliation = {Department of Mathematics University of Toronto Toronto ON M5S 2E4 Canada; Department of Mathematics University of Toronto Toronto ON M5S 2E4 CANADA and Korea Institute for Advanced Study},
author = {Cho, Peter J., Kim, Henry H.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {class number; dihedral and cyclic extensions; L-functions},
language = {eng},
month = {11},
number = {3},
pages = {583-603},
publisher = {Société Arithmétique de Bordeaux},
title = {Dihedral and cyclic extensions with large class numbers},
url = {http://eudml.org/doc/251027},
volume = {24},
year = {2012},
}

TY - JOUR
AU - Cho, Peter J.
AU - Kim, Henry H.
TI - Dihedral and cyclic extensions with large class numbers
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2012/11//
PB - Société Arithmétique de Bordeaux
VL - 24
IS - 3
SP - 583
EP - 603
AB - This paper is a continuation of [2]. We construct unconditionally several families of number fields with large class numbers. They are number fields whose Galois closures have as the Galois groups, dihedral groups $D_n$, $n=3,4,5$, and cyclic groups $C_n$, $n=4,5,6$. We first construct families of number fields with small regulators, and by using the strong Artin conjecture and applying some modification of zero density result of Kowalski-Michel, we choose subfamilies such that the corresponding $L$-functions are zero free close to 1. For these subfamilies, the $L$-functions have the extremal value at $s=1$, and by the class number formula, we obtain large class numbers.
LA - eng
KW - class number; dihedral and cyclic extensions; L-functions
UR - http://eudml.org/doc/251027
ER -

References

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