Dihedral and cyclic extensions with large class numbers

Peter J. Cho; Henry H. Kim

doi:10.5802/jtnb.812

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, Volume 24 (2012) no. 3, pp. 583-603.

Abstract
Résumé

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.

Cet article est la suite de [2]. Nous construisons inconditionnellement plusieurs familles de corps de nombres ayant un grand nombre de classes. Ce sont des corps de nombres dont la clôture galoisienne a pour groupe de Galois les groupes dièdraux $D_{n}$ , $n = 3, 4, 5$ , et les groupes cycliques $C_{n}$ , $n = 4, 5, 6$ . Nous construisons d’abord des familles de corps de nombres à petits régulateurs et, en utilisant la conjecture d’Artin forte et en appliquant une variante du résultat de densité nulle de Kowalski et Michel, nous choisissons des sous-familles telles que les fonctions $L$ correspondantes soient sans zéro près de $1$ . Pour ces sous-familles, la fonction $L$ prend une valeur extrémale en $s = 1$ et, par la formule du nombre de classes, nous obtenons un grand nombre de classes.

MR Zbl

DOI: 10.5802/jtnb.812

Author's affiliations:

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

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Peter J. Cho; Henry H. Kim. Dihedral and cyclic extensions with large class numbers. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 3, pp. 583-603. doi : 10.5802/jtnb.812. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.812/

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