Dihedral and cyclic extensions with large class numbers

Cho, Peter J.; Kim, Henry H.

Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 3, pp. 583-603.

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.

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.

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