Application of the Strong Artin Conjecture to the Class Number Problem

Cho, Peter J. ; Kim, Henry H.

Canadian journal of mathematics, Tome 65 (2013) no. 6, pp. 1201-1216

Résumé

We construct unconditionally several families of number fields with the largest possible class numbers. They are number fields of degree 4 and 5 whose Galois closures have the Galois group ${{A}_{4}},\,{{S}_{4}}$ , and ${{S}_{5}}$ . We first construct families of number fields with smallest regulators, and by using the strong Artin conjecture and applying the zero density result of Kowalski–Michel, we choose subfamilies of $L$ -functions that 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 the extreme class numbers.

Détail

DOI : 10.4153/CJM-2012-031-3

Mots-clés : 11R29, 11M41, class number, strong Artin conjecture

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