On Dirichlet biquadratic fields

Étienne Fouvry; Peter Koymans

doi:10.5802/jtnb.1220

Étienne Fouvry ¹; Peter Koymans ²

¹ Université Paris–Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405 Orsay, France
² Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany

Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 637-646.

Abstract
Résumé

We prove the existence of a subset, with positive natural density, of squarefree integers $n > 0$ such that the $4$ –rank of the ideal class group of $ℚ (\sqrt{- n}, \sqrt{n})$ is $ω_{3} (n) - 1$ , where $ω_{3} (n)$ is the number of prime divisors of $n$ that are $3$ modulo $4$ . Recall that for the class groups associated to $ℚ (\sqrt{n})$ or $ℚ (\sqrt{- n})$ an analogous subset of $n$ does not exist.

Nous prouvons l’existence d’un sous–ensemble, de densité positive, d’entiers $n > 0$ sans facteur carré, tels que le $4$ –rang du groupe de classes d’idéaux de $ℚ (\sqrt{- n}, \sqrt{n})$ vaut $ω_{3} (n) - 1$ . On a désigné par $ω_{3} (n)$ le nombre de diviseurs premiers de l’entier $n$ qui sont congrus à $3$ modulo $4$ . Rappelons que, pour les groupes de classes associés à $ℚ (\sqrt{n})$ et $ℚ (\sqrt{- n})$ , un sous–ensemble analogue d’entiers $n$ n’existe pas.

Received: 2021-04-15
Revised: 2021-05-31
Accepted: 2021-10-01
Published online: 2023-01-27

DOI: 10.5802/jtnb.1220

Classification: 11R29, 11R45
Keywords: Class groups, Dirichlet biquadratic fields

Author's affiliations:

Étienne Fouvry ¹; Peter Koymans ²

¹ Université Paris–Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405 Orsay, France
² Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {On {Dirichlet} biquadratic fields},
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     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Étienne Fouvry; Peter Koymans. On Dirichlet biquadratic fields. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 637-646. doi : 10.5802/jtnb.1220. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1220/

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