Generalized Pell’s equations and Weber’s class number problem

Hyuga Yoshizaki

doi:10.5802/jtnb.1249

Hyuga Yoshizaki ¹

¹ Department of Mathematics Graduate school of science and Technology Tokyo University of Science 2641, Yamazaki, Noda-shi, 278-8510, Chiba, Japan

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 2, pp. 373-391.

Résumé
Abstract

Nous étudions une généralisation de l’équation de Pell dont les coefficients sont certains entiers algébriques. Soient $X_{0} = 0$ et $X_{n} = \sqrt{2 + X_{n - 1}}$ pour chaque $n \in ℤ_{\geq 1}$ . Nous étudions les solutions de l’équation $x^{2} - X_{n}^{2} y^{2} = 1$ dans $ℤ [X_{n - 1}]$ . En imitant la solution de l’équation de Pell classique, nous introduisons de nouveaux développements en fraction continue de $X_{n}$ sur $ℤ [X_{n - 1}]$ et obtenons une solution explicite de l’équation de Pell généralisée. De plus, nous montrons que notre solution explicite génère toutes les solutions si et seulement si la réponse au problème du nombre de classes de Weber est affirmative. Nous obtenons également une congruence pour le rapport entre les nombres de classes dans la $ℤ_{2}$ -extension sur les rationnels et montrons la convergence de la suite des nombres de classes dans $ℤ_{2}$ .

We study a generalization of Pell’s equation, whose coefficients are certain algebraic integers. Let $X_{0} = 0$ and $X_{n} = \sqrt{2 + X_{n - 1}}$ for each $n \in ℤ_{\geq 1}$ . We study the $ℤ [X_{n - 1}]$ -solutions of the equation $x^{2} - X_{n}^{2} y^{2} = 1$ . By imitating the solution to the classical Pell’s equation, we introduce new continued fraction expansions for $X_{n}$ over $ℤ [X_{n - 1}]$ and obtain an explicit solution of the generalized Pell’s equation. In addition, we show that our explicit solution generates all the solutions if and only if the answer to Weber’s class number problem is affirmative. We also obtain a congruence relation for the ratios of the class numbers of the $ℤ_{2}$ -extension over the rationals and show the convergence of the class numbers in $ℤ_{2}$ .

Reçu le : 2021-07-10
Révisé le : 2023-01-11
Accepté le : 2023-01-31
Publié le : 2023-10-10

DOI : 10.5802/jtnb.1249

Classification : 11J70, 11D57, 11R29, 11R18, 11R27
Mots clés : Pell’s equation, Continued fraction, Weber’s class number problem.

Affiliations des auteurs :

Hyuga Yoshizaki ¹

¹ Department of Mathematics Graduate school of science and Technology Tokyo University of Science 2641, Yamazaki, Noda-shi, 278-8510, Chiba, Japan

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Generalized {Pell{\textquoteright}s} equations and {Weber{\textquoteright}s} class number problem},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {373--391},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Hyuga Yoshizaki. Generalized Pell’s equations and Weber’s class number problem. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 2, pp. 373-391. doi : 10.5802/jtnb.1249. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1249/

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