Coset diagrams of the modular group and continued fractions

Ayesha Rafiq; Qaiser Mushtaq

doi:10.1016/j.crma.2019.07.002

Group theory/Number theory

Coset diagrams of the modular group and continued fractions
[Diagrammes de classes du groupe modulaire et fractions continues]

Présenté par : the Editorial Board

Ayesha Rafiq ¹ ; Qaiser Mushtaq ²

¹ Institute of Space Technology, Islamabad, Pakistan
² The Islamia University of Bahawalpur, Bahawalpur, Pakistan

Comptes Rendus. Mathématique, Volume 357 (2019) no. 8, pp. 655-663.

Résumé
Abstract

Le diagramme des classes de chaque orbite de l'action du groupe modulaire sur $Q {(\sqrt{n})}^{⁎} = Q (\sqrt{n}) \cup {\infty}$ contient un circuit $C_{i}$ . Dans cette Note, pour tout $α \in Q (\sqrt{n})$ , le chemin menant au circuit $C_{i}$ et le circuit lui-même sont décrits en termes de fractions continues. Nous montrons que la structure des fractions continues des nombres quadratiques irrationnels réduits est liée à la structure ou au type du circuit. Les trois types de circuits de l'action de $V_{4}$ sur $Q {(\sqrt{n})}^{⁎}$ sont également reliés à la structure des fractions continues. L'action du groupe modulaire sur $Q {(\sqrt{5})}^{⁎}$ est choisie précisément, car un de ses circuits est lié au fait que les rapports des nombres de Fibonacci sont les convergents de la fraction continue du nombre d'or.

The coset diagram for each orbit under the action of the modular group on $Q {(\sqrt{n})}^{⁎} = Q (\sqrt{n}) \cup {\infty}$ contains a circuit $C_{i}$ . For any $α \in Q (\sqrt{n})$ , the path leading to the circuit $C_{i}$ and the circuit itself are obtained through continued fractions in this paper. We show that the structure of the continued fractions of a reduced quadratic irrational element is weaved with the structure or type of the circuit. The three types of circuits of the action of $V_{4}$ on $Q {(\sqrt{n})}^{⁎}$ are also interconnected with the structure of continued fractions. The action of the modular group on $Q {(\sqrt{5})}^{⁎}$ is chosen specifically because a circuit of it is related to the ratio of the Fibonacci numbers being the solution to the continued fractions of the golden ratio.

Reçu le : 2018-09-14
Accepté le : 2019-06-24
Publié le : 2019-08-01

DOI : 10.1016/j.crma.2019.07.002

Affiliations des auteurs :

Ayesha Rafiq ¹ ; Qaiser Mushtaq ²

¹ Institute of Space Technology, Islamabad, Pakistan
² The Islamia University of Bahawalpur, Bahawalpur, Pakistan

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     title = {Coset diagrams of the modular group and continued fractions},
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     pages = {655--663},
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     volume = {357},
     number = {8},
     year = {2019},
     doi = {10.1016/j.crma.2019.07.002},
     language = {en},
}

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Ayesha Rafiq; Qaiser Mushtaq. Coset diagrams of the modular group and continued fractions. Comptes Rendus. Mathématique, Volume 357 (2019) no. 8, pp. 655-663. doi : 10.1016/j.crma.2019.07.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.07.002/

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Cité par Sources :

^☆ This work was presented at the 13th International Pure Mathematics Conference 2012, Islamabad, Pakistan.

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