Ahmet Tekcan, Uludag University, Faculty of Science, Department of Mathematics, Gorukle, Bursa, Turkiye; email: tekcan@uludag.edu.tr

Abstract/Résumé:

Let \(\delta\) be a real quadratic irrational integer with trace \(t = \delta+\overline{\delta}\) and norm \(n = \delta.\overline{\delta}\). Then for a real quadratic irrational \(\gamma \in \mathbb{Q}(\delta)\), there are rational integers \(P\) and \(Q\) such that \(\gamma = \frac{P+\delta}{Q}\) with \(Q|(\delta+P) (\overline{\delta}+P)\). So for each \(\gamma\), we have an ideal \(I_{\gamma} = [Q,P+\delta]\) and an indefinite quadratic form \(F_{\gamma}(x,y) = Q(x+\delta y) (x+\overline{\delta}y)\) of discriminant \(\Delta = t^2-4n\). In this work, we derive some properties of \(I_{\gamma}\) and \(F_{\gamma}\) for some specific values of \(\delta\).

Soit \(\delta\) un entier irrationel quadratique réel de trace \(t = \delta+\overline{\delta}\) et norme \(n = \delta.\overline{\delta}\). Pour un irrationel quadratique réel \(\gamma \in \mathbb{Q}(\delta)\), il existe des entiers rationels \(P\) et \(Q\) tels que \(\gamma = \frac{P+\delta}{Q}\) avec \(Q|(\delta+P) (\overline{\delta}+P)\). Ainsi pour chaque \(\gamma\), on a un idéal \(I_{\gamma} = [Q,P+\delta]\) et une forme quadratique indéfinie \(F_{\gamma} (x,y) = Q(x+\delta y) (x+\overline{\delta}y)\) de discriminant \(\Delta = t^2-4n\). On déduit quelques propriétés de \(I_{\gamma}\) et \(F_{\gamma}\) pour certains valeurs de \(\delta\).

Keywords: cycles of forms, cycles of ideals, ideals, quadratic forms

AMS Subject Classification: Quadratic forms over general fields 11E04

PDF(click to download): On the cycles of indefinite binary quadratic forms and cycles of ideals III