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

Arzu Ozkoc, Uludag University, Faculty of Science, Department of Mathematics, Gorukle, Bursa, Turkiye; email: aozkoc@uludag.edu.tr

Hatice Alkan, Uludag University, Faculty of Science, Department of Mathematics, Gorukle, Bursa, Turkiye; email: halkan@uludag.edu.tr

Abstract/Résumé:

Let \(k\geq 2\) be an integer and let \(D = k^2+k+1\) be a positive non-square integer. In this work, we derive some properties (including cycles) of ideals \(I_1 = [k,k-1+\sqrt{D}]\), \(I_2 = [k+1,k+\sqrt{D}]\) and their product \(I\). In the last section, we consider the indefinite binary quadratic forms \(F_{I_1}\), \(F_{I_2}\) and \(F_I\) of discriminant \(\Delta=4D\) which correspond to \(I_1\), \(I_2\) and \(I\), respectively and we formulate the cycle of \(F_{I_1}\) and \(F_{I_2}\).

Soit \(k\ge 2\) un entier tel que \(D = k^2+k+1\) ne soit pas le carré d’un entier. Dans ce travail, on obtient quelques propriétés (incluant des cycles) des idéaux \(I_1 = [k,k-1+\sqrt{D}]\), \(I_2 = [k+1,k+\sqrt{D}]\) et de leur produit \(I\). Dans le dernier paragraphe, on considère les formes quadratiques binaires indéfinies \(F_{I_1}\), \(F_{I_2}\) et \(F_I\) de discriminant \(\Delta=4D\) qui correspondent respectivement aux idéaux \(I_1\), \(I_2\) et \(I\) à fin de formuler le cycle de \(F_{I_1}\) et \(F_{I_2}\).

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

AMS Subject Classification: Quadratic forms over general fields 11E04

PDF(click to download): On Cycles and Products of Ideals and Corresponding Indefinite Quadratic Forms

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

Arzu Ozkoc, Uludag University, Faculty of Science, Department of Mathematics, Gorukle, Bursa, Turkiye; email: aozkoc@uludag.edu.tr

Abstract/Résumé:

We consider some properties of positive definite binary quadratic forms \(F_{j}\) in the family \(\Omega \). We determine the number of integer solutions of quadratic congruences \(C_{F_{j}}\) and determine the number of rational points on singular curves \(E_{F_{j}}\) related to \(F_{j}\) over finite fields \(\mathbb{F}_{p}\).

On considère quelques propriétés des formes quadratiques binaires définies positives \(F_{j}\) dans la famille \(\Omega \). On détermine le nombre de solutions entières des congruences quadratiques \(C_{F_{j}}\), et le nombre de points rationnels sur des courbes singulières \(E_{F_{j}}\) reliées aux \(F_{j}\) sur des corps finis \(\mathbb{F}_{p}\).

Keywords: binary quadratic form, elliptic curve, quadratic congruence, singular curve

AMS Subject Classification: Quadratic forms over general fields 11E04

PDF(click to download): Positive Definite Binary Quadratic Forms, Quadratic Congruences, and Singular Curves

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