# Mathematical ReportsComptes rendus mathématiques

### 11E04 — 3 articles found.

On Cycles and Products of Ideals and Corresponding Indefinite Quadratic Forms

C. R. Math. Rep. Acad. Sci. Canada Vol. 32 (2) 2010, pp. 40–51
Vol.32 (2) 2010
Ahmet Tekcan; Arzu Ozkoc; Hatice Alkan Details
(Received: 2009-11-11 , Revised: 2010-02-02 )
(Received: 2009-11-11 , Revised: 2010-02-02 )

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 $$F_{I_2}$$.

AMS Subject Classification: Quadratic forms over general fields 11E04

C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (2) 2009, pp. 53–64
Vol.31 (2) 2009
Ahmet Tekcan; Arzu Ozkoc Details

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 $$\mathbb{F}_{p}$$.

AMS Subject Classification: Quadratic forms over general fields 11E04

On the cycles of indefinite binary quadratic forms and cycles of ideals III

C. R. Math. Rep. Acad. Sci. Canada Vol. 30 (1) 2008, pp. 22–32
Vol.30 (1) 2008
Ahmet Tekcan Details

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 $$\delta$$.

AMS Subject Classification: Quadratic forms over general fields 11E04