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Positive Definite Binary Quadratic Forms, Quadratic Congruences, and Singular Curves

C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (2) 2009, pp. 53–64
Vol.31 (2) 2009
Ahmet Tekcan; Arzu Ozkoc Details
(Received: 2008-09-08 )
(Received: 2008-09-08 )

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

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