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

C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (2) 2009, pp. 53–64

June 30, 2009

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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