Advances in Theoretical and Applied Mathematics
  • Year: 2007
  • Volume: 2
  • Issue: 2

Rational points on frey elliptic curves on finite fields1

  • Author:
  • Musa Demirci1,, Gökhan Soydan1,, Nazlı Yildiz İki Kardeş2, İsmail Naci Cangül1,
  • Total Page Count: 8
  • Page Number: 129 to 136

1Department of Mathematics, Uludaĝ University, 16059, Bursa, Turkey.

2Department of Mathematics, Balıkesir University, 10100, Balıkesir, Turkey. E-mail: nyildiz@balikesir.edu.tr

*E-mail: mdemirci@uludag.edu.tr

**E-mail: gsoydan@uludag.edu.tr

***E-mail: cangul@uludag.edu.tr

AMS Subject Classification: 11G20, 14H25, 14K15, 14G99.

Abstract

In this work, we study the rational points on elliptic curves of the form over finite fields Fp. These kind of elliptic curves are widely known as Frey curves. We show that there are two different possible cases where p ≡ 1 or 3 (mod 4). In the former case there is no general result which can be obtained about the rational points while in the latter case, we find exactly p + 1 points on these curves. We give results on the abscissae and ordinates of these points according to whether they are quadratic residues or not.

Keywords

Elliptic curves over finite fields, rational points