**Thesis Type:** Postgraduate

**Institution Of The Thesis:** Bursa Uludağ University, Fen Bilimleri Enstitüsü, Fen Bilimleri Enstitüsü, Turkey

**Approval Date:** 2011

**Thesis Language:** Turkish

**Student:** Buse Çapa

**Supervisor: **Osman Bizim

In this work, Tate normal forms of elliptic curves defined over finite fields are discussed and the structure of groups of points on these curves are given.In the first chapter, the concepts form the basis for the second and third chapters are given. The basic concepts and theorems of algebra and number theory are discussed in this chapter.In the second chapter, elliptic curves, singular curves and properties of these curves defined over finite fields are considered.Third chapter is the main part of the work. First, the concept of Tate normal form of elliptic curves are defined. Then by obtaining the points on the Tate normal form of the elliptic curves defined over finite fields Fp (where p is a prime), the orders of these curves are determined. According to these results curves are classified with respect to the orders. The group structures of the points on these curves are given by using the order of the points on the curves.