tc balıkesir üniversitesi fen bilimleri enstitüsü matematik anabilim ...

ABSTRACT
THE FREY ELLIPTIC CURVES ON FINITE FIELDS
Nazlı YILDIZ İKİKARDEŞ
Balıkesir University, Institute of Science, Department of Mathematics
(Ph. D. Thesis / Supervisor: Prof. Dr. İsmail Naci CANGÜL)
(Second Supervisor: Asst. Prof. Dr. Dilek NAMLI)
Balıkesir, 2006
In this thesis, the number of rational points, their orders, and the group
2 3 2
structure of them, on Frey elliptic curves y = x − n x which are the special case of
simplified Weierstrass equation over finite fields F p where p is prime, are studied.
This study consists of five chapters. In the first chapter, which is the
introductory chapter of this thesis, the thesis is introduced.
In the second chapter, the necessary definitions and results are recalled.
In the third chapter, some new results concerning the number of points on the
2 3 2
Frey elliptic curves y = x − n x are given.
In the fourth chapter, the group structures of the Frey elliptic curves
2 3 2
y = x − n x on finite fields F p are obtained. It is shown that the group structure of
these curves can be of two types according to that p is a prime congruent to 1 or 3
modulo 4. It has been shown that, in the former case, the group structure of rational
+
points on the curve En is isomorphic to En( Fp) ≅ Za× Z a. b,
where ab , ∈ . While
considering the group structure of these curves, the number of rational points is also
studied. Further, it has also been shown that whether the group has elements of order
4 according to whether n is in Q p or not.
In the fifth chapter, the results obtained in the thesis are recalled.
KEY WORDS: elliptic curves over finite fields / rational points / cubic
equations / Frey elliptic curves.
iii