Singular curves and elliptic divisibility sequences

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Thesis Type: Doctorate

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

Approval Date: 2009

Thesis Language: Turkish

Student: Betül Gezer

Supervisor: Osman Bizim

Abstract:

The aim of this work is to combine two topics of mathematics, “singular curves” and “elliptic divisibility sequences”. In the first part, the elliptic curve theory is discussed and some important properties of this theory are given. An elliptic curve is a curve E defined over F is given by an equation E : y 2 = x 3 + Ax + B where F is a field of characteristic not equal to 2 or 3. If the cubic x 3 + Ax + B = 0 has multiple roots, then the set of solution points form a singular curve. In the third chapter, the rational points on singular curves over finite fields Fp (where p > 3 is a prime) are considered. Some results concerning the number of the points on the singular curves are given by means of quadratic residue character, and the cubic residue character. Also some results are given on the sum of x and y coordinates of the points (x, y) on these curves. Then the structure of the group of the rational points and torsion points on these curves are determined and finally the results concerning the number of points in Fp is generalize to Fp n . In the second part, the theory of elliptic divisibility sequences which arise from elliptic curves and which contain a zero term is discussed. Elliptic divisibility sequences are described in detail in Morgan Ward’s paper “Memoir on elliptic divisibility sequences” published in 1948. In the fifth chapter, the techniques studied by Morgan Ward to characterize to sequences in certain ranks are developed. First of all, elliptic divisibility sequences over finite fields are defined. After that, general terms of these sequences over finite fields are given. Then elliptic curves and singular curves associated to this sequences are given and some results concerning of singular curves and singular elliptic divisibility sequences are given.