On the rank of elliptic curves

In this six-part study, some basic topics of elliptic curves theory are discussed. In the first part, elliptic curves are introduced and in the second part, the group structure of the elliptic curves is given. A special point addition rule on the points of elliptic curves is defined and the projective coordinates should be used to obtain "point at infinity" in order to have a commutative group. This constitutes the content of the third chapter. In the fourth chapter, it is seen that elliptic curves form a commutative group by showing the other group axioms, especially associativity which needs hard working. In the fifth chapter, the results of Mordell, Lutz-Nagell and Mazur which are important results for the determination of the algebraic structure of elliptic curves are investigated. In the last chapter, the concept of rank of elliptic curves is considered and some results are given on the rank of quadratic twist families. This study is compiled. Keywords: Elliptic curves; rank; elliptic curves group structure