Rational points of elliptic curves

6. SUMMARY It is well kowii that by suitable transformations a cubic equation of the form !(*?> y)= J2 aH x' V* ' = ° j+j<3 with rational coefficients can be transformed into an equation of the following form which is referred to as Weierstrass Normal Form: y2 = g(x) = x3 + ax2 + bx + c; a, b, c ? Q (6.1) If the equation g(x) = 0 has distinct complex roots, then the curve given by the equation (6.1) is called an elliptic curve over Q. If P and R are two points on an elliptic curve, then the line joining these two points intersects the curve in a uniquely determined third point. The symmetric partner of this point about the x- axis is defined to be the sum P + R of the points P and R. With this definition of addition, the set of points on the elliptic curve becomes an Abeliangroup whose identity element is the point at infinity of the elliptic curve. Given an elliptic curve and a point P = (x, y) on it; if a:, y ? Q, then P is callad a rational point of the curve. If the point at infinity is considered as a rational point, then the set of rational points constitutes s subgroup of the group of points on the elliptic curve. The aim of this thesis is to investigate the structure and properties of this subgroup. One of the questions that one might ask, in this context, is whether or not this subgroup finite or finitely generated. The answer to this question is Mordell's Theorem which states that the subgroup of rational points is a finitely generated Abelian group. In the first chapter of the thesis we give the preliminaries and introduce the notations. In the second chapter, we give the proof of Mordell's Theorem (under the assumption that there is a rational point of order two) where we use the the concept of height and also some homomorphic mappings. We also give some examples in connection with Mordell's Theorem. Since the Fundamental Theorem of Finitely Generated Abelian groups gives the structure of such groups completely; one can say that the structure of the group of rational points of an elliptic curve is completely described by Mordell's Theorem. By means of Nagel-Lutz Theorem and Mazur's Theorem the we have stated in the first chapter, it is possble to determine the subgroup of elements of finite order in the group of rational points of elliptic curves over Q. However, not much is known about the rank r of this group, and so far, no example is given for an elliptic curve with r > 15. In the third chapter of the thesis, we give the definition of elliptic curves over 74finite fields. We introduce the concept of rational points in that case also, and we reproduce the proof of a theorem of Gauss wliich gives the number of rational points of a particular elliptic curve over a finite field. 75