Pontos racionais em curvas elípticas

Abstract

We study Elliptic Curves. Initially we describe an operation on the curve which makes the set of points of an elliptic curve, over any eld, an abelian group. We introduce the Nagell-Lutz theorem which shows the necessary conditions for a rational point to have nite order. Next, we prove Mordell\'s theorem for curves dened by y2 = x3 + ax2 + bx. This theorem says that the set of rational points on an elliptic curve is a nitely generated abelian group. On the proof of this result, an algorithm is constructed. With this algorithm, it is possible, in some cases, to calculate the rank of the elliptic curve. We use this algorithm and the Nagell-Lutz theorem to study the Mordell-Weil Group of Elliptic Curves of the form y2 = x3 - px, where p is a prime number.