ABSTRACT
The aim of this article is to provide a brief introduction to elliptic curves along with some deep, ancient and still largely unsolved related problems in arithmetic. It is fair to say that the study of elliptic curves stretches over a vast mathematical canvas that straddles diverse areas such as complex analyis, topology, algebraic geometry, cryptography and even mathematical physics. However, we shall focus on specific aspects of this study related to number theory, where some of the deepest arithmetic problems occur. As usual, Z denotes the ring of integers and Q the field of rational numbers.
