Elliptic curves are plane curves that are the locus of points satisfying a cubic equation in two variables. The first instances of elliptic curves occur in the works of Diophantus and Fermat. Classically, the problem of computing the arc length of an ellipse gave rise to elliptic functions that satisfy cubic equations; hence plane cubic curves are called elliptic curves. In number theory, cubic equations arise naturally in many Diophantine problems, and many problems can be converted into a problem about elliptic curves.
In the first section, we introduce some of these classical problems related to elliptic curves such as the congruent number problem. In later sections, we discuss a few elementary properties of elliptic curves, particularly the fact that there is a way to add points on the curve so that the set of points becomes a group. The group law on the elliptic curve requires a discussion of projective spaces. This is done in the second section. We also include a discussion of elliptic curves modulo primes, and results on the determination of torsion and integer points on elliptic curves. The last section includes a detailed discussion of the elliptic curve factorization method, one of the most important modern factoring methods.
The material of the this chapter is the beginning of a deep and beautiful theory, which is also one of the most active areas of research in number theory. We will not be able to explore the recent applications of the theory of elliptic curves to Fermat's Last Theorem. The interested reader is invited to continue the study in the numerous texts that have recently appeared.