Abstract

This thesis provides a self-contained introduction to elliptic curves accessible to advanced undergraduates and graduate students in mathematics, with emphasis on the the theory of elliptic curves over finite fields. In Chapter 1, affine and projective planes are introduced. Chapter 2 introduces the theory of algebraic curves and the Weierstrass Normal Form of a cubic curve is derived. In Chapter 3 we define derivations on arbitrary polynomial rings, and prove the group law for elliptic curves. Chapter 4 discusses elliptic curves over finite fields and proves some results on counting points.