Finite Fields and Elliptic Curves

An equation in two variables can have infinitely many real solutions. The resulting geometric object is one-dimensional, i.e. a curve. If we replace the real numbers with a finite field, then there are only finitely many solutions. We consider a special class of curves known as elliptic curves and study the number of solutions as we vary both the finite field and curve. In this talk we will define finite fields with prime order and describe the counting problems we considered.