Elliptic curves over different fields

For better understanding of elliptic curves, we first define projective plane and points at infinity. Then we define elliptic curves and show some different forms of equations that represent them. Throughout the thesis, mostly we use Weierstrass form for elliptic curves, since every elliptic curve, with at least one point lying on it, can be transformed into it. We can look on elliptic curves as a set of points that solve the given equation of elliptic curve. That set of points, with point at infinity in which all lines parallel to y axis meet, represent an abelian group for adding points. Furthermore, we define the group structure of elliptic curves and adding points on them. As elliptic curves act differently depending on the field they are studied in, we discuss elliptic curves over real numbers, rational numbers and over finite field Z_p where p is a prime number. We also consider elliptic curves over integer numbers, even though a set of integer numbers is not a field and we cannot define a group structure with adding points, like we did before. When dealing with elliptic curves over real numbers, we focus on finding zeroes of elliptic curves while in other cases the focus on finding and counting points on elliptic curve.