Cubic Equations

Now assume that is cubic. The previous method no longer works. Indeed, if  we have a point and draw a line of slope , then equation

which we need to solve to find the points of intersection of the line with the curve will be cubic:

After factoring out , we are left with a quadratic equation, whose roots need not lie in .

Quadratic Equations

Consider a quadratic polynomial Assume, as usual, that it is irreducible, i.e., that the corresponding curve is not a union of two lines (or a double line). Then is a conic section: a circle, ellipsis, parabola, or a hyperbola.

Linear Equations

Let us begin with a curve

with . Without loss of generality, we may assume that . Then the map given by is a morphism defined over with an inverse morphism  given by . Therefore, every line in that is defined over is isomorphic to In particular, one can establish a bijection between and for all .

Basic Algebraic Geometry

The subject of Algebraic Geometry starts out as a study of solution of systems of polynomial equations in several variables. To make our life easier, we will restrict ourselves to solving just one equation in only two variables:

.

For now, let us denote this equation by . When the coefficients of belong to a field , we say that is defined over .

Welcome

This Fall, I will be teaching a course in Elliptic Curves with an emphasis on computations. I am thinking that posting some notes online would not hurt, hence this blog. Feel free to say hi.

Have a nice semester, everyone.