Elliptic Curves

 

ABSTRACT: This talk will cover two key properties of elliptic (cubic) curves in
projective space.  We will begin with the classification of cubic curves
under affine transformation.  By examining the properties of a curve's
flexes and undergoing a series of affine transformations, we will be able to
show that any smooth cubic curve may be written in a standard form, and will
furthermore show that two curves written in this form are affine equivalent
only under certain conditions.  We will then define an operation addition
on cubics, and show that the set of points E on such a curve are an affine
group under this operation.