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.`