Rational Points on elliptic curves

ABSTRACT: This presentation will cover the basic geometries and algebra of cubics as they pertain to rational points, and a little background on certain topics. I will establish that the set of rational solutions to a cubic equation form a group, which is abelian. The main focus will be to give a brief proof of Mordell's Theorem. Mordell's Theorem - If a non-singular plane cubic curve has a rational point, then the group of rational points is finitely generated.

I will prove some of the lemmas necessary to prove the theorem but not all of them will be proven. If time allows, concrete examples will be given.