by Whitney Montgomery

      At the beginning of this year, I decided to do a senior thesis.  I didn't really know much about the process at the time I decided this, but it has been one of the most fun and interesting things I've done as an undergraduate.  Partly what made this thesis so interesting was that the set up of doing a thesis is a lot different than a typical math course.  For example, the thesis topic is less general than most undergraduate classes and so there is more time to really understand the deeper theory involved.  I was fortunate to have Professor Koch agree to be my advisor, and we decided to study Mordell's theorem for the thesis topic.  For the purpose of this article, before I introduce Mordell's theorem, I should give some background information about polynomials over the field of rational numbers.  Linear equations are easily solvable over rational numbers.  The study behind solving quadratic equations over the rational  numbers is complete, but far from trivial.  Quadratic equations consist of the equations of familiar curves such as ellipses, parabolas, hyperbolas, and circles.  An interesting observation about quadratic equations is that if rational points exist, then they are dense on the curve, but they don't always exist.  For example, the ellipse 2x² + 7y² = 1 has rational solutions, but 3x² + 7y² = 1 does not.  Another interesting point is that finding rational points on a circle correspond to Pythagorean triangles which the Greeks studied.  We can see that once we start looking at polynomials with degree larger than one, the study of rational solutions more interesting.
    Cubic equation theory is still incomplete over the rational numbers.  One branch of this theory which is incomplete is the study of elliptic curves.  These are curves of the form y² = x³ + ax² + bx + c.  Elliptic curves were studied by the ancient Greeks; and many great mathematicians such as Fermat, Poincarè, and Abel dedicated time to them.  Elliptic curves are used in many parts of mathematics.  For example, they were

used to help prove Fermat's Last Theorem in 1993, and they are also used in modern cryptography. So, they are useful in an array of fields. But, elliptic curves still leave much to be discovered about themselves.  Perhaps further study could lead to other important uses in mathematics.
    One of the important facts about elliptic curves is that there is a natural group structure. This group structure was first observed by Abel,  and interestingly enough was one of the first examples of an 'abelian' group. But in order to understand this group structure, it is important to understand the point at infinity of the elliptic curve because this 'point' serves as the identity in the group. It is helpful to use Projective Space to do so because we are then given a familiar setting to observe the point at infinity. 
    Mordell's theorem is about elliptic curves that first have to meet a few requirements.  One of the requirements is that the coefficients of the elliptic curve need to be integers.  Mordell's theorem states that for an elliptic curve (with the necessary restrictions), the rational solutions to the equation are finitely generated.  The proof of Mordell's theorem depends heavily on the natural group structure of the curve.  So, let E denote the group of rational points on the elliptic curve.  In  Mordell's theorem, we consider a certain subgroup of  E, which we'll call 2E.  The key to proving Mordell's theorem is to show that E/2E  is finite.  I won't go into much more detail about the proof itself, because any more detail would require much more background information. 
    However, it is interesting to see the corresponding time line behind Mordell's theorem because it shows when and by whom the pieces came together.  This is especially interesting because it