Sungkon's Interest in Mathematics | |

Another example is solving equations for rational solutions. Maybe,
the Fermat equation: x | |

My Interest:
I have been interested in elliptic curves: y^{2} = x^{3}+ax + b (in its simplest form). The topology of its solutions in the complex numbers is rather simple, but the arithmetic of its rational solutions is immense and beyond our capability to be comprehensively understood.
Like the Fermat equation, this equation is known to many mathematicians, and a great of deal of work has been done while it still leaves so many
interesting questions unsolved. Mr. Andrew Wiles' attempt for a proof of Fermat's Last Theorem began when he heard that Ken Ribet at the University of California, Berkeley proved a connection, asserted by Frey, between the theory about this equation and Fermat's
Last Theorem. The set of all points (x,y) in the xy-coordinate plane for which y ^{2} = x^{3}+ax + b is true, is called an elliptic curve. Points with rational numbers x and y satisfying the above equation are called rational points on an elliptic curve, and it is the main subject of the arithmetic of elliptic curves.
One of the most (arithmetically) important results about elliptic curves is the fact that there are finitely many points (x _{1},y_{1}),...,(x_{n},y_{n}) with which we can "generate" all other rational points of an elliptic curve---"generating" means that there is a fixed algorithm in terms of these
x_{i},y_{i} by which all rational points of the elliptic curve are found. This result is called the Mordell-Weil Theorem.Finding a set of generators of a given elliptic curve has been one of the most challenging problems in the theory of elliptic curves, and if one is interested in a smallest set of generators, it gets far harder. The smallest number of generators is called the rank of an elliptic curve, and it is an open problem to compute the rank of an elliptic curve. I have been interested in many topics about elliptic curves, and had opportunities to study some of them such as finding elliptic curves with "large" rank, and twists of elliptic curves. Techniques used for studying the arithmetic of curves, in general, come from various fields of mathematics: commutative algebra, representation theory, algebraic geometry, complex analysis/geometry, algebraic/analytic number theory, and combinatorics. A great thing about the subject is that one has opportunities to study these various subjects of mathematics and appreciate their applications in number theory. |