Introduction to
General Audience

Introduction: Number Theory is the study of rational numbers such as 12/11 and 123/457; there are more complicated numbers, called irrational numbers, such as the square root of 2.  These days limiting the range of subjects of number theory to the study of rational numbers is not quite right any more, but a broader range of its subjects would be called that of number thoery. Among many subjects of number theory, prime numbers are a fundamental one. The integer 15 is not a prime number because 15=3×5, while 17 cannot be (nontrivially) decomposed as a product of smaller numbers, like 15, but 17=1×17 is the only possible decomposition. The numbers with this property (like 17) are called prime numbers.

Prime numbers are very important in number thoery. In most subjects of number theory, understanding prime numbers in the context of the subjects is always crucial for solving problems.

Another example is solving equations for rational solutions. Maybe, the Fermat equation: xn + yn = zn is one of the most popular ones. A British mathematician Andrew Wiles at the Princeton University is the main contributor of the proof of the puzzle in 1995, which had remained unsolved for nearly 360 years. It says that the above equation has no solutions other than trivial solutions such as x=0,y=1,z=1, if n>2. This one equation created a vast amount of history in mathematics, having defied many great mathematicians' attempts for more than 360 years. Upon the development and accumulation of modern mathematics, Wiles provided a way to prove that Fermat's assertion is true. In 1995, I as an undergraduate student was fascinated by the announcement in the newspaper, and it was the very moment that I decided to study number theory. My Interest: I have been interested in elliptic curves: y2 = x3+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 y2 = x3+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 (x1,y1),...,(xn,yn) 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 xi,yi 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 nearly equal to so-called the rank of an elliptic curve, and it is an open problem to find the rank of an elliptic curve.  In fact, number theorists found an algorithm for computing the rank, but they have not found a proof that the algorithm terminates for all elliptic curves.

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.