In the researcher’s words: Shaping the understanding of elliptic curves
Manjul Bhargava, the R. Brandon Fradd Professor of Mathematics
In many areas of mathematics, the following equation plays an extremely important role:
y² = x³ + ax + b.
The graph of such an equation is called an “elliptic curve” (see Figure A). Number theorists are particularly interested in the case where a and b are whole numbers, like 0, 1, 2, 3, etc., or -1, -2, -3, etc. Moreover, they are especially interested in finding rational solutions to this equation, i.e., those rational values of x and y that make the equation true. (Rational numbers are numbers that can be expressed as ratios of whole numbers, e.g., 1/2, -3/4, 7/3 are rational numbers.) For example, y²= x³ + 2x + 3 has the rational solution x = -1, y = 0, and also x = 3, y = 6, and also the less obvious rational solution x = 1/4, y = 15/8.
One reason elliptic curves are so structurally rich, and thus of particular interest to mathematicians, is that known solutions to their equations can be used to create new solutions by playing “connect-the-dots.” Here’s how it works: If you take any two rational points on an elliptic curve and draw a straight line between them, the line will always intersect the elliptic curve at a third rational point. Using this technique, you can start with some small set of rational points and use this procedure to find more and more (see Figures B and C).
According to Mordell’s Theorem -- proven in 1922 by Louis Mordell -- it is always possible to find some finite set of rational points on the curve, so that all of the rational points on the curve can be found from this finite set of points using the above connect-the-dots procedure. The minimal number of points needed to generate all the rational points on an elliptic curve is called the rank of the curve.
The question now arises as to whether the rank of the elliptic curve tends to increase, decrease, or stay the same as a and b get larger. In particular, does the rank approach infinity as a and b grow ever larger?
In our recent work, in collaboration with graduate student Arul Shankar, we have demonstrated that the rank, on average, actually has an upper limit as a and b tend to infinity. In fact, we find that the average rank of all elliptic curves is less than one. In particular, it follows that many -- indeed, we show at least 10 percent -- of all elliptic curves have no rational points!
Beyond advancing the subject of number theory in general, a heightened understanding of elliptic curves also has important implications in coding theory and cryptography. Encryption schemes, such as those used to protect our privacy when transmitting information online, often centrally involve the use of elliptic curves and the connect-the-dots construction.
Bhargava, Manjul and Arul Shankar. 2010. “Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves.” arXiv: 1006.1002.
Bhargava, Manjul and Arul Shankar. 2010. “Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0.” arXiv: 1007.0052.