This project is part of an ongoing research program. It revolves around understanding how various fundamental algebraic objects occurring in mathematics---such as rings, class groups, algebraic curves and varieties, and maps relating such structures---are parametrized. Our goals in understanding these parametrizations are at least fourfold: 1) to describe how these fundamental algebraic objects are distributed with respect to their most basic invariants; 2) to discover new invariants of these objects, and their applications; 3) to develop efficient and practical algorithms for performing computations with these algebraic objects; and, perhaps most importantly, 4) to discover and understand how various seemingly different algebraic structures are in fact closely related to each other.
Such parametrizations have been already used by the PI over the past few years to obtain precise information on the distribution of number fields with respect to basic invariants such as discriminant and class number divisibility. Applying refined counting methods to these parametrizations has led, for example, to a proof of the first known case of the Cohen-Lenstra-Martinet class number heuristics for higher degree number fields, and other theorems of this nature are forthcoming. We expect similar theorems for ranks of elliptic curves, and other data of this kind for algebraic curves and surfaces, in the near future.
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". 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. 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 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! In a complementary work with Christopher Skinner, we also show that the average rank has a strictly positive lower limit. In other words, a positive proportion of elliptic curves have infinitely many rational points! It is worth remarking that there is no known algorithm for determining all rational points on a given elliptic curve. However, the Birch and Swinnerton-Dyer Conjecture (one of the Clay Mathematics Institute's Millennium problems) provides such an algorithm, although it is an unsolved problem as to whether the algorithm actually works. In joint work with Christopher Skinner and Wei Zhang, building on the earlier work with Shankar and Skinner, we show that the BSD Conjecture is true for most values of a and b -- in fact, it is true for more than 66% of all a and b (and for these a and b, the rank of the elliptic curve is either zero or one). Thus for most elliptic curves, it is possible to find all the rational points on the elliptic curve! 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.
