L-functions, ranks of elliptic curves, and random matrix theory

The group of rational points on an elliptic curve is one of the more fascinating number theoretic objects studied in recent times. The description of this group in terms of the special value of the L-function, or a derivative of some order, at the center of the critical strip, as enunciated by Birch and Swinnerton-Dyer is surely one of the most beautiful relationships in all of mathematics; also it’s understanding carries a $1 million dollar reward! Random Matrix Theory (RMT) has recently been revealed to be an exceptionally powerful tool for expressing the finer structure of the value-distribution of L-functions. Initially developed in great detail by physicists interested in the statistical properties of energy levels of excited nuclei, RMT has proven to be capable of describing many complex phenomena, including average behavior of L-functions. The most important invariant of an elliptic curve is the rank of its (Mordell-Weil) group of rational points; it is a non-negative integer, believed to be 0 or 1 for almost all elliptic curves. The beginnings of the subject is a conjecture (see [1]) about how often the rank is greater than or equal to 2 for the family of quadratic twists of a given elliptic curve. Each elliptic curve has an L-function associated with it; this is an entire function which satisfies a functional equation. The Birch and Swinnerton-Dyer conjecture asserts, among other things, that the order of vanishing at the central point of the L-function associated with an elliptic curve is equal to the rank. It is generally conjectured that almost all elliptic curves have rank zero or one according to whether the sign of the functional equation of the related L-function is +1 or −1. Rank two curves should occur with L-functions that have a +1 sign of their functional equation but vanish nevertheless at the central point. These are expected to be rare; the question of how rare is the subject here. If the elliptic curve is given by E : y2 = x3 + Ax + B, and if d is a fundamental discriminant, then the quadratic twist of E by d is the elliptic curve Ed := dy2 = x3 +Ax+B. The conjecture, derived from RMT and number theory, is that Ed will have rank 2, or greater, for asymptotically cEx(log x)bE+ 3 8 values of d with |d| ≤ x; here bE is one of four values (see [7]): • bE = 1 when E has full rational 2-torsion, • bE = √ 2 2 when E has one rational 2-torsion point, • bE = 3 when E has no rational 2-torsion and the discriminant is a perfect square,