Ranks of elliptic curves and random matrix theory

Workshop theme

Purpose of the special week

The application of random matrix ideas to ranks of elliptic curves shows great promise for shedding light on longstanding problems. This focused week of seminars and discussion sessions is aimed at identifying the key issues needed to make further progress.

Spitalfields Day

The week incorporates a London Mathematical Society Spitalfields Day on Monday, February 9th entitled Random Matrix Theory and the Birch and Swinnerton-Dyer Conjecture. Spitalfields Days are an opportunity for recent developments in specialist topics to be made known to the general mathematical community. This Spitalfields Day concerns the Birch and Swinnerton-Dyer conjecture, which describes a deep connection between the rank of an elliptic curve and the order of vanishing of an L-function. Particular attention will be paid to recent work which uses random matrix theory to make precise predictions for the ranks of families of elliptic curves. The lectures are aimed at a general mathematical audience

Topics for discussion

The connection between ranks of elliptic curves and random matrix theory arises from random matrix models for the values of L-functions in families. These models have been found to predict accurately the small values of L-functions; in conjunction with formulas for special values of L-functions, these have been used to predict the frequency of rank two quadratic twists of a fixed elliptic curve. Recent work of David, Fearnley, and Kivilevsky has explored a similar scenario for cubic twists.

Random matrix theory gives an order of magnitude prediction for the number of vanishings mentioned above, but there is an unknown constant which is required to produce an asymptotic formula. There is evidence that Cohen-Lenstra like heuristics for Tate-Shafarevich groups (accomplished by C. Delaunay) will play a role in determining these constants. This is one area which will benefit from bringing together mathematicians with a variety of perspectives, and we expect this to be an active topic of discussion during the week.

It is hoped that random matrix theory can be used to predict the frequency with which curves in a family have a given rank. There has been some success with small ranks, and it is predicted that about x^(1/4) quadratic twists of a fixed elliptic curve should have rank 3. However, there is work by various authors suggesting that this frequency is higher in certain instances. So, it may be that x^(1/4) is the generic answer, but this may be exceeded for certain special families. Or it may be that the random matrix model doesn't work, or has been formulated incorrectly. One possibility is that the current model makes incorrect assumptions about the distribution of heights of generating points of rank 1 curves. It will be valuable to involve experts in elliptic curves in this discussion.

Finally, there will be some attention devoted to computation. Most of the interest is with quadratic and cubic twists. Formulas of Waldspurger and Kohnen-Zagier together with an algorithm of Gross and an implementation by Rodriguez-Villegas allow for the quick evaluation of critical values of twists of elliptic curve L-functions by imaginary quadratic characters; recent work of Mao and Baruch now offers the opportunity to extend this work to real quadratic characters. We would like to discuss how to go about compiling a large database of L-values for families. This will be a valuable resource for the field, comparable to the role played by databases of elliptic curves.