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.
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.
The schedule for Monday February 9th, 2004, is listed on the page for the LMS Spitalfields Day Random Matrix Theory and the Birch and Swinnerton-Dyer Conjecture. These are lectures aimed at a general mathematical audience. From the morning of Tuesday February 10th until late afternoon on Friday February 13th there will be a series of seminars and discussions of a more specialised nature.
Location and Cost
The special week will take place at the Newton Institute. There is no registration fee, but the support available for local expenses is also very limited.
The Institute has limited accommodation available. If you wish to participate in the special week, please fill in the application form below and on receipt of your formal workshop acceptance letter you are strongly advised to confirm your accommodation requirements immediately. Once all Institute accommodation has been allocated, you are likely to have to arrange your own. A list of local Guest Houses are available here.
The closing date for the receipt of applications has now passedScientific enquiries may be addressed to Nina Snaith (e-mail: firstname.lastname@example.org)