Introduction

Summary

The group of rational points on an elliptic curve is a fascinating number theoretic object. The description of this group, as enunciated by Birch and Swinnerton-Dyer in terms of the special value of the associated L-function, or a derivative of some order, at the center of the critical strip, is surely one of the most beautiful relationships in all of mathematics; and it's understanding also 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 atomic nuclei, RMT has proven to be capable of describing many complex phenomena, including average behavior of L-functions.

The purpose of this volume is to expose how RMT can be used to describe the statistics of some exotic phenomena such as the frequency of rank two elliptic curves. Many, but not all, of the papers here have origins in a workshop that took place at the Isaac Newton Institute in February of 2004 entitled “Clay Mathematics Institute Special week on Ranks of Elliptic Curves and Random Matrix Theory.” The workshop began with the Spittalsfield day of expository lectures, highlighted by reminiscences by Bryan Birch and Sir Peter Swinnerton-Dyer on the development of their conjecture. The week continued with a somewhat free-form workshop featuring discussion sessions, groups working on various problems, and spontaneous lectures.