Ranks of Elliptic Curves and Random Matrix Theory
Random Matrix Theory (RMT) was created by physicists in a study of statistical properties of energy levels of atomic nuclei. The book contains a series of 22 lectures on relations of RMT to problems in number theory and it clearly shows the amazing richness of the subject. Many papers are related to lectures given at a workshop on the topic organised at the Isaac Newton Institute in 2004. An important feature of the book is that it contains a number of excellent survey papers, including the paper by C. Delaunay (probabilistic group theory), D. W. Farmer (families of L-functions), A. Gamburd (symmetric functions and RMT), E. Kowalski (families of elliptic curves and random matrices), F. Rodrigues-Villegas (central values of L-functions), A. Silverberg (ranks of elliptic curves), P. Swinnerton-Dyer (2-descent on elliptic curves), D. Ulmer (functions fields and random matrices) and M. P. Young (analytic number theory and ranks of elliptic curves). Due to these expository lectures, the book may well be of help to newcomers to the field.