Ranks of Elliptic Curves and Random Matrix Theory [Paperback]

This comprehensive volume introduces elliptic curves and the fundamentals of modeling by a family of random matrices.This comprehensive volume highlights some of the most current results about ranks of elliptic curves, statistical properties of families of elliptic curves and their associated L-functions and the emerging uses of random matrix theory in this field. This is the only book to give an in-depth treatment of this subject.This comprehensive volume highlights some of the most current results about ranks of elliptic curves, statistical properties of families of elliptic curves and their associated L-functions and the emerging uses of random matrix theory in this field. This is the only book to give an in-depth treatment of this subject.Random matrix theory is an area of mathematics first developed by physicists interested in the energy levels of atomic nuclei, but it can also be used to describe some exotic phenomena in the number theory of elliptic curves. This book illustrates this interplay of number theory and random matrices. It begins with an introduction to elliptic curves and the fundamentals of modeling by a family of random matrices, and moves on to highlight the latest research. There are expositions of current research on ranks of elliptic curves, statistical properties of families of elliptic curves and their associated L-functions and the emerging uses of random matrix theory in this field. Most of the material here had its origin in a Clay Mathematics Institute workshop on this topic at the Newton Institute in Cambridge and together these contributions provide a unique in-depth treatment of the subject.Introduction J. B. Conrey, D. W. Farmer, F. Mezzadri and N. C. Snaith; Part I. Families: 1. Elliptic curves, rank in families and random matrices E. Kowalski; 2. Modeling families of L-functions D. W. Farmer; 3. Analytic number theory and ranks of elliptic curves M. P. Young; 4. The derivative of SO(2N +1) characteristic polynomials and rank 3 ells*