One of the striking developments in number theory over the past decade has been the emergence of using Random Matrix Theory to describe the statistics of various number-theoretic phenomena associated with elliptic curves and L-functions. This volume contains many excellent papers surveying recent results and the methods used, as well as open problems in the field. It presents a number of expository articles that would assist someone wishing to begin work in this field. Anyone interested in surveying recent progress in this field would benefit from reading this volume.
The book consists of 22 papers most of which were originally presented in the Clay Mathematics Institute’s workshop on “Ranks of Elliptic Curves and Random Matrix Theory” held at the Newton Institute in Cambridge, England in February 2004. The editors write that “the purpose of this volume is to expose how Random Matrix Theory can be used to describe the statistics of exotic number theory phenomena such as the frequency of rank two elliptic curves,” and the papers reflect this goal. They are divided into five subject areas:Families (6 papers), Ranks of Quadratic Twists (7 papers), Number Fields and Higher Twists (2 papers), Shimura Correspondence and Twists (4 papers), and Global Structure: Sha and Descent (3 papers). The first group of articles, on Families, is the heart of the volume, constituting over half the pages in the book.
In most of the areas, there is a mixture of expository surveys and research papers, but even the research papers are distinguished by their readability. Some of the expository surveys of note are by E. Kowalski (“Elliptic curves, rank in families, and random matrices”), who describes how random matrices are used to conjecturally understand the ranks of Mordell Weil groups via the Birch and Swinnerton-Dyer conjecture, and in studying the variation of ranks in families of elliptic curves. D. Farmer (“Modeling families of L-functions”) describes how to model the leading-order terms of L-functions in families by random matrix theory, and D. Ulmer (“Function Fields and Random Matrices”), aims to explain how families of L-functions for function fields of curves over finite fields give rise to well-distributed sets of matrices for classical groups, such as in the function field side of the Katz-Sarnak picture.
Other excellent surveys are F. Rodriguez-Villegas’s “Computing central values of L-functions”, which addresses the computational questions involved in efficiently calculating the central value of an L-function, A. Silverberg (“The distribution of ranks in families of quadratic twists in elliptic curves”), who surveys known results and conjectures on densities of ranks in families of quadratic twists of elliptic curves, and C. Delaunay (“Heuristics on class groups and on Tate-Shafarevich groups: The magic of the Cohen-Lenstra heuristics), who explains how the Cohen-Lenstra heuristics can be applied both to the cases of class groups and of Tate-Shafarevich groups.
In conclusion, “Ranks of Elliptic Curves and Random Matrix Theory” is a welcome addition to the number theory literature.Graduate students entering the field and all mathematicians wishing to understand this newly emerging area of number theory will enjoy reading this book.
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 elliptic curves N. C. Snaith; 5. Function fields and random matrices D. Ulmer; 6. Some applications of symmetric functions theory in random matrix theory A. Gamburd; Part II. Ranks of Quadratic Twists: 7. The distribution of ranks in families of quadratic twists of elliptic curves A. Silverberg; 8. Twists of elliptic curves of rank at least four K. Rubin and A. Silverberg; 9. The powers of logarithm for quadratic twists C. Delaunay and M. Watkins; 10. Note on the frequency of vanishing of L-functions of elliptic curves in a family of quadratic twists C. Delaunay; 11. Discretisation for odd quadratic twists J. B. Conrey, M. O. Rubinstein, N. C. Snaith and M. Watkins; 12. Secondary terms in the number of vanishings of quadratic twists of elliptic curve L-functions J. B. Conrey, A. Pokharel, M. O. Rubinstein and M. Watkins; 13. Fudge factors in the Birch and Swinnerton-Dyer Conjecture K. Rubin; Part III. Number Fields and Higher Twists: 14. Rank distribution in a family of cubic twists M. Watkins; 15. Vanishing of L-functions of elliptic curves over number fields C. David, J. Fearnley and H. Kisilevsky; Part IV. Shimura Correspondence, and Twists: 16. Computing central values of L-functions F. Rodriguez-Villegas; 17. Computation of central value of quadratic twists of modular L-functions Z. Mao, F. Rodriguez-Villegas and G. Tornaria; 18. Examples of Shimura correspondence for level p2 and real quadratic twists A. Pacetti and G. Tornaria; 19. Central values of quadratic twists for a modular form of weight H. Rosson and G. Tornaria; Part V. Global Structure: Sha and Descent: 20. Heuristics on class groups and on Tate-Shafarevich groups C. Delaunay; 21. A note on the 2-part of X for the congruent number curves D. R. Heath-Brown; 22. 2-Descent tThrough the ages P. Swinnerton-Dyer.