Ranks of Elliptic Curves and Random Matrix Theory

This comprehensive volume introduces elliptic curves and the fundamentals of modeling by a family of random matrices.

J. B. Conrey (Edited by), D. W. Farmer (Edited by), F. Mezzadri (Edited by), N. C. Snaith (Edited by)

9780521699648, Cambridge University Press

Paperback, published 8 February 2007

368 pages, 19 b/w illus. 40 tables
22.7 x 16.7 x 1.8 cm, 0.5 kg

'Due to these expository lectures, the book may well be of help to newcomers to the field.' European Mathematical Society Newsletter

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. The purpose of this book is to illustrate this interplay of number theory and random matrices. It begins with an introduction to elliptic curves and the fundamentals of modelling 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 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.

Subject Areas: Calculus & mathematical analysis [PBK], Number theory [PBH]