Ranks of Elliptic Curves and Random Matrix Theory

ISBN-13: 9780521699648 / Angielski / Miękka / 2007 / 368 str.

J. B. Conrey; D. W. Farmer; F. Mezzadri

Opis

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.

Recenzje

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

Spis treści

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.