Ranks of Elliptic Curves and Random Matrix Theory
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Part of London Mathematical Society Lecture Note Series
 Editors:
 J. B. Conrey, American Institute of Mathematics
 D. W. Farmer, American Institute of Mathematics
 F. Mezzadri, University of Bristol
 N. C. Snaith, University of Bristol
 Date Published: May 2012
 availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
 format: Adobe eBook Reader
 isbn: 9781139238984
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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 Lfunctions 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 indepth treatment of the subject.
Read more The first book on random matrix theory and elliptic curves
 Gives an overview of the entire subject, making it suitable as an introduction
 Many papers by leading experts in the field, presenting the very latest research findings
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×Product details
 Date Published: May 2012
 format: Adobe eBook Reader
 isbn: 9781139238984
 contains: 19 b/w illus. 40 tables
 availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
Table of Contents
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 Lfunctions 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 Lfunctions 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 Lfunctions J. B. Conrey, A. Pokharel, M. O. Rubinstein and M. Watkins
13. Fudge factors in the Birch and SwinnertonDyer Conjecture K. Rubin
Part III. Number Fields and Higher Twists:
14. Rank distribution in a family of cubic twists M. Watkins
15. Vanishing of Lfunctions of elliptic curves over number fields C. David, J. Fearnley and H. Kisilevsky
Part IV. Shimura Correspondence, and Twists:
16. Computing central values of Lfunctions F. RodriguezVillegas
17. Computation of central value of quadratic twists of modular Lfunctions Z. Mao, F. RodriguezVillegas 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 TateShafarevich groups C. Delaunay
21. A note on the 2part of X for the congruent number curves D. R. HeathBrown
22. 2Descent tThrough the ages P. SwinnertonDyer.
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