Ranks of Elliptic Curves and Random Matrix Theory

Couverture
J. B. Conrey, D. W. Farmer, F. Mezzadri, N. C. Snaith
Cambridge University Press, 8 févr. 2007 - 361 pages
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.
 

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Table des matières

Introduction
1
Families
7
Modeling families of Lfunctions
53
Analytic number theory and ranks of elliptic curves
71
The derivative of SO2N +1 characteristic polynomials
93
Function fields and random matrices
109
Some applications of symmetric functions theory in random
143
Ranks of quadratic twists
171
Twists of elliptic curves of rank at least four
177
The powers of logarithm for quadratic twists
189
Note on the frequency of vanishing of Lfunctions of elliptic
195
Discretisation for odd quadratic twists
201
Droits d'auteur

Expressions et termes fréquents

À propos de l'auteur (2007)

Brian Conrey is the Executive Director of the American Institute of Mathematics. He is also Professor of Mathematics at the University of Bristol.

David Farmer is the Associate Director of the American Institute of Mathematics.

Francesco Mezzadri is a Lecturer in the Department of Mathematics, University of Bristol.

Nina Snaith is a Lecturer in the Department of Mathematics, University of Bristol.

Informations bibliographiques