Ranks of Elliptic Curves and Random Matrix Theory

Name: Ranks of Elliptic Curves and Random Matrix Theory
Price: 69.52 USD
Availability: InStock
ISBN: 9780521699648

ISBN-10: 0521699649

ISBN-13: 9780521699648

Edition: 2007

Authors: F. Mezzadri, J. B. Conrey, D. W. Farmer, N. C. Snaith

List price: $70.99

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Description:

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…

Book details

List price: $70.99
Copyright year: 2007
Publisher: Cambridge University Press
Publication date: 2/8/2007
Binding: Paperback
Pages: 368
Size: 6.57" wide x 8.94" long x 0.71" tall
Weight: 1.100
Language: English

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

Nina Snaith is a Lecturer in Applied Mathematics at the University of Bristol.



Introduction




Families



Elliptic curves, rank in families and random matrices




Modeling families of L-functions




Analytic number theory and ranks of elliptic curves




The derivative of SO(2N +1) characteristic polynomials and rank 3 elliptic curves




Function fields and random matrices




Some applications of symmetric functions theory in random matrix theory




Ranks of Quadratic Twists



The distribution of ranks in families of quadratic twists of elliptic curves




Twists of elliptic curves of rank at least four




The powers of logarithm for quadratic twists




Note on the frequency of vanishing of L-functions of elliptic curves in a family of quadratic twists




Discretisation for odd quadratic twists




Secondary terms in the number of vanishings of quadratic twists of elliptic curve L-functions




Fudge factors in the Birch and Swinnerton-Dyer Conjecture




Number Fields and Higher Twists



Rank distribution in a family of cubic twists




Vanishing of L-functions of elliptic curves over number fields




Shimura Correspondence, and Twists



Computing central values of L-functions




Computation of central value of quadratic twists of modular L-functions




Examples of Shimura correspondence for level p2 and real quadratic twists




Central values of quadratic twists for a modular form of weight




Global Structure: Sha and Descent



Heuristics on class groups and on Tate-Shafarevich groups




A note on the 2-part of X for the congruent number curves




2-Descent tThrough the ages