Ranks of elliptic curves and random matrix theory
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...
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Format:  eBook 
Language:  English 
Published: 
Cambridge
Cambridge University Press
2007

Series:  London Mathematical Society lecture note series

Subjects:  
Online Access:  
Collection:  Cambridge Books Online  Collection details see MPG.ReNa 
Table of Contents:
 Introduction J.B. Conrey, D.W. Farmer, F. Mezzadri and N.C. Snaith
 Part I. Families: Elliptic curves, rank in families and random matrices E. Kowalski
 Modeling families of Lfunctions D.W. Farmer
 Analytic number theory and ranks of elliptic curves M.P. Young
 The derivative of SO(2N +1) characteristic polynomials and rank 3 elliptic curves N.C. Snaith
 Function fields and random matrices D. Ulmer
 Some applications of symmetric functions theory in random matrix theory A. Gamburd
 Part II. Ranks of Quadratic Twists
 The distribution of ranks in families of quadratic twists of elliptic curves A. Silverberg
 Twists of elliptic curves of rank at least four K. Rubin and A. Silverberg
 The powers of logarithm for quadratic twists C. Delaunay and M. Watkins
 Note on the frequency of vanishing of Lfunctions of elliptic curves in a family of quadratic twists C. Delaunay
 Discretisation for odd quadratic twists J.B. Conrey, M.O. Rubinstein, N.C. Snaith and M. Watkins
 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
 Fudge factors in the Birch and SwinnertonDyer Conjecture K. Rubin
 Part III. Number Fields and Higher Twists
 Rank distribution in a family of cubic twists M. Watkins
 Vanishing of Lfunctions of elliptic curves over number fields C. David, J. Fearnley and H. Kisilevsky
 Part IV. Shimura Correspondence, and Twists
 Computing central values of Lfunctions F. RodriguezVillegas
 Computation of central value of quadratic twists of modular Lfunctions Z. Mao, F. RodriguezVillegas and G. Tornaria
 Examples of Shimura correspondence for level p2 and real quadratic twists A. Pacetti and G. Tornaria
 Central values of quadratic twists for a modular form of weight H. Rosson and G. Tornaria
 Part V. Global Structure: Sha and Descent
 Heuristics on class groups and on TateShafarevich groups C. Delaunay
 A note on the 2part of X for the congruent number curves D.R. HeathBrown
 2Descent tThrough the ages P. SwinnertonDyer