Contents

pp v-vi

Introduction

pp 1-6

FAMILIES

Elliptic curves, rank in families and random matrices

pp 7-52

Modeling families of L-functions

pp 53-70

Analytic number theory and ranks of elliptic curves

pp 71-92

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

pp 93-108

Function fields and random matrices

pp 109-142

Some applications of symmetric functions theory in random matrix theory

pp 143-170

RANKS OF QUADRATIC TWISTS

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

pp 171-176

Twists of elliptic curves of rank at least four

pp 177-188

The powers of logarithm for quadratic twists

pp 189-194

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

pp 195-200

Discretisation for odd quadratic twists

pp 201-214

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

pp 215-232

Fudge Factors in the Birch and Swinnerton-Dyer Conjecture

pp 233-236

NUMBER FIELDS AND HIGHER TWISTS

Rank distribution in a family of cubic twists

pp 237-246

Vanishing of L-functions of elliptic curves over number fields

pp 247-259

SHIMURA CORRESPONDENCE, AND TWISTS

Computing central values of L-functions

pp 260-272

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

pp 273-288

Examples of Shimura correspondence for level p2 and real quadratic twists

pp 289-314

Central values of quadratic twists for a modular form of weight 4

pp 315-322

GLOBAL STRUCTURE: SHA AND DESCENT

Heuristics on class groups and on Tate-Shafarevich groups

pp 323-340

A Note on the 2-Part of Ш for the Congruent Number Curves

pp 341-344

2-Descent Through the Ages

pp 345-356