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 L-functions 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 L-functions 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 L-functions J. B. Conrey, A. Pokharel, M. O. Rubinstein and M. Watkins; 13. Fudge factors in the Birch and Swinnerton-Dyer Conjecture K. Rubin; Part III. Number Fields and Higher Twists: 14. Rank distribution in a family of cubic twists M. Watkins; 15. Vanishing of L-functions of elliptic curves over number fields C. David, J. Fearnley and H. Kisilevsky; Part IV. Shimura Correspondence, and Twists: 16. Computing central values of L-functions F. Rodriguez-Villegas; 17. Computation of central value of quadratic twists of modular L-functions Z. Mao, F. Rodriguez-Villegas 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 Tate-Shafarevich groups C. Delaunay; 21. A note on the 2-part of X for the congruent number curves D. R. Heath-Brown; 22. 2-Descent tThrough the ages P. Swinnerton-Dyer.
