Results and conjectures concerning orders of Tate-Shafarevich groups in families of elliptic curves

We plan to discuss numerical results and conjectures concerning orders of Tate-Shafarevich groups in
families of quadratic twists of elliptic curves, families of elliptic curves of Neumann-Setzer type and
families of cubic twists of X_0(27). We will give examples of elliptic curves with exceptionally large
orders of Tate-Shafarevich groups. We also calculate hypothetical orders of Tate-Shafarevich groups
for some quadratic twists of Gross curves. This is a joint work with Tomasz Jedrzejak and Lucjan
Szymaszkiewicz.