Joint distribution of L-values and orders of Sha groups of quadratic twists of elliptic curves

We study the joint distribution of central L-values and orders of Tate-Shafarevich groups of quadratic twists of elliptic curves. In particular, adapting Radziwill and Soundararajan's principles of establishing upper and lower bounds for the distribution of central values in families of L-functions, we obtain conditional upper and lower bounds for such a joint distribution for rank zero twists. These lead us to a conjecture on the full asymptotic for the joint distribution.