Behaviour of the order of Tate-Shafarevich groups for the quadratic twists of elliptic curves

Abstract: We present the results of our search for the orders of Tate-Shafarevich groups for the quadratic twists of elliptic curves. We formulate a general conjecture, giving for a fixed elliptic curve $E$ over $\Bbb Q$ and positive integer $k$, an asymptotic formula for the number of quadratic twists $E_d$, $d$ positive square-free integers less than $X$, with finite group $E_d(\Bbb Q)$ and $|\Sha(E_d(\Bbb Q))| = k^2$. This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of the curve $X_0(49)$. In section 8 we exhibit $88$ examples of rank zero elliptic curves with $|\Sha(E)| > 63408^2$, which was the largest previously known value for any explicit curve. Our record is an elliptic curve $E$ with $|\Sha(E)| = 1029212^2$.