The problem of estimating the number of imaginary quadratic fields
whose ideal class group has an element of order ℓ≥2$ is
classical in number theory. Analogous questions for quadratic twists
of elliptic curves have been the focus of recent interest. Whereas
works of C. Stewart and J. Top [ST], and of F. Gouvêa and
B. Mazur [GM] address the nontriviality of Mordell-Weil groups, less
is known about the nontriviality of Shafarevich-Tate groups. Here we
obtain a new type of result for the nontriviality of class groups of
imaginary quadratic fields using the circle method, and then we
combine it with works of G. Frey [F], V. Kolyvagin [K], and K. Ono
[O2] to obtain results on the nontriviality of Shafarevich-Tate groups
of certain elliptic curves. For E=X0 (11), these results imply
that
$\#\lbrace-X < D < 0 : D$ fundamental and $Ш(E(D),\mathbb {Q})[5]\neq \{0\}\rbrace\gg \frac {X\sp {3/5}}{\log \sp 2X}$