Elements of Class Groups and Shafarevich-tate Groups of Elliptic Curves

The problem of estimating the number of imaginary quadratic fields whose ideal class group has an element of order ` ≥ 2 is classical in number theory. Analogous questions for quadratic twists of elliptic curves have been the focus of recent interest. Whereas works of Stewart and Top [St-T], and of Gouvêa and Mazur [G-M] address the nontriviality of MordellWeil 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 Frey, Kolyvagin and the second author to obtain results on the nontriviality of Shafarevich-Tate groups of certain elliptic curves. For E = X0(11), these results imply that # ̆ −X < D < 0 : D fundamental and X(E(D), Q)[5] 6= {0} ̄ X3/5 log X .