Elements of class groups and Shafarevich-Tate groups of elliptic curves

Abstract

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