On Tate-Shafarevich Groups of some Elliptic Curves, by Franz Lemmermeyer

Generalizing results of Stroeker and Top we show that the $2$-ranks of the Tate-Shafarevich groups of the elliptic curves $y^2 = (x+k)(x^2+k^2)$ can become arbitrarily large. We also present a conjecture on the rank of the Selmer groups attached to rational $2$-isogenies of elliptic curves. This is an update of preprint number 166.

Franz Lemmermeyer <lemmerm@mpim-bonn.mpg.de>