On Elliptic Curves with Large Tate–Shafarevich Groups Original Research Article

Generalizing results of Lemmermeyer, we show that the 2-ranks of the Tate–Shafarevich groups of quadratic twists of certain elliptic curves with a rational point of order 2 can be arbitrarily large. We use only quadratic residue symbols in a quadratic field to obtain our results.