The Square-Free Sieve over Number Fields Original Research Article

For E an elliptic curve over a number field K, we give a lower bound, conditional on the "parity conjecture," for the number of quadratic twists of E whose Mordell-Weil rank is at least two. The main tool is a sieve-theoretic estimate of the number of square-free values of a homogeneous binary form. The results are a direct extension of those found by the author and B. Mazur (J. Amer. Math. Soc.4, 1991, 1-23).