ELLIPTIC CURVES WITH A LOWER BOUND ON 2-SELMER RANKS OF QUADRATIC TWISTS

For any number field K with a complex place, we present an infinite family of elliptic curves defined over K such that dim(F2)Sel(2)(E-F/K) >= dim(F2)E(F)(K)[2] + r(2) for every quadratic twist E-F of every curve E in this family, where r(2) is the number of complex places of K. This provides a counterexample to a conjecture appearing in work of Mazur and Rubin.