2-Selmer groups of quadratic twists of elliptic curves

In this paper we investigate families of quadratic twists of elliptic curves. Addressing a speculation of Ono, we identify a large class of elliptic curves for which the parities of the ``algebraic parts'' of the central values $L(E^{(d)}/\mathbb{D}{Q},1)$, as $d$ varies, have essentially the same multiplicative structure as the coefficients $a_d$ of $L(E/\mathbb{D}{Q},s)$. We achieve this by controlling the 2-Selmer rank (à la Mazur and Rubin) when the Tamagawa numbers do not already dictate the parity.