Local to global trace questions and twists of genus one curves

Let E be an elliptic curve defined over a number field F and K/F a quadratic extension. For a point P is an element of E(F) that is a local trace for every completion of K/F, we find necessary and sufficient conditions for P to lie in the image of the global trace map. These conditions can then be used to determine whether a quadratic twist of E, as a genus one curve, has rational points. In the case of quadratic twists of genus one modular curves X-0(N) with squarefree N, the existence of rational points corresponds to the existence of Q-curves of degree N defined over K.