On the independence of Heegner points on elliptic curves

Let E be an elliptic curve over Q of conductor N with no CM and O1, . . . ,Or be orders in distinct imaginary quadratic fields k1, . . . , kr, respectively, which satisfy the Heegner condition for N. Let P1, . . . , Pr be the Heegner points on E attached to O1, . . . ,Or, respectively. Silverman and Rosen proved that if Cond(O1) = · · · = Cond(Or) = 1, then there is a constant C such that if, for each i, #Pic(Oi)odd ≥ C then the points P1, . . . , Pr are independent in E(Q)/Etors(Q). In this paper we show that the condition Cond(O1) = · · · = Cond(Or) = 1 is not necessary and it can be removed