FINITENESS OF E(Q) AND Ø(E,Q) FOR A SUBCLASS OF WEIL CURVES

Let be an elliptic curve over , admitting a Weil parametrization , . Let be an imaginary quadratic extension of with discriminant , and let be a Heegner point. We show that if has infinite order ( must not belong to a finite set of fields that can be described in terms of ), then the Mordell-Weil group and the Tate-Shafarevich group of the curve (over ) are finite. For example, is finite. In particular, and are finite if and , where or is a rational prime such that and , where is the coefficient of in the -series of over . We indicate in terms of , , and a number annihilating and . Bibliography: 11 titles.