On the denominators of rational points on elliptic curves

Let x(P) = A_P/B²_P denote the x-coordinate of the rational pointP on an elliptic curve in Weierstrass form. We consider whenB_P can be a perfect power or a prime. Using Faltings' theorem,we show that for a fixed f > 1, there are only finitely manyrational points P with B_P equal to an fth power. Where descentvia an isogeny is possible, we show that there are only finitelymany rational points P with B_P equal to a prime, that thesepoints are bounded in number in an explicit fashion, and thatthey are effectively computable. Finally, we prove a strongerversion of this result for curves in homogeneous form.