Rational Points on Elliptic Curves Graham Everest, Jonathan Reynolds and Shaun Stevens

We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P ) = AP /B P denote the xcoordinate of the rational point P then we consider when BP can be a prime power. Using Faltings’ Theorem we show that for a fixed power greater than 1, there are only finitely many rational points with this property. Where descent via an isogeny is possible we show, with no restrictions on the power, that there are only finitely many rational points with this property, that these points are bounded in number in an explicit fashion, and that they are effectively computable. Let E denote an elliptic curve given by a Weierstrass equation (1) y + a1xy + a3y = x 3 + a2x 2 + a4x+ a6 with integral coefficients a1, . . . , a6. See [1] and [15] for background on elliptic curves. Let E(Q) denote the group of rational points on E. For an element P ∈ E(Q), the shape of the defining equation (1) requires that P be in the form (2) P = ( AP B P , CP B P ) where AP , BP , CP are integers with no common factor. In this paper we are concerned with the equation (3) BP = p f where p denotes a prime and f ≥ 1 denotes an integer. All of the methods extend to the equation BP = bp f , where b is fixed, with trivial adjustments. Gunther Cornelissen once remarked to the first author that the equation (3) has only finitely many solutions when f = 2, by re-arranging the equation and invoking Faltings’ Theorem. 1991 Mathematics Subject Classification. 11G05, 11A41.