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On the denominators of rational points on elliptic curves

2007; Wiley; Volume: 39; Issue: 5 Linguagem: Inglês

10.1112/blms/bdm061

1469-2120

Graham Everest, Jonathan Reynolds, Shaun Stevens,

Cryptography and Residue Arithmetic

Let x(P) = AP/B2P denote the x-coordinate of the rational point P on an elliptic curve in Weierstrass form. We consider when BP can be a perfect power or a prime. Using Faltings' theorem, we show that for a fixed f > 1, there are only finitely many rational points P with BP equal to an fth power. Where descent via an isogeny is possible, we show that there are only finitely many rational points P with BP equal to a prime, that these points are bounded in number in an explicit fashion, and that they are effectively computable. Finally, we prove a stronger version of this result for curves in homogeneous form.