On the denominators of rational points on elliptic curves |
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Authors: | Everest, Graham Reynolds, Jonathan Stevens, Shaun |
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Affiliation: | School of Mathematics University of East Anglia Norwich NR4 7TJ United Kingdom jonathan.reynolds{at}uea.ac.uk shaun.stevens{at}uea.ac.uk |
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Abstract: | Let x(P) = AP/B2P denote the x-coordinate of the rational pointP on an elliptic curve in Weierstrass form. We consider whenBP 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 BP equal to an fth power. Where descentvia an isogeny is possible, we show that there are only finitelymany rational points P with BP 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. |
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