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On the denominators of rational points on elliptic curves
Author(s) -
Everest Graham,
Reynolds Jonathan,
Stevens Shaun
Publication year - 2007
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdm061
Subject(s) - mathematics , isogeny , prime (order theory) , elliptic curve , rational point , bounded function , schoof's algorithm , pure mathematics , discrete mathematics , combinatorics , mathematical analysis , quarter period , algebraic number
Let x ( P ) = A P / B 2 P denote the x ‐coordinate of the rational point P on an elliptic curve in Weierstrass form. We consider when B P 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 B P equal to an f th power. Where descent via an isogeny is possible, we show that there are only finitely many rational points P with B P 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.