On the denominators of rational points on elliptic curves | Zendy

Everest Graham | Zendy; Reynolds Jonathan | Zendy; Stevens Shaun | Zendy

AI Assistant Blog Pricing

Home ZAIA Blog

Premium

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.

This content is not available in your region!

Continue researching here.

Having issues? You can contact us here

Accelerating Research