Rational points on elliptic curves

2006/05/31 by Graham Everest, Everest, Graham, Jonathan Reynolds +3 · 2 citations
Computer Science · Mathematics · #11G05 #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.math/0606003

openalex publication_date 2006/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P)=A_P/B_P² denote the x-coordinate of the rational point P then we consider when B_P 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. Where descent via an isogeny is possible we show, with no restrictions on the power, that there are only finitely many rational points, these points are bounded in number in an explicit fashion, and that they are effectively computable.

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