New York Journal of Mathematics
Volume 12 (2006) 257-268


Clayton Petsche

Small rational points on elliptic curves over number fields

Published: September 2, 2006
Keywords: Elliptic curves, heights, torsion points, Szpiro ratio, Lang's conjecture
Subject: 11G05, 11G07, 11G50

Let E/k be an elliptic curve over a number field. We obtain some quantitative refinements of results of Hindry-Silverman, giving an upper bound for the number of k-rational torsion points, and a lower bound for the canonical height of nontorsion k-rational points, in terms of expressions depending explicitly on the degree d=[k:Q] of k and the Szpiro ratio σ of E/k. The bounds exhibit only polynomial dependence on both d and σ.

Author information

Department of Mathematics, The University of Georgia, Athens, GA 30602-7403