Lower bounds for the canonical height on elliptic curves over abelian extensions
Number Theory
2007-05-23 v2 Algebraic Geometry
Abstract
Let K be a number field and let E/K be an elliptic curve. If E has complex multiplication, we show that there is a positive lower bound for the canonical height of non-torsion points on E defined over the maximal abelian extension K^ab of K. This is analogous to results of Amoroso-Dvornicich and Amoroso-Zannier for the multiplicative group. We also show that if E has non-integral j-invariant (so that in particular E does not have complex multiplication), then there exists C > 0 such that there are only finitely many points P in E(K^ab) of canonical height less than C. This strengthens a result of Hindry and Silverman.
Keywords
Cite
@article{arxiv.math/0212132,
title = {Lower bounds for the canonical height on elliptic curves over abelian extensions},
author = {Matthew Baker},
journal= {arXiv preprint arXiv:math/0212132},
year = {2007}
}
Comments
14 pages. Proof of Theorem 1.4 clarified