Lower bounds for the canonical height on elliptic curves over abelian extensions

TL;DR

This paper establishes lower bounds for the canonical height of points on elliptic curves over abelian extensions, differentiating between curves with and without complex multiplication, and extends previous finiteness results.

Contribution

It proves positive lower bounds for non-torsion points on CM elliptic curves and finiteness of points with small height on non-CM curves over abelian extensions.

Findings

Positive lower bound for non-torsion points on CM elliptic curves.

Finiteness of points with small height on non-CM elliptic curves.

Strengthens previous results by Hindry and Silverman.

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.