Explicit Uniform Lower Bounds for the Canonical Height on Elliptic Curves over Abelian Extensions

TL;DR

This paper provides an explicit, discriminant-independent lower bound for the Néron-Tate height on CM elliptic curves over abelian extensions, improving previous results with a new proof strategy.

Contribution

It introduces a fully explicit lower bound for the canonical height on CM elliptic curves over abelian extensions, independent of the base field's discriminant.

Findings

Established an explicit lower bound for the Néron-Tate height.

Provided an alternative proof to Baker's theorem.

The bound is independent of the discriminant of the base field.

Abstract

We establish an explicit lower bound for the N\'eron-Tate height on elliptic curves with complex multiplication, for nontorsion points defined over the maximal abelian extension of a number field. Building on a strategy developed by Amoroso, David, and Zannier, we provide an alternative proof of a theorem originally due to Baker. The novelty in our approach is that it produces a lower bound that is fully explicit and independent of the discriminant of the base field.