Explicit Uniform Lower Bounds for the Canonical Height on Elliptic Curves over Abelian Extensions
Jonathan Jenvrin
TL;DR
The paper proves an explicit, discriminant-free lower bound for the Néron–Tate height of non-torsion points on elliptic curves with complex multiplication over the maximal abelian extension of a number field. It adapts the Amoroso–David–Zannier strategy to the CM elliptic-curve setting, employing local height analysis and a Galois-representation lemma to produce a non-torsion linear combination of conjugates with controlled height. Key contributions include an elliptic analogue of a height-minoration result, a uniform lower bound under ramification conditions, and a technique to circumvent technical hypotheses without invoking Chebotarev, yielding a bound depending only on the degree $d=[F:\mathbb{Q}]$ and the curve’s $j$-invariant. The main result provides an explicit bound $\hat{h}(P) \ge 2^{-4(9+h(j_{\mathcal{E}}))d^{2}-32d-8}$ for all non-torsion $P \in \mathcal{E}(F^{\mathrm{ab}})$, with potential implications for effective height estimates in CM contexts.
Abstract
We establish an explicit lower bound for the Néron-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.