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

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.

💡 Research Summary

The paper addresses the problem of providing an explicit, uniform lower bound for the Néron–Tate canonical height of non‑torsion points on an elliptic curve with complex multiplication (CM) when those points are defined over the maximal abelian extension of a number field F. The classical result of Baker guarantees the existence of a positive constant C (depending on E/F) such that ˆh(P) ≥ C for all P ∈ E(F^ab) \ E_tors, but Baker’s proof uses the Chebotarev density theorem and yields a constant that also depends on the discriminant of F. The present work offers an alternative proof that eliminates the discriminant dependence and makes the constant completely explicit in terms of only two invariants: the degree d =