Lang’s conjecture on a uniform lower bound for heights on elliptic curves

Establish a uniform constant c > 0 such that for every elliptic curve E over Q with minimal discriminant Δ_E and for every nontorsion rational point P ∈ E(Q), the canonical Néron–Tate height satisfies ĥ(P) ≥ c · log |Δ_E|.

Background

The authors discuss height lower bounds in the case g = 1 and place their results in the context of classical conjectures. Lang’s conjecture provides a uniform lower bound for the canonical height of nontorsion points on elliptic curves in terms of the curve’s minimal discriminant.

They note that density-1 results are known that support instances where Lang’s conjecture holds, but the conjecture itself remains a central unresolved question in Diophantine geometry.