Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials

Abstract

AbstractAssuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/ℚ and non-torsion point P ∈ E(ℚ), there is at most one integral multiple [n]P such that n > C. The proof is a modification of a proof of Ingram giving an unconditional, but not uniform, bound. The new ingredient is a collection of explicit formulæ for the sequence v(Ψn) of valuations of the division polynomials. For P of non-singular reduction, such sequences are already well described in most cases, but for P of singular reduction, we are led to define a new class of sequences called elliptic troublemaker sequences, which measure the failure of the Néron local height to be quadratic. As a corollary in the spirit of a conjecture of Lang and Hall, we obtain a uniform upper bound on ĥ(P)/h(E) for integer points having two large integral multiples.