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 constantCsuch that for any elliptic curveE/ℚ and non-torsion pointP∈E(ℚ), there is at most one integral multiple [n]Psuch 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 sequencev(Ψn) of valuations of the division polynomials. ForPof non-singular reduction, such sequences are already well described in most cases, but forPof singular reduction, we are led to define a new class of sequences calledelliptic 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.