Integral points on elliptic curves and explicit valuations of division polynomials

TL;DR

Assuming Lang's conjecture, the paper proves a uniform bound on the number of integral multiples of a point on an elliptic curve, using explicit valuations of division polynomials and introducing elliptic troublemaker sequences.

Contribution

The paper introduces explicit valuation formulas for division polynomials and defines elliptic troublemaker sequences to analyze integral points on elliptic curves.

Findings

At most one large integral multiple of a non-torsion point exists under Lang's conjecture.

Established a uniform bound on the ratio of heights for certain integer points.

Extended understanding of valuation sequences for points with singular reduction.

Abstract

Assuming 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 in 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 formulae for the sequence 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 Neron 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 h(P)/h(E) for integer points having two large integral multiples.