Integral points on elliptic curves and explicit valuations of division polynomials

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.

41 pages; minor corrections and improvements to exposition

References in corpus (3)

• Primitive divisors of elliptic divisibility sequences
• Terms in elliptic divisibility sequences divisible by their indices
• Elliptic nets and elliptic curves

Cited by in corpus (7)

• Orders of Vanishing and U(1) Charges in F-theory
• Common valuations of division polynomials
• Some effectivity results for primitive divisors of elliptic divisibility sequences
• Integral points on the congruent number curve
• The greatest common valuation of ${\varphi }_n$ and ${\psi }_n^2$ at points on elliptic curves
• Explicit valuation of elliptic nets for elliptic curves with complex multiplication
• Remarks on a theorem of Silverman