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# Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials

Published:2016-07-14
Printed: Oct 2016
• Katherine E. Stange,
Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder, CO, 80309, USA
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## 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 $E/\mathbb{Q}$ and non-torsion point $P \in E(\mathbb{Q})$, there is at most one integral multiple $[n]P$ such that $n \gt C$. 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 $v(\Psi_n)$ 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 \emph{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 $\widehat{h}(P)/h(E)$ for integer points having two large integral multiples.
 Keywords: elliptic divisibility sequence, Lang's conjecture, height functions
 MSC Classifications: 11G05 - Elliptic curves over global fields [See also 14H52] 11G07 - Elliptic curves over local fields [See also 14G20, 14H52] 11D25 - Cubic and quartic equations 11B37 - Recurrences {For applications to special functions, see 33-XX} 11B39 - Fibonacci and Lucas numbers and polynomials and generalizations 11Y55 - Calculation of integer sequences 11G50 - Heights [See also 14G40, 37P30] 11H52 - unknown classification 11H52

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