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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 NĂ©ron 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 elliptic divisibility sequence, Lang's conjecture, height functions
MSC Classifications: 11G05, 11G07, 11D25, 11B37, 11B39, 11Y55, 11G50, 11H52 show english descriptions Elliptic curves over global fields [See also 14H52]
Elliptic curves over local fields [See also 14G20, 14H52]
Cubic and quartic equations
Recurrences {For applications to special functions, see 33-XX}
Fibonacci and Lucas numbers and polynomials and generalizations
Calculation of integer sequences
Heights [See also 14G40, 37P30]
unknown classification 11H52
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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