Abstract view
Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials


Published:20160714
Printed: Oct 2016
Katherine E. Stange,
Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder, CO, 80309, USA
Abstract
Assuming Lang's conjectured lower bound on the heights of nontorsion
points on an elliptic curve, we show that there exists an absolute
constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ and nontorsion
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 nonsingular 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.
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 33XX} 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 33XX} 11B39  Fibonacci and Lucas numbers and polynomials and generalizations 11Y55  Calculation of integer sequences 11G50  Heights [See also 14G40, 37P30] 11H52  unknown classification 11H52
