Integral points on elliptic curves and explicit valuations of division polynomials

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.Comment: 41 pages; minor corrections and improvements to expositio

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By observing students at Leeds Metropolitan University, Katherine Everest, Debbie Morris and colleagues were able to provide library spaces for the way they actually work, not the way we think they ought to work

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What makes a good academic library? Those of us who work in academic libraries know that we need to be able to demonstrate value for money. The Library and Information Research Group (LIRG) has recently organised two seminars on the effective academic library. These have been concerned with how we measure the performance of an academic library