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Hall–Lang conjecture on integral points of elliptic curves

Let E/QE/\mathbb{Q} be an elliptic curve in minimal Weierstrass form, let P∈E(Q)P \in E(\mathbb{Q}) be an integral point of infinite order, let h^(P)\widehat{h}(P) denote its canonical height, and let h(E)h(E) denote the height of EE. Hall–Lang conjecture. There is a uniform constant CHLC_{HL} such that

h^(P)<CHLh(E).\widehat{h}(P) < C_{HL} h(E).

This conjecture gives an upper bound for the height of an integral point in terms of the height of the curve, complementing Lang's lower bound for arbitrary nontorsion rational points. The source presents it as another conjecture relevant to uniform height bounds; no resolution is supplied.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Hall–Lang conjecture on integral points of elliptic curves

    Let A,B,x,yA,B,x,y be integers satisfying

    y2=x3+Ax+B.y^{2}=x^{3}+Ax+B.

    Hall–Lang conjecture. There are two constants K,M>0K,M>0 such that

    max⁡{∣x∣,∣y∣}≤Kmax⁡{∣A∣,∣B∣}M.\max\{|x|,|y|\}\le K\max\{|A|,|B|\}^{M}.

    This is the second conjectural input used in the source, alongside Lang's height conjecture, to obtain uniform bounds for indices of prime power terms in magnified elliptic divisibility sequences. The source does not state a resolution or give further qualifications on the constants.

    source: Valéry Mahé, “Prime power terms in elliptic divisibility sequences”, arXiv:1002.4202 (2010).

Sources & referencesView supporting material

Primary source

Katherine E. Stange, “Integral points on elliptic curves and explicit valuations of division polynomials”, arXiv:1108.3051 (2014).

diophantine-equationsdiophantine-geometryelliptic-curvesheight-theoryintegral-points

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