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The canonical height and integral points on elliptic curves. (English) Zbl 0657.14018
In the present paper two important conjectures of Lang and Szpiro are related.
Lang’s conjecture predicts a lower bound for the canonical height \(\hat h(P)\) of non-torsion points \(P\in E(K)\) of an elliptic curve E defined over a number field K, which depends only on K and on the minimal discriminant \(D_{E/K}\) of E. - Szpiro’s conjecture measures the extent to which the discriminant of E/K is divisible by large powers. It is true when K is a function field of characteristic zero.
The authors prove a precise version of Lang’s conjecture for function fields and a weaker version of it for number fields. - In the number field case, the lower bound for \(\hat h(P)\) obtained by the authors depends on the Szpiro ratio \(\sigma_{E/K}\), so that they show, in particular, that Lang’s conjecture on heights would follow from Szpiro’s conjecture.
Frey had showed before that Szpiro’s conjecture implies an uniform bound for the number of torsion points on elliptic curves. The authors give in the paper an explicit estimate of \(| E(K)_{tor}|\) in terms of the degree of K and \(\sigma_{E/K}\). In order to get the lower bound for the canonical heights, the minimal discriminant ideal \(D_{E/K}\) is split into two parts. The archimedean contribution to \(\hat h(P)\) is controlled by using the box principle and an explicit formula for local heights in terms of theta functions, due to the second author. To compensate the negative contribution arising from the “large part” of \(D_{E/K}\), some weighted sums on heights play a role. They are estimated by making use of a theorem of Blanksby and Montgomery which gives a lower bound for certain weighted average sums of Bernoulli polynomials.
The paper, which is nicely written, includes also a proof of Szpiro’s conjecture in the function field case.
Reviewer: P.Bayer

MSC:
14H25 Arithmetic ground fields for curves
14H52 Elliptic curves
14G25 Global ground fields in algebraic geometry
14H45 Special algebraic curves and curves of low genus
14G05 Rational points
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