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Zbl 0657.14018
Hindry, M.; Silverman, J.H.
The canonical height and integral points on elliptic curves. (English)
[J] Invent. Math. 93, No.2, 419-450 (1988). ISSN 0020-9910; ISSN 1432-1297/e

In the present paper two important conjectures of Lang and Szpiro are related. \par 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 $D\sb{E/K}$ 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. \par 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 $\sigma\sb{E/K}$, so that they show, in particular, that Lang's conjecture on heights would follow from Szpiro's conjecture. \par 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 $\vert E(K)\sb{tor}\vert$ in terms of the degree of K and $\sigma\sb{E/K}$. In order to get the lower bound for the canonical heights, the minimal discriminant ideal $D\sb{E/K}$ 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 $D\sb{E/K}$, 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. \par The paper, which is nicely written, includes also a proof of Szpiro's conjecture in the function field case.
[P.Bayer]

MSC 2000:: *14H25 Arithmetic ground fields (curves)
14H52 Elliptic curves
14G25 Global ground fields
14H45 Special curves and curves of low genus
14G05 Rationality questions, rational points

Keywords: integral points; Lang's conjecture; canonical height; Szpiro's conjecture; discriminant; bound for the number of torsion points on elliptic curves

Cited in: Zbl 1239.11071 Zbl 1106.11041 Zbl 1116.14025 Zbl 0981.11021 Zbl 1021.11021 Zbl 0992.14008 Zbl 0993.11028 Zbl 0898.11020 Zbl 0741.14013 Zbl 0714.14023 Zbl 0714.11034

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