# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Apostol, T.: Introduction to analytic number theory. Berlin-Heidelberg-New York: Springer 1976 · Zbl 0335.10001 [2] Apostol, T.: Modular functions and Dirichlet series in number theory. Berlin-Heidelberg-New York: Springer 1976 · Zbl 0332.10017 [3] Blanksby, P.E., Montgomery, H.L.: Algebraic integers near the unit circle. Acta Arith.18, 355-369 (1971) · Zbl 0221.12003 [4] Cox, D., Zucker, S.: Intersection numbers of sections of elliptic surfaces. Invent. Math.53, 1-44 (1979) · Zbl 0444.14004 [5] Dem’janenko, V.A.: Estimate of the remainder term in Tate’s formula. Mat. Zametki.3, 271-278 (1968) [6] Dem’janenko, V.A.: On Tate height and the representation of numbers by binary forms. Math. USSR, Izv.8, 463-476 (1974) · Zbl 0309.14024 [7] Frey, G.: Links between stable elliptic curves and certain Diophantine equations. Ann. Univ. Sar., Ser. Math.1, 1-40 (1986) · Zbl 0586.10010 [8] Frey, G.: Letter to Serge Lang. Sept. 3, 1986 [9] Lang, S.: Elliptic curves: Diophantine analysis. (Grundlehren der Math. Wissenschaften, Vol. 231). Berlin-Heidelberg-New York: Springer 1978 · Zbl 0388.10001 [10] Mason, R.C.: The hyperelliptic equation over function fields. Math. Proc. Camb. Philos. Soc.93, 219-230 (1983) · Zbl 0513.10016 [11] Ogg, A.: Elliptic curves and wild ramification. Am. J. Math.89, 1-21 (1967) · Zbl 0147.39803 [12] Rosser, J.B., Schoenfeld, L.: Sharper bounds for the Chebyshev functions ?(x) and ?(x). Math. Comput.29, 243-269 (1975) · Zbl 0295.10036 [13] Schmidt, W.: Thue’s equation over function fields. Aust. Math. Soc. Gaz.25, 385-422 (1978) · Zbl 0389.10019 [14] Serre, J.-P.: A course in arithmetic. Berlin-Heidelberg-New York: Springer 1973 · Zbl 0256.12001 [15] Serre, J.-P., Tate, J.: Good reduction of abelian varieties. Ann. Math.68, 492-517 (1968) · Zbl 0172.46101 [16] Shimura, G.: Introduction to the arithmetic theory of automorphic forms. Princeton N.J.: Princeton University Press 1971 · Zbl 0221.10029 [17] Siegel, C.L.: ?ber Gitterpunkte in convexen K?rpern und ein damit zusammenh?ngendes Extremalproblem. Acta Math.65, 307-323 (1935) · Zbl 0012.39502 [18] Silverman, J.H.: Lower bound for the canonical height on elliptic curves. Duke Math. J.48, 633-648 (1981) · Zbl 0475.14033 [19] Silverman, J.H.: Integer points and the rank of Thue elliptic curves. Invent. Math.66, 395-404 (1982) · Zbl 0494.14008 [20] Silverman, J.H.: TheS-unit equation over function fields. Math. Proc. Camb. Philos. Soc.95, 3-4 (1984) · Zbl 0533.10013 [21] Silverman, J.H.: The arithmetic of elliptic curves. Berlin-Heidelberg-New York: Springer 1986 · Zbl 0585.14026 [22] Silverman, J.H.: Heights and elliptic curves, In: (Cornell, G., Silverman, J., (eds.) Arithmetic geometry). Berlin-Heidelberg-New York: Springer 1986 · Zbl 0596.00007 [23] Silverman, J.H.: A quantitative version of Siegel’s theorem. J. Reine Angew. Math.378, 60-100 (1987) · Zbl 0608.14021 [24] Silverman, J.H.: Computing heights on elliptic curves. Math. Comput., to appear · Zbl 0656.14016 [25] Szpiro, L.: S?minaire sur les pinceaux de courbes de genre au moins deux. Ast?risque86, 44-78 (1981) [26] Tate, J.: Modular functions of one variable IV. (Lecture Notes in Math., Vol. 476, Birch, B., Kuyk, W. (eds.)). Berlin-Heidelberg-New York: Springer 1975 · Zbl 0333.01015 [27] Vojta, P.: Diophantine approximations and value distribution theory. (Lecture Notes in Math., Vol. 1239). Berlin-Heidelberg-New York: Springer 1987 · Zbl 0609.14011 [28] Zimmer, H.: On the difference of the Weil height and the N?ron-Tate height. Math. Z.147, 35-51 (1976) · Zbl 0311.14003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.