# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $\stackrel{^}{h}\left(P\right)$ of non-torsion points $P\in E\left(K\right)$ 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 $\stackrel{^}{h}\left(P\right)$ 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 ${|E\left(K\right)}_{tor}|$ 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 $\stackrel{^}{h}\left(P\right)$ 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 (curves) 14H52 Elliptic curves 14G25 Global ground fields 14H45 Special curves and curves of low genus 14G05 Rational points
##### References:
 [1] Apostol, T.: Introduction to analytic number theory. Berlin-Heidelberg-New York: Springer 1976 [2] Apostol, T.: Modular functions and Dirichlet series in number theory. Berlin-Heidelberg-New York: Springer 1976 [3] Blanksby, P.E., Montgomery, H.L.: Algebraic integers near the unit circle. Acta Arith.18, 355-369 (1971) [4] Cox, D., Zucker, S.: Intersection numbers of sections of elliptic surfaces. Invent. Math.53, 1-44 (1979) · Zbl 0444.14004 · doi:10.1007/BF01403189 [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 · doi:10.1070/IM1974v008n03ABEH002114 [7] Frey, G.: Links between stable elliptic curves and certain Diophantine equations. Ann. Univ. Sar., Ser. Math.1, 1-40 (1986) [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 [10] Mason, R.C.: The hyperelliptic equation over function fields. Math. Proc. Camb. Philos. Soc.93, 219-230 (1983) · Zbl 0513.10016 · doi:10.1017/S0305004100060497 [11] Ogg, A.: Elliptic curves and wild ramification. Am. J. Math.89, 1-21 (1967) · Zbl 0147.39803 · doi:10.2307/2373092 [12] Rosser, J.B., Schoenfeld, L.: Sharper bounds for the Chebyshev functions ?(x) and ?(x). Math. Comput.29, 243-269 (1975) [13] Schmidt, W.: Thue’s equation over function fields. Aust. Math. Soc. Gaz.25, 385-422 (1978) · doi:10.1017/S1446788700021406 [14] Serre, J.-P.: A course in arithmetic. Berlin-Heidelberg-New York: Springer 1973 [15] Serre, J.-P., Tate, J.: Good reduction of abelian varieties. Ann. Math.68, 492-517 (1968) · doi:10.2307/1970722 [16] Shimura, G.: Introduction to the arithmetic theory of automorphic forms. Princeton N.J.: Princeton University Press 1971 [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 · doi:10.1007/BF02420949 [18] Silverman, J.H.: Lower bound for the canonical height on elliptic curves. Duke Math. J.48, 633-648 (1981) · Zbl 0475.14033 · doi:10.1215/S0012-7094-81-04834-1 [19] Silverman, J.H.: Integer points and the rank of Thue elliptic curves. Invent. Math.66, 395-404 (1982) · Zbl 0494.14008 · doi:10.1007/BF01389220 [20] Silverman, J.H.: TheS-unit equation over function fields. Math. Proc. Camb. Philos. Soc.95, 3-4 (1984) · Zbl 0533.10013 · doi:10.1017/S0305004100061235 [21] Silverman, J.H.: The arithmetic of elliptic curves. Berlin-Heidelberg-New York: Springer 1986 [22] Silverman, J.H.: Heights and elliptic curves, In: (Cornell, G., Silverman, J., (eds.) Arithmetic geometry). Berlin-Heidelberg-New York: Springer 1986 [23] Silverman, J.H.: A quantitative version of Siegel’s theorem. J. Reine Angew. Math.378, 60-100 (1987) · Zbl 0608.14021 · doi:10.1515/crll.1987.378.60 [24] Silverman, J.H.: Computing heights on elliptic curves. Math. Comput., to appear [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 [27] Vojta, P.: Diophantine approximations and value distribution theory. (Lecture Notes in Math., Vol. 1239). Berlin-Heidelberg-New York: Springer 1987 [28] Zimmer, H.: On the difference of the Weil height and the N?ron-Tate height. Math. Z.147, 35-51 (1976) · doi:10.1007/BF01214273