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Integral points on elliptic curves over function fields of positive characteristic

Published online by Cambridge University Press:  17 April 2009

Amílcar Pacheco
Amílcar Pacheco
Affiliation:
Rua Guaiaquil 83, Cachambi, 20785-050 Rio de Janeiro, RJ, Brasil e-mail: amilcar@impa.br
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Abstract

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Let K be a one variable function field of genus g defined over an algebraically closed field k of characteristic p > 0. Let E/K be a non-constant elliptic curve. Denote by MK the set of places of K and let S ⊂ MK be a non-empty finite subset.

Mason in his paper “Diophantine equations over function fields” Chapter VI, Theorem 14 and Voloch in “Explicit p-descent for elliptic curves in characteristic p” Theorem 5.3 proved that the number of S-integral points of a Weiertrass equation of E/K defined over RS is finite. However, no explicit upper bound for this number was given. In this note, under the extra hypotheses that E/K is semi-stable and p > 3, we obtain an explicit upper bound for this number for a certain class of Weierstrass equations called S-minimal.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 58 , Issue 3 , December 1998 , pp. 353 - 357
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]David, S., ‘Points de petite hauteur sur les courbes elliptiques’, J. Number Theory 64 (1995), 104–129.CrossRefGoogle Scholar
[2]Goldfeld, D. and Szpiro, L., ‘Bounds for the Tate-Shafarevich group’, Compositio Math. 86 (1995), 71–87.Google Scholar
[3]Hindry, M. and Silverman, J., ‘The canonical height and integral points on elliptic curves’, Invent. Math. 93 (1988), 419–450.CrossRefGoogle Scholar
[4]Hindry, M. and Silverman, J., ‘On Lehmer's conjecture for elliptic curves’, in Séminaire de Théorie des Nombres Paris 1988–89 (1989), Progr. Math. 91 (Birkhaüser, Boston, MA, 1990), pp. 103–116.Google Scholar
[5]Lang, S., Fundamentals of diophantine geometry (Springer-Verlag, Berlin, Heidelberg, New York, 1983).CrossRefGoogle Scholar
[6]Mason, R.C., Diophantine equations over function fields, Lecture Notes of the London Math. Soc. 96 (Cambridge University Press, Cambridge, London, 1984).CrossRefGoogle Scholar
[7]Silverman, J., ‘A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves’, J. Reine Angew. Math. 378 (1987), 60–100.Google Scholar
[8]Silverman, J., Advanced topics in the arithmetic of elliptic curves (Springer-Verlag, Berlin, Heidelberg, New York, 1994).CrossRefGoogle Scholar
[9]Szpiro, L., ‘Discriminant et conducteur d'une courbe elliptique’, Astérisque 86 (1990), 7–18.Google Scholar
[10]Voloch, J.F., ‘Explicit p-descent for elliptic curves in characteristic p’, Compositio Math. 74 (1990), 247–258.Google Scholar