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On the size of integer solutions of elliptic equations

Published online by Cambridge University Press:  17 April 2009

Yann Bugeaud
Yann Bugeaud
Affiliation:
Université Louis PasteurMathématiques7, rue René Descartes67084 Strasbourg, CedexFrance e-mail: bugeaud@math.u-strasbg.fr
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Abstract

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We improve upon earlier effective bounds for the magnitude of integer points on an elliptic curve ε defined over a number field K. We slightly refine the dependence on the discriminant of K. In most of the previous papers, the estimates obtained are exponential in the height of ε. In this work, taking also into consideration the prime ideals dividing the discriminant of ε, we provide a totally explicit bound which is only polynomial in the height.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 57 , Issue 2 , April 1998 , pp. 199 - 206
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Baker, A., ‘The diophantine equation y 2 = ax 3 + bx 2 + cx + d’ J. London Math. Soc. 43 (1968), 1–9.CrossRefGoogle Scholar
[2]Bugeaud, Y., ‘Bounds for the solutions of superelliptic equations’, Compositio Math. 107 (1997), 187–219.CrossRefGoogle Scholar
[3]Bugeaud, Y. and Győry, K., ‘Bounds for the solutions of unit equations’, Acta Arith. 74 (1996), 67–80.Google Scholar
[4]Chabauty, C., ‘Démonstration de quelques lemmes de rehaussement’, C.R. Acad. Sci. Paris 217 (1943), 413–415.Google Scholar
[5]Hajdu, L. and Herendi, T., ‘Explicit bounds for the solutions of elliptic equations with rational coefficients’, (submitted).Google Scholar
[6]Lang, S., Elliptic curves: Diophantine analysis (Springer-Verlag, Berlin, Heidelberg, New York, 1978).CrossRefGoogle Scholar
[7]Narkiewicz, W., Elementary and analytic theory of algebraic numbers (Springer-Verlag, Berlin, Heidelberg, New York, 1990).Google Scholar
[8]Pintér, A., ‘On the magnitude of integer points on elliptic curves’, Bull. Austral. Math. Soc. 52 (1995), 195–199.CrossRefGoogle Scholar
[9]Poulakis, D., ‘Integer points on algebraic curves with exceptional units’, J. Austral. Math. Soc. (to appear).Google Scholar
[10]Schmidt, W.M., ‘Integer points on curves of genus 1’, Compositio Math. 81 (1992), 33–59.Google Scholar
[11]Siegel, C.L. (Under the pseudonym X), ‘The integer solutions of the equation y 2 = axn + bx n−1 +…+ k’, J. London Math. Soc. 1 (1926), 66–68.Google Scholar
[12]Silverman, J.H., The arithmetic of elliptic curves, Graduate Texts in Math. 106 (Springer-Verlag, Berlin, Heidelberg, New York, 1986).CrossRefGoogle Scholar
[13]Sprindžuk, V.G., Classical Diophantine equations, Lecture Notes in Math. 1559 (Springer-Verlag, Berlin, Heidelberg, New York, 1993).Google Scholar
[14]Voutier, P.M., ‘An upper bound for the size of integer solutions to Ym = f(X)’, J. Number Theory 53 (1995), 247–271.Google Scholar
[15]Waldschmidt, M., ‘Minorations de combinaisons linéaires de logarithmes de nombres algébriques’, Canad. J. Math. 45 (1993), 176–224.CrossRefGoogle Scholar
[16]Yu, Kunrui, ‘Linear forms in p-adic logarithms III’, Compositio Math. 91 (1994), 241–276.Google Scholar