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Effective analysis of integral points on algebraic curves

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Abstract

LetK be an algebraic number field,S⊇S \t8 a finite set of valuations andC a non-singular algebraic curve overK. Letx∈K(C) be non-constant. A pointP∈C(K) isS-integral if it is not a pole ofx and |x(P)| v >1 impliesv∈S. It is proved that allS-integral points can be effectively determined if the pair (C, x) satisfies certain conditions. In particular, this is the case if

  1. (i)

    x:C→P1 is a Galois covering andg(C)≥1;

  2. (ii)

    the integral closure of\(\bar Q\)[x] in\(\bar Q\)(C) has at least two units multiplicatively independent mod\(\bar Q\)*.

This generalizes famous results of A. Baker and other authors on the effective solution of Diophantine equations.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Ben-Gurion University, 84105, Beer Sheva, Israel

    Yuri Bilu

  2. Mathématiques Stochastiques, Université Bordeaux 2, F-33076, Bordeaux Cedex, France

    Yuri Bilu

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  1. Yuri Bilu
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Bilu, Y. Effective analysis of integral points on algebraic curves. Israel J. Math. 90, 235–252 (1995). https://doi.org/10.1007/BF02783215

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  • DOI: https://doi.org/10.1007/BF02783215

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