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
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(i)
x:C→P1 is a Galois covering andg(C)≥1;
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(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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References
A. Baker,Linear forms in the logarithms of algebraic numbers I, Mathematica13 (1966), 204–216; II, ibid.14 (1967), 102–107; III, ibid.14 (1967); 220–224; IV, ibid.15 (1968), 204–216.
A. Baker,Contributions to the theory of Diophantine equations. I, II, Philosophical Transactions of the Royal Society of London263 (1967–68), 173–208.
A. Baker,The Diophantine equation y 2=ax 3+bx 2+cx+d, Journal of the London Mathematical Society43 (1968), 1–9.
A. Baker,Bounds for the solutions of the hyperelliptic equations, Proceedings of the Cambridge Philosophical Society65 (1969), 439–444.
A. Baker and J. Coates,Integer points on curves of genus 1, Proceedings of the Cambridge Philosophical Society67 (1970), 592–602.
A. Baker and G. Wüstholz,Logarithmic forms and Group Varieties, Journal für die Reine und Angewandte Mathematik442 (1993), 19–62.
F. Beukers,Ternary form equations, preprint No. 771, University of Utrecht, 1993.
Yu. Bilu (Belotserkovski),Effective analysis of a class of Diophantine equations (Russian), Vestsi Akademii Navuk BSSR, Ser. Fiz.-Math. Navuk3 (1988), 111–115.
Yu. Bilu (Belotserkovski),Effective analysis of a new class of Diophantine equations (Russian), Vestsi Akad. Navuk BSSR, Ser. Fiz.-Math. Navuk6 (1988), 34–39, 125.
Yu. Bilu,Effective analysis of integral points on algebraic curves, Thesis, Beer Sheva, 1993.
J. Coates,Construction of rational functions on a curve, Proceedings of the Cambridge Philosophical Society68 (1970), 105–123.
W. L. Chow,The Jacobian variety of an algebraic curve, American Journal of Mathematics76 (1954), 453–476.
N. G. Chebotarev,The Theory of of Algebraic Functions (Russian), Moscow-Leningrad, 1948.
G. Faltings,Eindlichkeitssätze für abelche Varietäten über Zahlkörpern, Inventiones Mathematicae3 (1983), 349–366; Erratum:75 (1984), p. 381.
A. O. Gelfond,Transcendent and Algebraic Numbers, (Russian), Moscow 1952; English transl.: New York, Dover, 1960.
Y. Ihara, a letter to the author from 28.09.92.
H. Kleiman,On the Diophantine equation f(x, y)=0, Journal für die Reine und Angewandte Mathematik286/287 (1976), 124–131.
S. L. Kleiman and D. Laksov,Another proof of the existence of special divisors, Acta Mathematica132 (1974), 163–176.
D. Kubert and S. Lang,Modular Units, Springer, Berlin, 1981.
S. V. Kotov and L. A. Trellina,S-ganze Punkte auf elliptischen Kurven, Journal für die Reine und Angewandte Mathematik306 (1979), 28–41.
S. Lang,Fundamentals of Diophantine Geometry, Springer, Berlin, 1983.
S. Lang,Elliptic Curves: Diophantine Analysis, Springer, Berlin, 1978.
S. Lang,Introduction to Modular Forms, Springer, Berlin, 1977.
S. Lang,Elliptic Functions, Addison-Wesley, 1973.
D. W. Masser,Linear relations on algebraic groups, inNew Advances in Transcendence Theory, Cambridge Univ. Press, 1988, pp. 248–263.
A. P. Ogg,Modular Forms and Dirichlet Series, Benjamin, 1969.
D. Poulakis,Points entiers sur les courbes de genre 0, Colloquium Mathematicum66 (1983), 1–7.
P. Philippon and M. Waldschmidt,Lower bounds for linear forms in logarithms, inNew Advances in Transcendence Theory, Cambridge Univ. Press, 1988, pp. 280–312.
C. L. Siegel,Über einige Anwendungen Diophantischer Approximationen, Abh. Preuss Akad. Wiss. Phys.-Math. Kl., 1929, Nr. 1.
W. M. Schmidt,Construction and Estimation of Bases in Function Fields, Journal of Number Theory39 (1991), 181–224.
W. M. Schmidt,Integer points on curves of genus 1, Compositio Mathematica81 (1992), 33–59.
G. Shimura,Introduction to the Arithmetic Theory of Automorphic Function, Iwanami Shoten and Princeton Univ. Press, 1971.
T. N. Shorey and R. Tijdeman,Exponential Diophantine equations, Cambridge Univ. Press, Cambridge, 1986.
V. G. Sprindžuk,Classical Diophantine Equations in Two Unknowns (Russian), Nauka, Moscow, 1982; English transl.: Lecture Notes in Math.1559, Springer, Berlin, 1994.
P. Vojta,Diophantine Approximation and Value Distribution Theory, Lecture Notes in Math.1239, Springer, Berlin, 1987.
G. Wüstholz,A new approach to Baker’s theorem on linear forms in logarithms III, inNew Advances in Transcendence Theory, Cambridge Univ. Press, 1988, pp. 399–410.
Kunrui Yu,Linear forms in p-adic logarithms II, Compositio Mathematica74 (1990), 15–113.
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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