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Integer points on curves of genus 1

Published online by Cambridge University Press:  24 October 2008

A. Baker  and
J. Coates
A. Baker
Affiliation:
Trinity College, Cambridge
J. Coates
Affiliation:
Trinity College, Cambridge
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Extract

1. Introduction. A well-known theorem of Siegel(5) states that there exist only a finite number of integer points on any curve of genus ≥ 1. Siegel's proof, published in 1929, depended, inter alia, on his earlier work concerning rational approximations to algebraic numbers and on Weil's recently established generalization of Mordell's finite basis theorem. Both of these possess a certain non-effective character and thus it is clear that Siegel's argument cannot provide an algorithm for determining all the integer points on the curve. The purpose of the present paper is to establish such an algorithm in the case of curves of genus 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 67 , Issue 3 , May 1970 , pp. 595 - 602
Copyright
Copyright © Cambridge Philosophical Society 1970

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References

REFERENCES

(1)Baker, A.Contributions to the theory of Diophantine equations: I. On the representation of integers by binary forms; II. The Diophantine equation y 2 = x 3 + k. Philos. Trans. Roy. Soc. London Ser. A. 263 (1968), 173–208.Google Scholar
(2)Baker, A.Bounds for the solutions of the hyperelliptic equation. Proc. Cambridge Philos. Soc. 65 (1969), 439–444.Google Scholar
(3)Chevalley, C.Introduction to the theory of algebraic functions of one variable. American Math. Soc. (New York, 1951).Google Scholar
(4)Coates, J.Construction of rational functions on curves. Proc. Cambridge Philos. Soc. (to appear).Google Scholar
(5)Siegel, C. L. Über einige Anwendungen diophantischer Approximationen. Abh. Preuss. Akad. Wiss. 1929, No. 1.Google Scholar
(6)Van der Waerden, B.Modern algebra (New York, 1953).Google Scholar