Abstract
Siegel [Si 2] proved that on an arbitrary affine curve of genus ≥ 1 there exist only a finite number of integral points. In dealing with hyperelliptic equations [Si 1] he uses a somewhat different principle. He first reduces the existence of integral points on the curve to the existence of solutions of another equation
where a,b are fixed coefficients, and α,α′ are units in a number field. He then selects an integer n ≥ 3, and coset representatives for the factor group U/ U n, where U is the group of units. Using these, the above equation becomes equivalent with the equation
again to be solved in units a, a′’ This makes a further reduction to a curve of higher genus. He then uses a theorem on diophantine approximations which ultimately became known as the Thue-Siegel-Roth theorem, to conclude the proof.
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© 1978 Springer-Verlag Berlin Heidelberg
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Lang, S. (1978). Integral Points. In: Elliptic Curves. Grundlehren der mathematischen Wissenschaften, vol 231. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-07010-9_6
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DOI: https://doi.org/10.1007/978-3-662-07010-9_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05717-5
Online ISBN: 978-3-662-07010-9
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