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Integer points on curves of genus two and their Jacobians


Author: David Grant
Journal: Trans. Amer. Math. Soc. 344 (1994), 79-100
MSC: Primary 11G10; Secondary 11G30, 14G25, 14K15
DOI: https://doi.org/10.1090/S0002-9947-1994-1184116-2
MathSciNet review: 1184116
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Abstract: Let C be a curve of genus 2 defined over a number field, and $ \Theta $ the image of C embedded into its Jacobian J. We show that the heights of points of J which are integral with respect to $ {[2]_\ast}\Theta $ can be effectively bounded. As a result, if P is a point on C, and $ \bar P$ its image under the hyperelliptic involution, then the heights of points on C which are integral with respect to P and $ \bar P$ can be effectively bounded, in such a way that we can isolate the dependence on P, and show that if the height of P is bigger than some bound, then there are no points which are S-integral with respect to P and $ \bar P$.

We relate points on C which are integral with respect to P to points on J which are integral with respect to $ \Theta $, and discuss approaches toward bounding the heights of the latter.


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DOI: https://doi.org/10.1090/S0002-9947-1994-1184116-2
Article copyright: © Copyright 1994 American Mathematical Society

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