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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Integer points on curves of genus two and their Jacobians
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by David Grant PDF
Trans. Amer. Math. Soc. 344 (1994), 79-100 Request permission

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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Additional Information
  • © Copyright 1994 American Mathematical Society
  • 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