## 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