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Let C be a curve of genus 2 defined over a number field, and e 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].6 can be effectively bounded. As a result, if P is a point on C , and P its image under the hyperelliptic involution, then the heights of points on C which are integral with respect to P and 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 P . We relate points on C which are integral with respect to P to points on J which are integral with respect to 6, and discuss approaches toward bounding the heights of the latter.

Integer points on curves of genus two and their Jacobians