Abstract
For varieties over global fields, weak approximation in the space of adelic points can fail. For a subvariety of an abelian variety one expects this failure is always explained by a finite descent obstruction, in the sense that the rational points should lie dense in the set of unobstructed (modified) adelic points. We show that this follows from a priori weaker assumptions concerning descent obstructions to the Hasse principle, i.e., to the existence of rational points. We also prove a similar statement for the obstruction coming from the Mordell–Weil sieve.
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Acknowledgements
This research was supported by the Marsden Fund Council, managed by Royal Society Te Apārangi. The author thanks the anonymous referee for a number of helpful comments which have improved the article, in particular for identifying gaps in the proof of a function field analogue of Theorem 1.1 appearing in an earlier version of this paper.
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Creutz, B. Weak approximation versus the Hasse principle for subvarieties of abelian varieties. Math. Z. 306, 11 (2024). https://doi.org/10.1007/s00209-023-03407-2
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DOI: https://doi.org/10.1007/s00209-023-03407-2