Counting rational points on smooth hypersurfaces with high degree
Verzobio M. 2025. Counting rational points on smooth hypersurfaces with high degree. International Mathematics Research Notices. 2025(16), rnaf249.
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Abstract
Let X be a smooth projective hypersurface defined over Q. We provide new bounds for rational points of bounded height on X. In particular, we show that if X is a smooth projective hypersurface in Pn with n 4 and degree d 50, then the set of rational points on X of height bounded by B have cardinality On,d,ε (Bn−2+ε ). If X is smooth and has degree d 6, we improve the dimension growth conjecture bound. We achieve an analogue result for affine hypersurfaces whose projective closure is smooth.
Publishing Year
Date Published
2025-08-01
Journal Title
International Mathematics Research Notices
Publisher
Oxford University Press
Acknowledgement
While working on this paper, the author was supported by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 101034413. The author is very grateful to Tim Browning for suggesting the problem and for many useful discussions. We thank the anonymous referees for their many helpful comments, which improved the exposition of the paper. We are also grateful to Gal Binyamini for their interest in this work and for drawing our attention to the aforementioned paper [1].
We shared an early version of this paper with Per Salberger, who mentioned that he announced a new bound for smooth threefolds in P4 during a talk in 2019 (see [7] for the abstract). This result has not been published.
Volume
2025
Issue
16
Article Number
rnaf249
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eISSN
IST-REx-ID
Cite this
Verzobio M. Counting rational points on smooth hypersurfaces with high degree. International Mathematics Research Notices. 2025;2025(16). doi:10.1093/imrn/rnaf249
Verzobio, M. (2025). Counting rational points on smooth hypersurfaces with high degree. International Mathematics Research Notices. Oxford University Press. https://doi.org/10.1093/imrn/rnaf249
Verzobio, Matteo. “Counting Rational Points on Smooth Hypersurfaces with High Degree.” International Mathematics Research Notices. Oxford University Press, 2025. https://doi.org/10.1093/imrn/rnaf249.
M. Verzobio, “Counting rational points on smooth hypersurfaces with high degree,” International Mathematics Research Notices, vol. 2025, no. 16. Oxford University Press, 2025.
Verzobio M. 2025. Counting rational points on smooth hypersurfaces with high degree. International Mathematics Research Notices. 2025(16), rnaf249.
Verzobio, Matteo. “Counting Rational Points on Smooth Hypersurfaces with High Degree.” International Mathematics Research Notices, vol. 2025, no. 16, rnaf249, Oxford University Press, 2025, doi:10.1093/imrn/rnaf249.
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