Integral points on cubic surfaces: heuristics and numerics
Browning TD, Wilsch FA. 2025. Integral points on cubic surfaces: heuristics and numerics. Selecta Mathematica New Series. 31(4), 81.
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Abstract
We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be adjusted to handle singular cubic surfaces. We compare our heuristic to Heath-Brown’s prediction for sums of three cubes, as well as to asymptotic formulae in the literature around Zagier’s work on the Markoff cubic surface, and work of Baragar and Umeda on further surfaces of Markoff-type. We also test our heuristic against numerical data for several families of cubic surfaces.
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Date Published
2025-09-01
Journal Title
Selecta Mathematica New Series
Publisher
Springer Nature
Acknowledgement
The authors owe a debt of thanks to Yonatan Harpaz for asking about circle method heuristics for log K3 surfaces. His contribution to the resulting discussion is gratefully acknowledged. Thanks are also due to Andrew Sutherland for help with numerical data for the equation x^3 + y^3 + z^3 = 1, together with Alex Gamburd, Amit Ghosh, Peter Sarnak and Matteo Verzobio for their interest in this paper. Special thanks are due to Victor Wang for helpful conversations about the circle method heuristics and to the anonymous referee for several useful comments. While working on this paper, the authors were supported by a FWF grant (DOI 10.55776/P32428), and the first author was supported by a further FWF grant (DOI 10.55776/P36278) and a grant from the School of Mathematics at the Institute for Advanced Study in Princeton.
Open access funding provided by Institute of Science and Technology (IST Austria).
Volume
31
Issue
4
Article Number
81
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Cite this
Browning TD, Wilsch FA. Integral points on cubic surfaces: heuristics and numerics. Selecta Mathematica New Series. 2025;31(4). doi:10.1007/s00029-025-01074-1
Browning, T. D., & Wilsch, F. A. (2025). Integral points on cubic surfaces: heuristics and numerics. Selecta Mathematica New Series. Springer Nature. https://doi.org/10.1007/s00029-025-01074-1
Browning, Timothy D, and Florian Alexander Wilsch. “Integral Points on Cubic Surfaces: Heuristics and Numerics.” Selecta Mathematica New Series. Springer Nature, 2025. https://doi.org/10.1007/s00029-025-01074-1.
T. D. Browning and F. A. Wilsch, “Integral points on cubic surfaces: heuristics and numerics,” Selecta Mathematica New Series, vol. 31, no. 4. Springer Nature, 2025.
Browning TD, Wilsch FA. 2025. Integral points on cubic surfaces: heuristics and numerics. Selecta Mathematica New Series. 31(4), 81.
Browning, Timothy D., and Florian Alexander Wilsch. “Integral Points on Cubic Surfaces: Heuristics and Numerics.” Selecta Mathematica New Series, vol. 31, no. 4, 81, Springer Nature, 2025, doi:10.1007/s00029-025-01074-1.
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