On a question of Davenport and diagonal cubic forms over Fq(t)
Glas J, Hochfilzer L. 2024. On a question of Davenport and diagonal cubic forms over Fq(t). Mathematische Annalen.
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Author
Glas, JakobISTA;
Hochfilzer, Leonhard
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Abstract
Given a non-singular diagonal cubic hypersurface X⊂Pn−1 over Fq(t) with char(Fq)≠3, we show that the number of rational points of height at most |P| is O(|P|3+ε) for n=6 and O(|P|2+ε) for n=4. In fact, if n=4 and char(Fq)>3 we prove that the number of rational points away from any rational line contained in X is bounded by O(|P|3/2+ε). From the result in 6 variables we deduce weak approximation for diagonal cubic hypersurfaces for n≥7 over Fq(t) when char(Fq)>3 and handle Waring's problem for cubes in 7 variables over Fq(t) when char(Fq)≠3. Our results answer a question of Davenport regarding the number of solutions of bounded height to x31+x32+x33=x34+x35+x36 with xi∈Fq[t].
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Date Published
2024-12-11
Journal Title
Mathematische Annalen
Publisher
Springer Nature
Acknowledgement
Open Access funding enabled and organized by Projekt DEAL.
The authors would like to thank Tim Browning for suggesting this project. Further they are grateful for his and Damaris Schindler’s helpful comments. We would also like to thank Efthymios Sofos for bringing Davenport’s question to our attention and Keith Matthews for providing us with scanned copies of the original correspondence. Finally we would like to thank the reviewer for helpful comments.
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Cite this
Glas J, Hochfilzer L. On a question of Davenport and diagonal cubic forms over Fq(t). Mathematische Annalen. 2024. doi:10.1007/s00208-024-03035-z
Glas, J., & Hochfilzer, L. (2024). On a question of Davenport and diagonal cubic forms over Fq(t). Mathematische Annalen. Springer Nature. https://doi.org/10.1007/s00208-024-03035-z
Glas, Jakob, and Leonhard Hochfilzer. “On a Question of Davenport and Diagonal Cubic Forms over Fq(T).” Mathematische Annalen. Springer Nature, 2024. https://doi.org/10.1007/s00208-024-03035-z.
J. Glas and L. Hochfilzer, “On a question of Davenport and diagonal cubic forms over Fq(t),” Mathematische Annalen. Springer Nature, 2024.
Glas J, Hochfilzer L. 2024. On a question of Davenport and diagonal cubic forms over Fq(t). Mathematische Annalen.
Glas, Jakob, and Leonhard Hochfilzer. “On a Question of Davenport and Diagonal Cubic Forms over Fq(T).” Mathematische Annalen, Springer Nature, 2024, doi:10.1007/s00208-024-03035-z.
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