On a question of Davenport and diagonal cubic forms over ${\mathbb{F}}_q(t)$

Abstract

Given a non-singular diagonal cubic hypersurface $X\subset {\mathbb{P}}^{n-1}$ over ${\mathbb{F}}_q(t)$ with $\mathrm{char}({\mathbb{F}}_q)\ne 3$, we show that the number of rational points of height at most $|P|$ is $O(|P{|}^{3+\epsilon })$ for $n=6$ and $O(|P{|}^{2+\epsilon })$ for $n=4$. In fact, if $n=4$ and $\mathrm{char}({\mathbb{F}}_q)>3$ we prove that the number of rational points away from any rational line contained in $X$ is bounded by $O(|P{|}^{3/2+\epsilon })$. From the result in $6$ variables we deduce weak approximation for diagonal cubic hypersurfaces for $n\ge 7$ over ${\mathbb{F}}_q(t)$ when $\mathrm{char}({\mathbb{F}}_q)>3$ and handle Waring's problem for cubes in $7$ variables over ${\mathbb{F}}_q(t)$ when $\mathrm{char}({\mathbb{F}}_q)\ne 3$. Our results answer a question of Davenport regarding the number of solutions of bounded height to $x_1^3+x_2^3+x_3^3=x_4^3+x_5^3+x_6^3$ with $x_i\in {\mathbb{F}}_q[t]$.

39 pages. Comments are welcome