On the Smallest Point on a Diagonal Cubic Surface

Abstract

For diagonal cubic surfaces $S$, we study the behavior of the height $\m(S)$ of the smallest rational point versus the Tamagawa-type number $\tau(S)$ introduced by E. Peyre. We determined both quantities for a sample of $849{,}781$ diagonal cubic surfaces. Our methods are explained in some detail. The results suggest an inequality of the type $\smash{\m(S) < C(\varepsilon)/\tau(S)^{1+\varepsilon}}$. We conclude the article with the construction of a sequence of diagonal cubic surfaces showing that the inequality $\m(S) < C/\tau(S)$ is false in general.