On the Smallest Point on a Diagonal Cubic Surface

Abstract

For diagonal cubic surfaces S, we study the behavior of theheight m(S) of the smallest rational point versus the Tamagawatypenumber τ(S) introduced by E. Peyre. We determined bothquantities for a sample of 849,781 diagonal cubic surfaces. Ourmethods are explained in some detail. The results suggest aninequality of the type m(S) < C(ϵ)/τ(S)1+ϵ. We conclude thearticle with the construction of a sequence of diagonal cubicsurfaces showing that the inequality m(S) < C/τ(S) is false ingeneral.