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 Tamagawatype number τ(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 m(S) < C(ϵ)/τ(S)^1+ϵ. We conclude the article with the construction of a sequence of diagonal cubic surfaces showing that the inequality m(S) < C/τ(S) is false in general.

Keywords:

• 11G35, 11G50, 11G40, 14J20, 14J26
• Diagonal cubic surface
• Diophantine equation
• smallest solution
• naive height
• E. Peyre's Tamagawa-type number