On the Smallest Point on a Diagonal Cubic Surface
Elsenhans, Andreas-Stephan ; Jahnel, Jörg
Experiment. Math., Tome 19 (2010) no. 1, p. 181-193 / Harvested from Project Euclid
Text on Project Euclid
PDF
Alt PDF
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.
Publié le : 2010-05-15
Classification:  Diagonal cubic surface,  Diophantine equation,  smallest solution,  naive height,  E. Peyre's Tamagawa-type number,  11G35,  11G50,  11G40 
@article{1276784789,
     author = {Elsenhans, Andreas-Stephan and Jahnel, J\"org},
     title = {On the Smallest Point on a Diagonal Cubic Surface},
     journal = {Experiment. Math.},
     volume = {19},
     number = {1},
     year = {2010},
     pages = { 181-193},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1276784789}
}

Elsenhans, Andreas-Stephan; Jahnel, Jörg. On the Smallest Point on a Diagonal Cubic Surface. Experiment. Math., Tome 19 (2010) no. 1, pp.  181-193. http://gdmltest.u-ga.fr/item/1276784789/