Experimental Mathematics

On the Smallest Point on a Diagonal Cubic Surface

Andreas-Stephan Elsenhans and Jörg Jahnel

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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.

Article information

Experiment. Math., Volume 19, Issue 2 (2010), 181-193.

First available in Project Euclid: 17 June 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G35: Varieties over global fields [See also 14G25] 11G50: Heights [See also 14G40, 37P30] 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]

Diagonal cubic surface Diophantine equation smallest solution naive height E. Peyre's Tamagawa-type number


Elsenhans, Andreas-Stephan; Jahnel, Jörg. On the Smallest Point on a Diagonal Cubic Surface. Experiment. Math. 19 (2010), no. 2, 181--193. https://projecteuclid.org/euclid.em/1276784789

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