## Experimental Mathematics

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

### On the Smallest Point on a Diagonal Cubic Surface

Andreas-Stephan Elsenhans and Jörg Jahnel

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 17 June 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1276784789

**Mathematical Reviews number (MathSciNet)**

MR2676747

**Zentralblatt MATH identifier**

1233.11072

**Subjects**

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]

**Keywords**

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

#### Citation

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