## Minimal cubic surfaces over finite fields.(English. Russian original)Zbl 1426.11065

Sb. Math. 208, No. 9, 1399-1419 (2017); translation from Mat. Sb. 208, No. 9, 148-170 (2017).
Summary: Let $$X$$ be a minimal cubic surface over a finite field $$\mathbb{F}_q$$. The image $$\Gamma$$ of the Galois group $$\operatorname{Gal}(\overline{\mathbb{F}}_q / \mathbb{F}_q)$$ in the group $$\operatorname{Aut}(\operatorname{Pic}(\overline{X}))$$ is a cyclic subgroup of the Weyl group $$W(E_6)$$. There are $$25$$ conjugacy classes of cyclic subgroups in $$W(E_6)$$, and five of them correspond to minimal cubic surfaces. It is natural to ask which conjugacy classes come from minimal cubic surfaces over a given finite field. In this paper we give a partial answer to this question and present many explicit examples.

### MSC:

 11G25 Varieties over finite and local fields 14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties 14G05 Rational points

### Keywords:

finite field; cubic surface; zeta function; del Pezzo surface
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### References:

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