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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
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