Geodesic
Minimal cubic surfaces over finite fields
Sbornik. Mathematics, Tome 208 (2017) no. 9, pp. 1399-1419 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. Bibliography: 11 titles.
Keywords: finite field, cubic surface, zeta function
Mots-clés : del Pezzo surface. 
@article{SM_2017_208_9_a6,
     author = {S. Yu. Rybakov and A. S. Trepalin},
     title = {Minimal cubic surfaces over finite fields},
     journal = {Sbornik. Mathematics},
     pages = {1399--1419},
     year = {2017},
     volume = {208},
     number = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2017_208_9_a6/}
}
TY  - JOUR
AU  - S. Yu. Rybakov
AU  - A. S. Trepalin
TI  - Minimal cubic surfaces over finite fields
JO  - Sbornik. Mathematics
PY  - 2017
SP  - 1399
EP  - 1419
VL  - 208
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/SM_2017_208_9_a6/
LA  - en
ID  - SM_2017_208_9_a6
ER  - 
%0 Journal Article
%A S. Yu. Rybakov
%A A. S. Trepalin
%T Minimal cubic surfaces over finite fields
%J Sbornik. Mathematics
%D 2017
%P 1399-1419
%V 208
%N 9
%U http://geodesic.mathdoc.fr/item/SM_2017_208_9_a6/
%G en
%F SM_2017_208_9_a6
S. Yu. Rybakov; A. S. Trepalin. Minimal cubic surfaces over finite fields. Sbornik. Mathematics, Tome 208 (2017) no. 9, pp. 1399-1419. http://geodesic.mathdoc.fr/item/SM_2017_208_9_a6/

[1] Théorie des topos et cohomologie étale des schémas, Séminaire de géométrie algébrique du Bois-Marie 1963–1964 (SGA 4). Dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de P. Deligne, B. Saint-Donat, v. 1, Lecture Notes in Math., 269, Springer-Verlag, Berlin–New York, 1972, xix+525 pp. ; v. 2, Lecture Notes in Math., 270, 1972, iv+418 pp. ; v. 3, Lecture Notes in Math., 305, 1973, vi+640 pp. | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[2] B. Banwait, F. Fité, D. Loughran, Del Pezzo surfaces over finite fields and their Frobenius traces, arXiv: 1606.00300

[3] R. W. Carter, “Conjugacy classes in the Weyl group”, Compositio Math., 25:1 (1972), 1–59 | MR | Zbl

[4] M. Deuring, “Die Typen der Multiplikatorenringe elliptischer Funktionenkörper”, Abh. Math. Sem. Hansischen Univ., 14 (1941), 197–272 | DOI | MR | Zbl

[5] I. V. Dolgachev, Classical algebraic geometry. A modern view, Cambridge Univ. Press, Cambridge, 2012, xii+639 pp. | DOI | MR | Zbl

[6] Yu. I. Manin, “On the arithmetic of rational surfaces”, Soviet Math. Dokl., 4 (1963), 1243–1247 | MR | Zbl

[7] Yu. I. Manin, M. Hazewinkel, Cubic forms: algebra, geometry, arithmetic, North-Holland Math. Library, 4, North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., New York, 1974, vii+292 pp. | MR | MR | Zbl | Zbl

[8] S. Rybakov, “Zeta functions of conic bundles and del Pezzo surfaces of degree 4 over finite fields”, Mosc. Math. J., 5:4 (2005), 919–926 | MR | Zbl

[9] H. P. F. Swinnerton-Dyer, “The zeta function of a cubic surface over a finite field”, Proc. Cambridge Philos. Soc., 63 (1967), 55–71 | DOI | MR | Zbl

[10] P. Swinnerton-Dyer, “Cubic surfaces over finite fields”, Math. Proc. Cambridge Philos. Soc., 149:3 (2010), 385–388 | DOI | MR | Zbl

[11] W. C. Waterhouse, “Abelian varieties over finite fields”, Ann. Sci. École Norm. Sup. (4), 2 (1969), 521–560 | DOI | MR | Zbl