Geodesic
Square-tiled surfaces and curves over number fields
Fundamentalʹnaâ i prikladnaâ matematika, Tome 25 (2024) no. 2, pp. 197-203 
G. B. Shabat. Square-tiled surfaces and curves over number fields. Fundamentalʹnaâ i prikladnaâ matematika, Tome 25 (2024) no. 2, pp. 197-203. http://geodesic.mathdoc.fr/item/FPM_2024_25_2_a10/
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     title = {Square-tiled surfaces and curves over number fields},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The paper presents an analog of an old result by the author and V. Voevodsky, according to which a Riemann surface admits a conformal structure defined by an equilateral triangulation if and only if the corresponding algebraic curve can be defined over the field of the algebraic numbers; a similar result is obtained for the square-tiled surfaces.

[1] Voevodskii V. A., Shabat G. B., “Ravnostoronnie triangulyatsii rimanovykh poverkhnostei i krivye nad polyami algebraicheskikh chisel”, DAN SSSR, 304:2 (1989), 265–268 | MR | Zbl

[2] Grothendieck A., “Esquisse d'un Programme”, Geometric Galois Actions 1. Around Grothendieck's Esquisse d'un Programme, London Math. Soc. Lect. Note Ser., 242, eds. P. Lochak, L. Schneps, Cambridge Univ. Press, Cambridge, 1997, 5–48 | MR | Zbl

[3] Kontsevich M., “Intersection theory on the moduli space of curves and the matrix Airy function”, Commun. Math. Phys., 147 (1992), 1–23 | DOI | MR | Zbl

[4] Nehari Z., Conformal Mapping, Dover, New York, 1982 | MR

[5] Shabat G., “Counting Belyi pairs over finite fields”, 2016 MATRIX Annals, eds. D. R. Wood, J. de Gier, Ch. E. Praeger, T. Tao, Springer, Cham, 2018, 305–322 | DOI | MR | Zbl

[6] Shabat G., Square-tiled surfaces and curves over number fields, 2022, arXiv: 2212.07755

[7] Strebel K., Quadratic Differentials, A Ser. Modern Surveys Math., 5, Springer, Berlin, 1984 | MR | Zbl

[8] Zorich A., “Flat surfaces”, Frontiers in Number Theory, Physics, and Geometry, v. I, eds. P. Cartier, B. Julia, P. Moussa, P. Vanhove, Springer, Berlin, 2006, 439–586 | DOI | MR