×

Fuchsian triangle groups and Grothendieck dessins. Variations on a theme of Belyi. (English) Zbl 0811.14030

According to a theorem of Belyi, a smooth projective algebraic curve \(X\) is defined over a number field if and only if there is a non-constant element of its function field ramified only over 0, 1, and \(\infty\). The existence of such a Belyi function \(\beta\) is equivalent to that of a representation of the curve as a possibly compactified quotient of the Poincaré upper half plane by a subgroup of finite index in a Fuchsian triangle group. If in particular \(\beta\) ramifies in all points of \(\beta^{-1}(1)\) with ramification order 2, the graph \(\beta^{-1} ([0,1])\) on \(X\) is called a “Grothendieck dessin”, and it is shown that \(X\) is uniquely determined up to isomorphisms already by the topology of its Grothendieck dessins.
Examples such as the Platonic solids on the sphere and natural triangulations of the Fermat curves are discussed in detail, the interplay with combinatorial objects as the “cartographic group” of the dessin is described, and connections with Shimura varieties and parts of physics like rational triangular billards are indicated.

MSC:

14H30 Coverings of curves, fundamental group
30F10 Compact Riemann surfaces and uniformization
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] [AhlSa] Ahlfors, L.V., Sario, L. Riemann Surfaces. Princeton, New Jersey: Princeton Univ. Press, 1960
[2] [AuItz] Aurell, E., Itzykson, C.: Rational billiards and algebraic curves. JGP5, n. 2, 191–208 (1988) · Zbl 0688.70017 · doi:10.1016/0393-0440(88)90004-6
[3] [BauItz] Bauer, M., Itzykson, C.: Triangulations. Coll. en hommage à P. Cartier, 54ème Renc. Strasbourg. 1992
[4] [Be] Belyi, G.: On Galois extensions of a maximal cyclotomic field. Math. USSR Izv.14, No. 2, 247–256 (1980) · Zbl 0429.12004 · doi:10.1070/IM1980v014n02ABEH001096
[5] [Beh] Behr, H.: Über die endliche Definierbarkeit von Gruppen, J.f.d. reine u. angew. Math.211, 116–122 (1962) · Zbl 0107.26201 · doi:10.1515/crll.1962.211.116
[6] [Ber] Berger, M.: La mathématique du billard. Pour La Science, N.163, 76–85 Mai, 1991
[7] [CoWo1] Cohen, P., Wolfart, J.: Modular Embeddings for some non-arithmetic Fuchsian groups. Acta Arithmetica56, 93–110 · Zbl 0717.14014
[8] [CoWo2] Beazley Cohen, P., Wolfart, J.: Dessins de Grothendieck et variétés de Shimura, C.R. Acad. Sci. Paris, t. 315, Série I, 1025–1028 (1992) · Zbl 0787.14010
[9] [Cn] Cohn, H.: Conformal Mapping on Riemann Surfaces. London: Dover Pub., 1967 · Zbl 0175.08202
[10] [De] Deligne, P.: Le groupe fondamental de la droite projective moins trois points. In: Galois Groups over \(\mathbb{Q}\), ed. Y. Ihara et al., MSRI Publ.16, Berlin-Heidelberg-New York: Springer, 1989, pp. 79–297 · Zbl 0742.14022
[11] [Dic] Encyc. Dic. of Math., Math. Soc. Japan, Cambridge, Mass. and London, Eng.: The MIT Press, 1980
[12] [Ge] Gerstenhaber, M.: On the algebraic structure of discontinuous groups. Proc. Am. Math. Soc.4, 745–750 (1953) · Zbl 0051.40205 · doi:10.1090/S0002-9939-1953-0058602-1
[13] [Gr] Grothendieck, A.: Esquisse d’un programme. 1984, unpublished
[14] [Hob] Hobson, A.: Ergodic properties of a particle moving inside a polygon. J. Math. Phys.16, No. 11, (Nov. 1975)
[15] [LuWe] Lundell, A.T., Weingram, S.: The Topology of CW Complexes. van Nostrand Reinhold, 1969 · Zbl 0207.21704
[16] [MKS] Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory, London: Dover 1976
[17] [P] Poincaré, H.: Théorie des groupes Fuchsiens. Acta Math.1, 1–62 (1882) · JFM 14.0338.01 · doi:10.1007/BF02592124
[18] [RiBe] Richens, P.J., Berry, M.V.: Pseudo-integrable systems in classical and quantum mechanics. Physica2D, 495–512 (1981) · Zbl 1194.37150
[19] [RoSa] Rourke, C.P., Sanderson, B.J.: Introduction to Piecewise-Linear Topology. Ergeb. der Math., Band69, Berlin, Heidelberg, New York: Springer 1972 · Zbl 0254.57010
[20] [ShVo] Shabat, G.B., Voevodsky, V.A.: Drawing curves over number fields. In: The Grothendieck Festschrift, Vol. III, ed. P. Cartier et al., Progress in Math.88, Basel, Boston: Birkhäuser, 1990, pp. 199–227
[21] [Shi] Shimura, G.: Construction of class fields and zeta functions of algebraic curves. Ann. Math.85, 58–159 (1967) · Zbl 0204.07201 · doi:10.2307/1970526
[22] [VoSh] Voevodsky, V.A., Shabat, G.B.: Equilateral triangulations of Riemann surfaces, and curves over algebraic number fields. Soviet Math. Dokl.39, No. 1, 38–41 (1989) · Zbl 0697.14017
[23] [Ta] Takeuchi, K.: Arithmetic triangle groups. J. Math. Soc. Japan29, 91–106 (1977) · Zbl 0344.20035 · doi:10.2969/jmsj/02910091
[24] [Vi] Vignéras, M-F.: Arithmétique des Algèbres de Quaternions. LN in Math.800, Berlin, Heidelberg, New York: Springer 1980
[25] [We] Weil, A.: The field of definition of a variety. Am. J. Math.78, 509–524 (1956) · Zbl 0072.16001 · doi:10.2307/2372670
[26] [Wo1] Wolfart, J.: Mirror-invariant triangulations of Riemann surfaces, triangle groups and Grothendieck dessins: Variations on a thema of Belyi. Preprint of the Dept. Math. Joh. Wolf. Goethe-Univers., Frank./Main
[27] [Wo2] Wolfart, J.: Eine arithmetische Eigenschaft automorpher Formen zu gewissen nichtarithmetischen Gruppen. Math. Ann.262, 1–21, (1983) · Zbl 0486.10020 · doi:10.1007/BF01474165
[28] [Wo3] Wolfart, J.: Werte hypergeometrische Funktionen. Invent. Math.92, 187–216 (1988) · Zbl 0649.10022 · doi:10.1007/BF01393999
[29] [Wo4] Wolfart, J.: Diskrete Deformationen Fuchsscher Gruppen und ihrer automorphen Formen. J.f.d. reine u. angew. Math.348, 203–220 (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.