Abstract
According to a theorem of Belyi, a smooth projective algebraic curve is defined over a number field if and only if there exists a non-constant element of its function field ramified only over 0, 1 and ∞. The existence of such a Belyi function is equivalent to that of a representation of the curve as a possibly compactified quotient space of the Poincaré upper half plane by a subgroup of finite index in a Fuchsian triangle group. On the other hand, Fuchsian triangle groups arise in many contexts, such as in the theory of hypergeometric functions and certain triangular billiard problems, which would appear at first sight to have no relation to the Galois problems that motivated the above discovery of Belyi. In this note we review several results related to Belyi's theorem and we develop certain aspects giving examples. For preliminary accounts, see the preprint [Wo1], the conference proceedings article [BauItz] and the “Comptes Rendus” note [CoWo2].
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References
[AhlSa] Ahlfors, L.V., Sario, L. Riemann Surfaces. Princeton, New Jersey: Princeton Univ. Press, 1960
[AuItz] Aurell, E., Itzykson, C.: Rational billiards and algebraic curves. JGP5, n. 2, 191–208 (1988)
[BauItz] Bauer, M., Itzykson, C.: Triangulations. Coll. en hommage à P. Cartier, 54ème Renc. Strasbourg. 1992
[Be] Belyi, G.: On Galois extensions of a maximal cyclotomic field. Math. USSR Izv.14, No. 2, 247–256 (1980)
[Beh] Behr, H.: Über die endliche Definierbarkeit von Gruppen, J.f.d. reine u. angew. Math.211, 116–122 (1962)
[Ber] Berger, M.: La mathématique du billard. Pour La Science, N.163, 76–85 Mai, 1991
[CoWo1] Cohen, P., Wolfart, J.: Modular Embeddings for some non-arithmetic Fuchsian groups. Acta Arithmetica56, 93–110
[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)
[Cn] Cohn, H.: Conformal Mapping on Riemann Surfaces. London: Dover Pub., 1967
[De] Deligne, P.: Le groupe fondamental de la droite projective moins trois points. In: Galois Groups over ℚ, ed. Y. Ihara et al., MSRI Publ.16, Berlin-Heidelberg-New York: Springer, 1989, pp. 79–297
[Dic] Encyc. Dic. of Math., Math. Soc. Japan, Cambridge, Mass. and London, Eng.: The MIT Press, 1980
[Ge] Gerstenhaber, M.: On the algebraic structure of discontinuous groups. Proc. Am. Math. Soc.4, 745–750 (1953)
[Gr] Grothendieck, A.: Esquisse d'un programme. 1984, unpublished
[Hob] Hobson, A.: Ergodic properties of a particle moving inside a polygon. J. Math. Phys.16, No. 11, (Nov. 1975)
[LuWe] Lundell, A.T., Weingram, S.: The Topology of CW Complexes. van Nostrand Reinhold, 1969
[MKS] Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory, London: Dover 1976
[P] Poincaré, H.: Théorie des groupes Fuchsiens. Acta Math.1, 1–62 (1882)
[RiBe] Richens, P.J., Berry, M.V.: Pseudo-integrable systems in classical and quantum mechanics. Physica2D, 495–512 (1981)
[RoSa] Rourke, C.P., Sanderson, B.J.: Introduction to Piecewise-Linear Topology. Ergeb. der Math., Band69, Berlin, Heidelberg, New York: Springer 1972
[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
[Shi] Shimura, G.: Construction of class fields and zeta functions of algebraic curves. Ann. Math.85, 58–159 (1967)
[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)
[Ta] Takeuchi, K.: Arithmetic triangle groups. J. Math. Soc. Japan29, 91–106 (1977)
[Vi] Vignéras, M-F.: Arithmétique des Algèbres de Quaternions. LN in Math.800, Berlin, Heidelberg, New York: Springer 1980
[We] Weil, A.: The field of definition of a variety. Am. J. Math.78, 509–524 (1956)
[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
[Wo2] Wolfart, J.: Eine arithmetische Eigenschaft automorpher Formen zu gewissen nichtarithmetischen Gruppen. Math. Ann.262, 1–21, (1983)
[Wo3] Wolfart, J.: Werte hypergeometrische Funktionen. Invent. Math.92, 187–216 (1988)
[Wo4] Wolfart, J.: Diskrete Deformationen Fuchsscher Gruppen und ihrer automorphen Formen. J.f.d. reine u. angew. Math.348, 203–220 (1986)
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Cohen, P.B., Itzykson, C. & Wolfart, J. Fuchsian triangle groups and Grothendieck dessins. Variations on a theme of Belyi. Commun.Math. Phys. 163, 605–627 (1994). https://doi.org/10.1007/BF02101464
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DOI: https://doi.org/10.1007/BF02101464