Belyi's theorem

Belyi's theorem In mathematics, Belyi's theorem on algebraic curves states that any non-singular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only.

This is a result of G. V. Belyi from 1979. At the time it was considered surprising, and it spurred Grothendieck to develop his theory of dessins d'enfant, which describes non-singular algebraic curves over the algebraic numbers using combinatorial data.

Contents 1 Quotients of the upper half-plane 2 Belyi functions 3 Applications 4 References 5 Further reading Quotients of the upper half-plane It follows that the Riemann surface in question can be taken to be the quotient H/Γ (where H is the upper half-plane and Γ is a subgroup of finite index in the modular group) compactified by cusps. Since the modular group has non-congruence subgroups, it is not the conclusion that any such curve is a modular curve.

Belyi functions A Belyi function is a holomorphic map from a compact Riemann surface S to the complex projective line P1(C) ramified only over three points, which after a Möbius transformation may be taken to be {displaystyle {0,1,infty }} . Belyi functions may be described combinatorially by dessins d'enfants.

Belyi functions and dessins d'enfants – but not Belyi's theorem – date at least to the work of Felix Klein; he used them in his article (Klein 1879) to study an 11-fold cover of the complex projective line with monodromy group PSL(2,11).[1] Applications Belyi's theorem is an existence theorem for Belyi functions, and has subsequently been much used in the inverse Galois problem.

References ^ le Bruyn, Lieven (2008), Klein's dessins d'enfant and the buckyball. Serre, Jean-Pierre (1997). Lectures on the Mordell-Weil theorem. Aspects of Mathematics. Vol. 15. Translated from the French by Martin Brown from notes by Michel Waldschmidt (Third ed.). Friedr. Vieweg & Sohn, Braunschweig. doi:10.1007/978-3-663-10632-6. ISBN 3-528-28968-6. MR 1757192. Klein, Felix (1879). "Über die Transformation elfter Ordnung der elliptischen Functionen" [On the eleventh order transformation of elliptic functions]. Mathematische Annalen (in German). 15 (3–4): 533–555. doi:10.1007/BF02086276. Belyĭ, Gennadiĭ Vladimirovich (1980). Translated by Neal Koblitz. "Galois extensions of a maximal cyclotomic field". Math. USSR Izv. 14 (2): 247–256. doi:10.1070/IM1980v014n02ABEH001096. MR 0534593. Further reading Girondo, Ernesto; González-Diez, Gabino (2012), Introduction to compact Riemann surfaces and dessins d'enfants, London Mathematical Society Student Texts, vol. 79, Cambridge: Cambridge University Press, ISBN 978-0-521-74022-7, Zbl 1253.30001 Wushi Goldring (2012), "Unifying themes suggested by Belyi's Theorem", in Dorian Goldfeld; Jay Jorgenson; Peter Jones; Dinakar Ramakrishnan; Kenneth A. Ribet; John Tate (eds.), Number Theory, Analysis and Geometry. In Memory of Serge Lang, Springer, pp. 181–214, ISBN 978-1-4614-1259-5 hide vte Topics in algebraic curves Rational curves Five points determine a conicProjective lineRational normal curveRiemann sphereTwisted cubic Elliptic curves Analytic theory Elliptic functionElliptic integralFundamental pair of periodsModular form Arithmetic theory Counting points on elliptic curvesDivision polynomialsHasse's theorem on elliptic curvesMazur's torsion theoremModular elliptic curveModularity theoremMordell–Weil theoremNagell–Lutz theoremSupersingular elliptic curveSchoof's algorithmSchoof–Elkies–Atkin algorithm Applications Elliptic curve cryptographyElliptic curve primality Higher genus De Franchis theoremFaltings's theoremHurwitz's automorphisms theoremHurwitz surfaceHyperelliptic curve Plane curves AF+BG theoremBézout's theoremBitangentCayley–Bacharach theoremConic sectionCramer's paradoxCubic plane curveFermat curveGenus–degree formulaHilbert's sixteenth problemNagata's conjecture on curvesPlücker formulaQuartic plane curveReal plane curve Riemann surfaces Belyi's theoremBring's curveBolza surfaceCompact Riemann surfaceDessin d'enfantDifferential of the first kindKlein quarticRiemann's existence theoremRiemann–Roch theoremTeichmüller spaceTorelli theorem Constructions Dual curvePolar curveSmooth completion Structure of curves Divisors on curves Abel–Jacobi mapBrill–Noether theoryClifford's theorem on special divisorsGonality of an algebraic curveJacobian varietyRiemann–Roch theoremWeierstrass pointWeil reciprocity law Moduli ELSV formulaGromov–Witten invariantHodge bundleModuli of algebraic curvesStable curve Morphisms Hasse–Witt matrixRiemann–Hurwitz formulaPrym varietyWeber's theorem Singularities AcnodeCrunodeCuspDelta invariantTacnode Vector bundles Birkhoff–Grothendieck theoremStable vector bundleVector bundles on algebraic curves Categories: Algebraic curvesTheorems in algebraic geometry