Belyi map

introduction

• Belyi's theorem on algebraic curves
  • 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 \(\{0,1,\infty\}\) only.
• Belyi map gives rise to a projective curve

Belyi maps of degree 2

• Belyi map \(f:\mathbb{P}^1\to \mathbb{P}^1\) defined by \(z\mapsto z^2\)

Grobner techniques

• start with three permutations \((12), (23), (132)\). They generate \(S_3\).
• Riemann-Hurwitz formula gives the genus \(g=1-3+(1+1+2)/2=0\)

complex analytic method

• using modular forms

related items

• Dessin d'enfant

expositions

• Sijsling, Jeroen, and John Voight. 2013. “On Computing Belyi Maps.” arXiv:1311.2529 [math] (November 11). http://arxiv.org/abs/1311.2529.
• Zvonkin, Belyi functions: examples, properties, and applications
• Magot, Nicolas, and Alexander Zvonkin. 2000. “Belyi Functions for Archimedean Solids.” Discrete Mathematics 217 (1–3) (April 28): 249–271. doi:10.1016/S0012-365X(99)00266-6.

articles

• Ayberk Zeytin, Belyi Lattes Maps, arXiv:1011.5644[math.AG], November 25 2010, http://arxiv.org/abs/1011.5644v3
• Van Hoeij, Mark, and Raimundas Vidunas. “Belyi Functions for Hyperbolic Hypergeometric-to-Heun Transformations.” Physical Review Letters 111, no. 10 (September 2013). doi:10.1103/PhysRevLett.111.107802.
• Klug, Michael, Michael Musty, Sam Schiavone, and John Voight. 2013. “Numerical Calculation of Three-Point Branched Covers of the Projective Line.” arXiv:1311.2081 [math] (November 8). http://arxiv.org/abs/1311.2081.
• Köck, Bernhard. “Belyi’s Theorem Revisited.” arXiv:math/0108222, August 31, 2001. http://arxiv.org/abs/math/0108222.

encyclopedia

• http://en.wikipedia.org/wiki/Dessin_d%27enfant

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• [{'LOWER': 'dessin'}, {'LEMMA': "d'enfant"}]