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

 

 

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

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

• 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.

encyclopedia

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