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 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=