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* Belyi's theorem on algebraic curves<br> | * Belyi's theorem on algebraic curves<br> | ||
| − | ** 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. | + | ** 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 map gives rise to a projective curve | ||
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==related items== | ==related items== | ||
| − | + | * [[Dessin d'enfant]] | |
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* http://en.wikipedia.org/wiki/Dessin_d%27enfant | * http://en.wikipedia.org/wiki/Dessin_d%27enfant | ||
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[[분류:개인노트]] | [[분류:개인노트]] | ||
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[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:math]] | [[분류:math]] | ||
2013년 12월 3일 (화) 05:12 판
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
p-adic method
history
encyclopedia