"Belyi map"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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==Belyi maps of degree 2== | ==Belyi maps of degree 2== | ||
| − | * Belyi map f:\mathbb{P}^1\to \mathbb{P}^1 defined by z\mapsto z^2 | + | * Belyi map $f:\mathbb{P}^1\to \mathbb{P}^1$ defined by $z\mapsto z^2$ |
| 19번째 줄: | 19번째 줄: | ||
==Grobner techniques== | ==Grobner techniques== | ||
| − | * start with three permutations (12), (23), (132). They generate S_3. | + | * 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 | + | * Riemann-Hurwitz formula gives the genus $g=1-3+(1+1+2)/2=0$ |
2013년 12월 3일 (화) 05:13 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