"Belyi map"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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* Belyi map gives rise to a projective curve | * Belyi map gives rise to a projective curve | ||
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==Belyi maps of degree 2== | ==Belyi maps of degree 2== | ||
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* Belyi map <math>f:\mathbb{P}^1\to \mathbb{P}^1</math> defined by <math>z\mapsto z^2</math> | * Belyi map <math>f:\mathbb{P}^1\to \mathbb{P}^1</math> defined by <math>z\mapsto z^2</math> | ||
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==Grobner techniques== | ==Grobner techniques== | ||
| 22번째 줄: | 22번째 줄: | ||
* Riemann-Hurwitz formula gives the genus <math>g=1-3+(1+1+2)/2=0</math> | * Riemann-Hurwitz formula gives the genus <math>g=1-3+(1+1+2)/2=0</math> | ||
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==complex analytic method== | ==complex analytic method== | ||
| 30번째 줄: | 30번째 줄: | ||
* using modular forms | * using modular forms | ||
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==p-adic method== | ==p-adic method== | ||
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==related items== | ==related items== | ||
* [[Dessin d'enfant]] | * [[Dessin d'enfant]] | ||
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==expositions== | ==expositions== | ||
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* http://en.wikipedia.org/wiki/Dessin_d%27enfant | * http://en.wikipedia.org/wiki/Dessin_d%27enfant | ||
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2020년 12월 28일 (월) 04:02 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
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.