"Belyi map"의 두 판 사이의 차이

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==introduction==
  
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*  Belyi's theorem on algebraic curves
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** 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 <math>\{0,1,\infty\}</math> only.
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* Belyi map gives rise to a projective curve
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==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>
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==Grobner techniques==
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* start with three permutations <math>(12), (23), (132)</math>. They generate <math>S_3</math>.
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* Riemann-Hurwitz formula gives the genus <math>g=1-3+(1+1+2)/2=0</math>
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==complex analytic method==
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* using modular forms
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==p-adic method==
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==related items==
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* [[Dessin d'enfant]]
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==expositions==
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* Sijsling, Jeroen, and John Voight. 2013. “On Computing Belyi Maps.” arXiv:1311.2529 [math] (November 11). http://arxiv.org/abs/1311.2529.
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* Zvonkin, [https://www.labri.fr/perso/zvonkin/Research/belyi.pdf Belyi functions: examples, properties, and applications]
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* 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.
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==articles==
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* Ayberk Zeytin, Belyi Lattes Maps, arXiv:1011.5644[math.AG], November 25 2010, http://arxiv.org/abs/1011.5644v3
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* 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.
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* 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.
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* Köck, Bernhard. “Belyi’s Theorem Revisited.” arXiv:math/0108222, August 31, 2001. http://arxiv.org/abs/math/0108222.
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==encyclopedia==
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* http://en.wikipedia.org/wiki/Dessin_d%27enfant
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[[분류:개인노트]]
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[[분류:math and physics]]
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[[분류:math]]
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[[분류:migrate]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q3024615 Q3024615]
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===Spacy 패턴 목록===
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* [{'LOWER': 'dessin'}, {'LEMMA': "d'enfant"}]

2021년 2월 17일 (수) 03:14 기준 최신판

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

related items


expositions


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.

encyclopedia

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'dessin'}, {'LEMMA': "d'enfant"}]
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