"Belyi map"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
||
| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
| − | * Belyi's theorem on algebraic curves | + | * 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 | + | ** 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. |
* Belyi map gives rise to a projective curve | * Belyi map gives rise to a projective curve | ||
| 11번째 줄: | 11번째 줄: | ||
==Belyi maps of degree 2== | ==Belyi maps of degree 2== | ||
| − | * Belyi map | + | * Belyi map <math>f:\mathbb{P}^1\to \mathbb{P}^1</math> defined by <math>z\mapsto z^2</math> |
| 19번째 줄: | 19번째 줄: | ||
==Grobner techniques== | ==Grobner techniques== | ||
| − | * start with three permutations | + | * start with three permutations <math>(12), (23), (132)</math>. They generate <math>S_3</math>. |
| − | * Riemann-Hurwitz formula gives the genus | + | * Riemann-Hurwitz formula gives the genus <math>g=1-3+(1+1+2)/2=0</math> |
2020년 11월 13일 (금) 08:43 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
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.
encyclopedia