"L-values of elliptic curves"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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==introduction== | ==introduction== | ||
| − | * Computing | + | * Computing <math>L(E;1)</math> is easy: it is either 0 or the period of elliptic curve <math>E</math> |
| − | * Computing | + | * Computing <math>L(E;k)</math> for <math>k\geq 2</math> is highly non-trivial. The already mentioned results of Beilinson generalised later by Denninger-Scholl show that any such L-value can be expressed as a period. |
| − | * Several examples are explicitly given for | + | * Several examples are explicitly given for <math>k=2</math>, mainly motivated by showing particular cases of Beilinson's conjectures in K-theory and Boyd's (conjectural) evaluations of Mahler measures. |
| − | * In spite of the algorithmic nature of Beilinson's method and in view of its complexity, no examples were produced so far for a single | + | * In spite of the algorithmic nature of Beilinson's method and in view of its complexity, no examples were produced so far for a single <math>L(E;3)</math>. |
* Rogers and Zudilin in 2010-11 created an elementary alternative to Beilinson-Denninger-Scholl to prove some conjectural Mahler evaluations. | * Rogers and Zudilin in 2010-11 created an elementary alternative to Beilinson-Denninger-Scholl to prove some conjectural Mahler evaluations. | ||
| 9번째 줄: | 9번째 줄: | ||
==elliptic curve of conductor 32== | ==elliptic curve of conductor 32== | ||
* {{수학노트|url=타원곡선_y²%3Dx³-x}} | * {{수학노트|url=타원곡선_y²%3Dx³-x}} | ||
| − | * elliptic curve | + | * elliptic curve <math>E_{32} : y^2=x^3-x</math> |
* 모듈라 형식 | * 모듈라 형식 | ||
:<math> | :<math> | ||
| 18번째 줄: | 18번째 줄: | ||
</math> | </math> | ||
* L-values | * L-values | ||
| − | + | :<math> | |
L(E_{32},1) | L(E_{32},1) | ||
=\frac{\beta}{4} | =\frac{\beta}{4} | ||
| − | + | </math> | |
| − | where | + | where <math>\beta=\int_1^{\infty } \frac{1}{\sqrt{x^3-x}} \, dx=2.6220575543\cdots</math> |
| − | + | :<math> | |
L(E_{32},2) | L(E_{32},2) | ||
=\frac\pi8\int_0^1\frac{x}{\sqrt{1-x^4}}\,\log\frac{1+x}{1-x}\,d x. | =\frac\pi8\int_0^1\frac{x}{\sqrt{1-x^4}}\,\log\frac{1+x}{1-x}\,d x. | ||
| − | + | </math> | |
==related items== | ==related items== | ||
| 39번째 줄: | 39번째 줄: | ||
==expositions== | ==expositions== | ||
| − | * Zudilin, Wadim. ‘Transformations of | + | * Zudilin, Wadim. ‘Transformations of <math>L</math>-Values’. arXiv:1202.5630 [math], 25 February 2012. http://arxiv.org/abs/1202.5630. |
* Zudilin, Wadim [http://carma.newcastle.edu.au/wadim/PS/NewDelhi-slides.pdf Hypergeometric evaluations of L-values of an elliptic curve] | * Zudilin, Wadim [http://carma.newcastle.edu.au/wadim/PS/NewDelhi-slides.pdf Hypergeometric evaluations of L-values of an elliptic curve] | ||
2020년 11월 16일 (월) 05:29 기준 최신판
introduction
- Computing \(L(E;1)\) is easy: it is either 0 or the period of elliptic curve \(E\)
- Computing \(L(E;k)\) for \(k\geq 2\) is highly non-trivial. The already mentioned results of Beilinson generalised later by Denninger-Scholl show that any such L-value can be expressed as a period.
- Several examples are explicitly given for \(k=2\), mainly motivated by showing particular cases of Beilinson's conjectures in K-theory and Boyd's (conjectural) evaluations of Mahler measures.
- In spite of the algorithmic nature of Beilinson's method and in view of its complexity, no examples were produced so far for a single \(L(E;3)\).
- Rogers and Zudilin in 2010-11 created an elementary alternative to Beilinson-Denninger-Scholl to prove some conjectural Mahler evaluations.
elliptic curve of conductor 32
- 틀:수학노트
- elliptic curve \(E_{32} : y^2=x^3-x\)
- 모듈라 형식
\[ \begin{aligned} f(\tau)&={\eta(4\tau)^2\eta(8\tau)^2}=q\prod_{n=1}^{\infty} (1-q^{4n})^2(1-q^{8n})^2\\ {}&=\sum_{n=1}^{\infty}c_nq^n=q - 2 q^{5 }-3q^9+6q^{13}+2q^{17}+\cdots \end{aligned} \]
- L-values
\[ L(E_{32},1) =\frac{\beta}{4} \] where \(\beta=\int_1^{\infty } \frac{1}{\sqrt{x^3-x}} \, dx=2.6220575543\cdots\)
\[ L(E_{32},2) =\frac\pi8\int_0^1\frac{x}{\sqrt{1-x^4}}\,\log\frac{1+x}{1-x}\,d x. \]
- Mahler measures and L-values of elliptic curves
- Introduction to Elliptic Curves and Modular Forms by Koblitz
computational resources
expositions
- Zudilin, Wadim. ‘Transformations of \(L\)-Values’. arXiv:1202.5630 [math], 25 February 2012. http://arxiv.org/abs/1202.5630.
- Zudilin, Wadim Hypergeometric evaluations of L-values of an elliptic curve
articles
- Martin, Kimball. “The Jacquet-Langlands Correspondence, Eisenstein Congruences, and Integral L-Values in Weight 2.” arXiv:1601.03284 [math], January 13, 2016. http://arxiv.org/abs/1601.03284.
- [Z2013] Zudilin, Wadim. 2013. “Period(d)ness of L-Values.” In Number Theory and Related Fields, edited by Jonathan M. Borwein, Igor Shparlinski, and Wadim Zudilin, 381–395. Springer Proceedings in Mathematics & Statistics 43. Springer New York. http://link.springer.com/chapter/10.1007/978-1-4614-6642-0_20.