"L-values of elliptic curves"의 두 판 사이의 차이
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* 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 $L(E;3)$. | * 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. | * Rogers and Zudilin in 2010-11 created an elementary alternative to Beilinson-Denninger-Scholl to prove some conjectural Mahler evaluations. | ||
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| + | ==elliptic curve of conductor 32== | ||
| + | * {{수학노트|url=타원곡선_y²%3Dx³-x}} | ||
| + | * 모듈라 형식 | ||
| + | :<math> | ||
| + | \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} | ||
| + | </math> | ||
2015년 1월 2일 (금) 02:49 판
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
- 모듈라 형식
\[ \begin{aligned} f(\tau)&={\eta(4\tau)^2\eta(8\tau)^2}=q\prod_{n=1}^{\infty} (1-q^{4n})^2(1-q^{8n})^2\\ {}&=\sum_{n=1}^{\infty}c_nq^n=q - 2 q^{5 }-3q^9+6q^{13}+2q^{17}+\cdots \end{aligned} \]
expositions
- Zudilin, Wadim Hypergeometric evaluations of L-values of an elliptic curve
articles
- [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.