L-values of elliptic curves

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

\[ \begin{aligned} f(\tau)&={\eta(4\tau)^2\eta(8\tau)^2}=q\prod_{n=1}^{\infty} (1-q^{4n})^2(1-q^{8n})^2\\ {}&=\sum_{n=1}^{\infty}c_nq^n=q - 2 q^{5 }-3q^9+6q^{13}+2q^{17}+\cdots \end{aligned} \]

• L-values

$$ L(E_{32},2) =\frac\pi8\int_0^1\frac{x}{\sqrt{1-x^4}}\,\log\frac{1+x}{1-x}\,d x. $$

related items

• Mahler measures and L-values of elliptic curves

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

• [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.