"L-values of elliptic curves"의 두 판 사이의 차이

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 \(\beta=\int_1^{\infty } \frac{1}{\sqrt{x^3-x}} \, dx=2.6220575543\cdots\)

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

related items

computational resources

https://docs.google.com/file/d/0B8XXo8Tve1cxOFgydVRmWEh3N2s/edit

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.

@@ 1번째 줄: / 1번째 줄: @@
+==introduction==
+* Computing <math>L(E;1)</math> is easy: it is either 0 or the period of elliptic curve <math>E</math>
+* 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 <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 <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.
+==elliptic curve of conductor 32==
+* {{수학노트|url=타원곡선_y²%3Dx³-x}}
+* elliptic curve <math>E_{32} : y^2=x^3-x</math>
+*  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>
+* L-values
+:<math>
+L(E_{32},1)
+=\frac{\beta}{4}
+</math>
+where <math>\beta=\int_1^{\infty } \frac{1}{\sqrt{x^3-x}} \, dx=2.6220575543\cdots</math>
+:<math>
+L(E_{32},2)
+=\frac\pi8\int_0^1\frac{x}{\sqrt{1-x^4}}\,\log\frac{1+x}{1-x}\,d x.
+</math>
 ==related items==
 * [[Mahler measures and L-values of elliptic curves]]
+* [[Introduction to Elliptic Curves and Modular Forms by Koblitz]]
+==computational resources==
+* https://docs.google.com/file/d/0B8XXo8Tve1cxOFgydVRmWEh3N2s/edit
+==expositions==
+* 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]
 ==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.
 [[분류:L-functions and L-values]]
+[[분류:migrate]]