论文标题
Landen的Trilogarithm功能方程和$ \ Ell $ -Adic Galois多个小聚集体
Landen's trilogarithm functional equation and $\ell$-adic Galois multiple polylogarithms
论文作者
论文摘要
Pro-$ \ ell $étale的基本群体的Galois动作在投影线的三分位点上减去了三分，从而在两个带有$ \ ell $ addic系数的变量中增加了一个非交通的正式功率序列，称为$ \ ell $ \ ell $ - ell $ - ell $ aidic-adic Galois Associator。在本文中，我们专注于Landen的Trilogarithm的功能方程式及其$ \ Ell $ -ADIC GALOIS类似物可以从$ S_3 $ -SMEMEMETRY的代数衍生而来，这是Projective Line Minus减去三分的三分。将介绍功能方程的双重证明，一个基于Zagier在分级Lie代数框架中设计的Zagier张量标准，另一个基于Associator Power Series的链条规则。在第二个证据的过程中，我们被带到研究$ \ ell $ - ad-adic galois多个多聚体，作为$ \ ell $ - $ -ADIC GALOIS Associator的常规系数。作为一个应用程序，我们在$ li_ {1，\ dots，1,2}（1-z）$和$ li_k（z）$'s $（k = 1,2，...）$之间显示了Oi-ueno功能方程的$ \ ell $ -adic galois类似物。
The Galois action on the pro-$\ell$ étale fundamental groupoid of the projective line minus three points with rational base points gives rise to a non-commutative formal power series in two variables with $\ell$-adic coefficients, called the $\ell$-adic Galois associator. In the present paper, we focus on how Landen's functional equation of trilogarithms and its $\ell$-adic Galois analog can be derived algebraically from the $S_3$-symmetry of the projective line minus three points. Twofold proofs of the functional equation will be presented, one is based on Zagier's tensor criterion devised in the framework of graded Lie algebras and the other is based on the chain rule for the associator power series. In the course of the second proof, we are led to investigate $\ell$-adic Galois multiple polylogarithms appearing as regular coefficients of the $\ell$-adic Galois associator. As an application, we show an $\ell$-adic Galois analog of Oi-Ueno's functional equation between $Li_{1,\dots,1,2}(1-z)$ and $Li_k(z)$'s $(k=1,2,...)$ .
