Descent for $l$-adic polylogarithms

Abstract

Let $L$ be a finite Galois extension of a number field $K$. Let $G := Gal(L/K)$. Let $z_{1}, \dots, z_{N} \in L^{*} \setminus \{1\}$ and let $m_{1}, \dots, m_{N} \in \mathbb{Q}_{l}$. Let us assume that the linear combination of $l$-adic polylogarithms $c_{n} := \sum_{i=1}^{N} m_{i}l_{n}(z_{i})_{\gamma_{i}}$ (constructed in some given way) is a cocycle on $G_{L}$ and that the formal sum $\sum_{i=1}^{N} m_{i}[z_{i}]$ is $G$-invariant. Then we show that $c_{n}$ determines a unique cocycle $s_{n}$ on $G_{K}$. We also prove a weak version of Zagier conjecture for $l$-adic dilogarithm. Finally we show that if $c_{2}$ is "motivic" ($m_{1}, \dots, m_{N} \in \mathbb{Q}$) then $s_{2}$ is also "motivic".