Finite and p-adic Polylogarithms
Finite and p-adic Polylogarithms
复制标题
有限和p进多对数
作者：
Amnon Besser
有限n重对数lin（z）∈ n/p（z）定义为∑k= 1 p − 1 zk/kn。我们陈述并证明了下面的定理。设Lik：Lip → Lip是由科尔曼定义的p-adic多项式。则多项式和多项式的乘积的线性组合Fn，其系数与p无关，具有p1−nDFn（z）模p>n+1约化为lin−1（σ（z））的性质，其中D是Cathelineau算子z（1−z）d/dz，σ是p幂映射的逆。这个定理的一个稍微修改的版本是由Kontsevich证明的。这个定理被Elbaz-Vincent和Gangl用来从复多项式的函数方程推导出有限多项式的函数方程。
The finite nth polylogarithm lin(z) ∈ ℤ/p(z) is defined as ∑k=1p−1zk/kn. We state and prove the following theorem. Let Lik: ℂp → ℂp be the p-adic polylogarithms defined by Coleman. Then a certain linear combination Fn of products of polylogarithms and logarithms, with coefficients which are independent of p, has the property that p1−nDFn(z) reduces modulo p>n+1 to lin−1(σ(z)), where D is the Cathelineau operator z(1−z)d/dz and σ is the inverse of the p-power map. A slightly modified version of this theorem was conjectured by Kontsevich. This theorem is used by Elbaz-Vincent and Gangl to deduce functional equations of finite polylogarithms from those of complex polylogarithms.