Variations of Mixed Hodge Structures of Multiple Polylogarithms
Variations of Mixed Hodge Structures of Multiple Polylogarithms
复制标题
多重多对数的混合Hodge结构的变体
DOI：
10.4153/cjm-2004-057-2
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
Jianqiang Zhao
中科院分区：
文献类型：
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作者：
Jianqiang Zhao
摘要众所周知，多重多重多项式会产生混合Hodge-Tate结构的良好幂幺变体。在本文中，我们将明确地确定这些结构有关的多重图和其他一些多重多项式的较低的重量。这种显式构造的目的是给出一些重要的应用。首先研究了混合Hodge-Tate结构的极限，并将多重多项式的混合Hodge-Tate结构的变化与一般多重多项式的混合Hodge-Tate结构的变化联系起来。然后根据德利涅和贝林森，我们描述了一种方法来定义单值真实的解析版本的多重多项式，它推广了著名的结果Zagier的经典多项式。在这个过程中，我们发现一些有趣的身份有关的单值多重多项式相同的重量$K$当$K=2$和3。在本文的最后，由Zagier猜想的动机，我们提出了一个问题，涉及的特殊值的多个Dedekind zeta函数的数域的多个多项式的单值版本。
Abstract It is well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall explicitly determine these structures related to multiple logarithms and some other multiple polylogarithms of lower weights. The purpose of this explicit construction is to give some important applications. First we study the limit of mixed Hodge-Tate structures and make a conjecture relating the variations of mixed Hodge-Tate structures of multiple logarithms to those of general multiple polylogarithms. Then following Deligne and Beilinson we describe an approach to defining the single-valued real analytic version of the multiple polylogarithms which generalizes the well-known result of Zagier on classical polylogarithms. In the process we find some interesting identities relating single-valued multiple polylogarithms of the same weight $K$ when $K=2$ and 3. At the end of this paper, motivated by Zagier's conjecture we pose a problem which relates the special values of multiple Dedekind zeta functions of a number field to the single-valued version of multiple polylogarithms.