Shuffle Relations for Function Field Multizeta Values
Shuffle Relations for Function Field Multizeta Values
复制标题
函数域 Multizeta 值的随机关系
DOI：
10.1093/imrn/rnp202
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发表时间：
2009
影响因子：
1
通讯作者：
D. Thakur
中科院分区：
文献类型：
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作者：
D. Thakur
尽管和洗牌或积分洗牌关系的朴素类比失败，但我们证明了作者在函数域环境中引入的多重值（对于一般a，在无穷远处有有理位）的“洗牌”关系的存在性。这使得多ζ值的fp -张成一个代数。我们有效地确定并证明了所有的Fp-系数恒等式（但不是Fp(t)-系数恒等式）。0. 最初由欧拉引入和研究的多ζ值最近又重新引起了人们的兴趣，因为它们出现在数学和数学物理的研究中，连接了不同的观点。如见。[T09b]的介绍和参考文献。这篇论文是[T09b]的续篇。作者定义并研究了函数域的两种类型的multizeta [T04， sec5.10]，一种是复值（推广Artin-Weil zeta函数），另一种是有限域上Laurent级数环上的值（推广Carlitz zeta值）。（关于函数域算法的一般背景，我们参考[G96， T04]）对于Fq[t]案例，在[T04]中对第一种类型进行了完全评估（参见[M06]，对高等属案例进行了更详细的研究）。对于第二种类型，注意到和和积分洗牌恒等式的失效，但在[T04， T09b]以及硕士论文工作[L09， L？何塞·亚历杭德罗·劳拉·罗德里格斯在亚利桑那大学完成的研究。此外，根据Carlitz-Tate t动机的显式迭代扩展，[AT09]给出了这些多ζ值的周期解释。与经典的收敛值与发散值（归一化）的区分相反，在我们的例子中，所有的值都是收敛的。代替求和或积分洗牌关系，我们有不同类型的洗牌关系：具有Fp-系数的洗牌型关系和具有Fp(t)-系数的洗牌型关系。（当然，在经典中，没有这样的区别，在这种情况下，有理数域是素数域）。本文证明了shuffle型关系的存在性，证明了多重值的积也可以表示为若干多重值的和，从而证明了所有多重值的fp -张成空间是一个代数。[T09b, L09， L?]]推测并证明了许多这样有趣的关系（在特殊情况下A = Fq[t]），这些关系与经典情况不同，在组合中很难描述，这里我们直接证明存在性（对于一般的A，定义如下），而不是证明这些猜想。日期：2009年10月16日作者获美国国家安全局资助项目H98230-08-1-0049。
Despite the failure of naive analogs of the sum shuffle or the integral shuffle relations, we prove the existence of ‘shuffle’ relations for the multizeta values (for a general A, with a rational place at infinity) introduced [T04] by the author in the function field context. This makes the Fp-span of the multizeta values into an algebra. We effectively determine and prove all the Fp-coefficients identities (but not the Fp(t)-coefficients identities). 0. Introduction Multizeta values introduced and studied originally by Euler have been pursued recently again with renewed interest because of their emergence in studies in mathematics and mathematical physics connecting diverse viewpoints. See eg. introduction to [T09b] and references there. This paper is sequel to [T09b]. The author defined and studied two types of multizeta [T04, Sec 5.10] for function fields, one complex valued (generalizing the Artin-Weil zeta function) and the other with values in Laurent series ring over finite fields (generalizing the Carlitz zeta values). (For general background on function field arithmetic, we refer to [G96, T04]. ) For the Fq[t] case, the first type was completely evaluated in [T04] (see [M06] for more detailed study in the higher genus case). For the second type, the failure of sum and integral shuffle identities was noted, but different combinatorially involved identities were established or conjectured in [T04, T09b] as well as in the Masters thesis work [L09, L?] of Jose Alejandro Lara Rodriguez done at the University of Arizona. Also, period interpretation for these multizeta values was given in [AT09] in terms of explicit iterated extensions of the Carlitz-Tate t-motives. In contrast to the classical division between the convergent versus the divergent (normalized) values, all the values are convergent in our case. In place of the sum or the integral shuffle relations, we have different kinds of relations: the shuffle type relations with Fp-coefficients and the relations with Fp(t)-coefficients. (Classically, of course, there is no such distinction, the rational number field being the prime field in that case). In this paper, we show the existence of shuffle type relations proving that the product of multizeta values can also be expressed as a sum of some multizeta values, so that the Fp-span of all multizeta values is an algebra. While [T09b, L09, L?] conjectured and proved many such interesting relations (in the special case A = Fq[t]), which are combinatorially quite involved to describe unlike the classical case, here we prove the existence directly (for general A, defined below) rather than proving those conjectures. Date: October 16, 2009. The author was supported by NSA grant H98230-08-1-0049.