Multizeta in function field arithmetic
Multizeta in function field arithmetic
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函数域算术中的Multizeta
DOI：
10.4171/198-1/9
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
D. Thakur
中科院分区：
文献类型：
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作者：
D. Thakur
欧拉的多重zeta值最近再次引起人们的兴趣，因为它们的出现，例如在Grothendieck-Ihara计划中通过投影线减去三点的基本群来研究绝对伽罗瓦群，以及泰特动机的迭代扩展的相关研究。函数域中定义了两种类型的multizeta [T04，Sec 5.10]，一种是复值的（推广Artin-Weil zeta函数），另一种是有限域上的Laurent级数的值（推广Carlitz zeta值）。为了你的幸福！例中，第一种类型在[T04]中进行了完整评价（更详细的高等属病例研究见[M06]）。我们在本报告中重点介绍第二个类似物。与收敛值与发散（归一化）值之间的经典划分相反，在我们的情况下，所有值都是收敛的。代替和或积分混洗关系，我们有不同类型的关系：Fp系数的混洗型关系和Fp.t/-系数的关系。（当然，经典上没有这样的区别，在这种情况下，有理数域是素域）。第一种关系已经被理解（尽管没有令人满意的结构描述），并表明多zeta值的乘积也可以表示为一些多zeta值的和，因此所有多zeta值的FP-跨度是一个代数。而[T09，Lr 09，Lr 10]在特殊情况下证明了A D Fqqt！许多这样的有趣的关系，组合非常涉及到描述不同于经典的情况下，证明[T10]直接给出存在性（对于generalA，定义如下），而不是证明这些命题。到目前为止，我们只有第二种关系的例子。对于第一段中提到的互连的类似物，我们只能通过混合动机[A86，AT 09]连接到Grothendieck-Ihara程序中的绝对Galois群（通过Ihara幂级数[ATp]的类似物）和基本群方法。我们描述了这些动机的一些方面和关系，最近的工作V. Lafforgue和L。塔尔曼
Euler’s multizeta values have been pursued recently again with renewed interest because of their emergence, for example in Grothendieck-Ihara program to study the absolute Galois group through the fundamental group of projective line minus three points and related studies of iterated extensions of Tate motives. Two types of multizeta were defined [T04, Sec 5.10] for function fields, one complex valued (generalizing Artin-Weil zeta function) and the other with values in Laurent series over finite fields (generalizing Carlitz zeta values). For the Fq Œt ! case, the first type was completely evaluated in [T04] (see [M06] for more detailed study in the higher genus case). We focus on the second analog in this report. 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). The first kind of relations have been understood (though not with a satisfying structural description) and show 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 [T09, Lr09, Lr10] conjectured and proved, in the special case A D FqŒt !, many such interesting relations, combinatorially quite involved to describe unlike the classical case, the proofs [T10] give the existence directly (for generalA, defined below) rather than proving those conjectures. We only have examples of second kind of relations so far. As for the analogs of interconnections mentioned in the first paragraph, we can connect to absolute Galois group (through analog of Ihara power series [ATp]) and fundamental group approach in the Grothendieck-Ihara program only through the mixed motives [A86, AT09]. We describe some of these motivic aspects and relation with recent work of V. Lafforgue and L. Taelman.