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On Zagier-Hoffman’s conjectures in positive characteristic | Annals of Mathematics
Annals of Mathematics ( IF 4.9 ) Pub Date : 2021-06-23 , DOI: 10.4007/annals.2021.194.1.6
Tuan Ngo Dac 1
Affiliation  

We study Todd-Thakur’s analogues of Zagier-Hoffman’s conjectures in positive characteristic. These conjectures predict the dimension and an explicit basis $\mathcal{T}_w$ of the span of characteristic $p$ multiple zeta values of fixed weight $w$ which were introduced by Thakur as analogues of classical multiple zeta values of Euler. \par In the present paper we first establish the algebraic part of these conjectures which states that the span of characteristic $p$ multiple zeta values of weight $w$ is generated by the set $\mathcal{T}_w$. As a consequence, we obtain upper bounds for the dimension. This is the analogue of Brown’s theorem and also those of Deligne-Goncharov and Terasoma.

We then prove two results towards the transcendental part of these conjectures. First, we establish the linear independence for a large subset of $\mathcal{T}_w$ and yield lower bounds for the dimension. Second, for small weights we prove the linear independence for the whole set $\mathcal{T}_w$ and completely solve Zagier-Hoffman’s conjectures in positive characteristic. Our key tool is the Anderson-Brownawell-Papanikolas criterion for linear independence in positive characteristic.



中文翻译:

关于正特征中的扎吉尔-霍夫曼猜想 数学年鉴

我们研究了扎吉尔-霍夫曼猜想的正面特征的托德-塔库尔类似物。这些猜想预测了特征 $p$ 固定权重 $w$ 的多个 zeta 值的跨度的维度和显式基础 $\mathcal{T}_w$,这些值由 Thakur 引入,作为欧拉的经典多个 zeta 值的类似物。\par 在本文中,我们首先建立了这些猜想的代数部分,它指出特征 $p$ 多个 zeta 值的权重 $w$ 的跨度是由集合 $\mathcal{T}_w$ 生成的。因此,我们获得了维度的上限。这是布朗定理的类似物,也是德利涅-冈察洛夫和 Terasoma 的类似物。

然后,我们针对这些猜想的先验部分证明了两个结果。首先,我们为 $\mathcal{T}_w$ 的大子集建立线性独立性并产生维度的下限。其次,对于小权重,我们证明了整个集合 $\mathcal{T}_w$ 的线性独立性,并完全解决了 Zagier-Hoffman 的正特征猜想。我们的关键工具是正特性线性独立性的 Anderson-Brownawell-Papanikolas 标准。

更新日期:2021-06-24
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