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研究生： 黃惟晟
Huang, Wei-Cheng
論文名稱： 多重zeta值洗牌關係之 t-動機詮釋
A t-motivic interpretation of shuffle relations for multizeta values
指導教授： 張介玉
Chang, Chieh-Yu
口試委員: Brownawell, W. Dale
Brownawell, W. Dale
王姿月
Wang, Tzu-Yueh
于靖
Yu, Jing
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
論文出版年： 2017
畢業學年度： 105
語文別： 英文
論文頁數： 26
中文關鍵詞： 多重 zeta 值 、洗牌關係 、t-模
外文關鍵詞： multizeta values, shuffle relations, t-modules
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    • Thakur [Tha10] 證明：予兩正整數 $r$ 和 $s$ ，兩個 Carlitz zeta 值 $\zeta_A(r)$ 和 $\zeta_A(s)$ 的乘積可以表示成 $\zeta_A(r+s)$ 和所有權重為 $r+s$ 雙重 zeta 值以係數 $\mathbb{F}_p$ 的線性組合。 Thakur 稱這種表示法為洗牌關係。予兩正整數 $r$ 和 $s$ ，我們建構一個 $\mathbb{F}_q[t]$-模 $X$。予一係數 $\mathbb{F}_q(\theta)$ 的 $n$ 元組，我們亦在模 $X$ 中建構一個點 $v$ 對應於此 $n$ 元組。為了有效的地判別給定 $n$ 元組是否滿足洗牌關係，我們將它連結到點 $v$ 的 $\mathbb{F}_q[t]$-扭性質。我們亦提供一套對於點 $v$ 的 $\mathbb{F}_q[t]$-扭性質有效的判別法。

    Thakur [Tha10] showed that, for $r,$ $s\in \mathbb{N}$, a product of two Carlitz zeta values $\zeta_A(r)$ and $\zeta_A(s)$ can be expressed as an $\mathbb{F}_p$-linear combination of $\zeta_A(r+s)$ and double zeta values of weight $r+s$. Such an expression is called shuffle relation by Thakur. Fixing $r,$ $s\in \mathbb{N}$, we construct an $\mathbb{F}_q[t]$-module $X$. To determine effectively whether an $n$-tuple of coefficients in $\mathbb{F}_q(\theta)$ satisfies a shuffle relation, we relate it to the $\mathbb{F}_q[t]$-torsion property of the point $v\in X$ constructed with respect to the given coefficients. We also provide an effective criterion for the $\mathbb{F}_q[t]$-torsion property of the point $v$.

    1 Introduction 1 2 Preliminaries and The Main Theorems 2 2.1 Some notations and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Frobenius modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 The $\Ext^1$-module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Proof of the Theorem 2.6 5 3.1 Some important properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 A key lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 The Anderson-Brownawell-Papanikolas Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 Proof of the Theorem 2.6(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.5 Proof of the Theorem 2.6(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 A necessary condition for the SR-property 14 5 An effective criterion for the $\mathbb{F}_q[t]$-torsion property of $M_\mathfrak{C}$ 14 5.1 Anderson t-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.2 Identification of $\Ext_{\mathscr{F}}^1(\mathbf{1},M')$ and the Anderson t-module E . . . . . . . . . . . . . . . . . . . . 16 5.3 The structure of $\mathbf{C}^{\otimes n}(k)_{\tor}$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.4 A criterion for the $\mathbb{F}_q[t]$-torsion property of $M_\mathfrak{C}$ in $\Ext_{\mathscr{F}}^1(\mathbf{1},M')$ . . . . 18 6 Algorithm 19 7 Another method 20 Appendix A The proof of the Theorem 5.8 23 A.1 Two crucial properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 References 26

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