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研究生： 陳冠廷
Chen, Guan-Ting
論文名稱： 探討函數體上的Drinfeld-Siegel可逆模函數
On Drinfeld-Siegel Units over Function Fields
指導教授： 魏福村
Wei, Fu-Tsun
口試委員: 于靖
Yu, Jing
張介玉
Chang, Chieh-Yu
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： 2021
畢業學年度： 109
語文別： 英文
論文頁數： 33
中文關鍵詞： Drinfeld模曲線 、Drinfeld可逆模函數 、Drinfeld-Siegel可逆模函數 、尖點因子子群 、Stickelberger分佈
外文關鍵詞： Drinfeld modular curve, Drinfeld modular unit, Drinfeld-Siegel unit, cuspidal divisor subgroup, Stickelberger distribution
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    • 我們參考Kubert和Lang的做法，以Stickelberger分佈去分析Drinfeld-Siegel可逆模函數在Drinfeld模曲線尖點處的零根數。我們證明在給定的等級下Drinfeld-Siegel可逆模函數能夠生成出對於所有Drinfeld可逆模函數的一個有限指數子群，並且能推論出對於Drinfeld模曲線的Picard群中尖點因子子群的有限性。

    We follow Kubert and Lang's method on Stickelberger distribution to analyze the orders of vanishing of Drinfeld-Siegel units at cusps of Drinfeld modular curves. Our result shows that for a given level, the Drinfeld-Siegel units generate a finite index subgroup of all Drinfeld modular units. A direct consequence is the finiteness of the cuspidal divisor subgroup of the Picard group of a Drinfeld modular curve.

    Abstract 誌謝辭 Contents Chapter 1. Introduction----------------------------------------1 Chapter 2. The Cartan Groups-----------------------------------5 Chapter 3. Distributions---------------------------------------7 3.1. Stickelberger Distributions-------------------------------8 3.2. Bernoulli Distributions----------------------------------10 3.3. Hurwitz Zeta Function------------------------------------12 Chapter 4. Drinfeld Modular Units-----------------------------17 4.1. Drinfeld Modular Forms-----------------------------------17 4.2. Klein Forms----------------------------------------------19 4.3. Drinfeld Modular Units and Cusps-------------------------21 4.4. Drinfeld-Siegel Units------------------------------------24 4.5. Examples of Indices--------------------------------------26 4.6. Drinfeld Modular Curves and Cuspidal Divisor Subgroups---27 Appendix A. Code----------------------------------------------29 List of Symbols-----------------------------------------------31 Bibliography--------------------------------------------------33

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