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研究生： 郭坤宗
Kun-Tsung Kuo
論文名稱： 秩為一的 Drinfeld 模上的一個密度問題
On a density problem for Rank One Drinfeld Modules ( Carlitz Type )
指導教授： 于靖
Jing Yu
口試委員:
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： 2004
畢業學年度： 92
語文別： 英文
論文頁數： 32
中文關鍵詞： 阿廷猜想
外文關鍵詞： Artin's conjecture
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    • 在此篇論文中，我們將研究在Carlitz模上的Artin猜想以及秩為二的 Drinfeld 模上的一個密度問題。

    Let Fq be the finite field of q elements ( q , some power of a fixed prime number ) ; Fq[t] be the polynomial ring with one variable over Fq and Fq(t) the quotient field of Fq[t] . For any element in Fq[t] , we will study Artin's conjecture for the Carlitz module in the case q=2 , and for a Drinfeld Fq[t]-modules of rank 2 over Fq(t) . Each such Drinfeld module gives a new Fq[t]-modules structure on Fq(t) . It also induces a Fq[t]-modules structure on Fq[t]/(p) for almost all irreducible polynomial p in Fq[t] . We will study the Dirichlet density of all the monic irreducible polynomials in Fq[t] which make the induced Fq[t]-module on Fq[t]/(p) cyclic .

    1. Introduction ..................................................1 2. Explicit formula of the density in the case q>2 ..................................................2 3. Effective version of the Chebotarev density therem for function filds ..................................................8 4. A generalization of Artin's problem for function fields ..................................................14 5. Artin's conjecture for Carlitz modules in the case q=2 ..................................................23 6. On a cyclicity density problem for rank two Drinfeld modules ..................................................29 7. References ..................................................31

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    [12] W.-C. Chi and Anly Li : Kummer theory of divison points over Drinfeld modules of rank one . Joural of Pure and Applied Algebra 156 (2001) 171-185 .

    [13] C.-N. Hsu : On Drinfeld Modules of Carlitz type . Preprint (1994) .

    [14] J. P. Serre : Linear Representations of Finite Groups , Springer Verlag .

    [15] J.-Z. Li : On a cyclicity problem for Drinfeld moule. 2003 . N.T.H.U. paper of master .

    [16] D. A. Clark and M. Kuwata : Generalized Artin's conjecture for primitive roots and cycicity mod $p$ of Elliptic curves over function fields , Canad. Math. Bull. 38 (2) (1995) 167-173 .

    [17] C.-N. Hsu and J. Yu : On Artin's Conjecture for Rank One Drinfeld Modules . Journal of Number Theory 88 , 157-174 (2001).

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