在本論文中，我們利用卡利茲模的直和來建構最簡單的高維度希爾伯特布魯蒙
索t模。並且透過海斯所提出的明確類體論去探討其伽羅瓦表示。

In the thesis, we construct the simplest example of Hilbert-Blumenthal t-modules in the higher dimensional case by considering the direct sum of Carlitz modules. Also, we study the Galois representation associated with the Hilbert-Blumenthal t-modules in our case via the explicit class field theory developed by Hayes.

1 Introduction 1
2 Preliminaries 4
2.1 NotationsandDefinitions 4
2.2 Class Field Theory and Chebotarev Density Theorem 4
2.3 RealQuadraticExtension 7
2.3.1 NumberFields 7
2.3.2 FunctionFields 9
3 Anderson t-modules 10
3.1 Background on Anderson t-modules 10
3.2 Hilbert-Blumenthal O_K-module 13
3.3 Tate Modules and Galois Representations 14
3.4 Cyclotomic Function Fields 16
4 2-dimensional H-B-C module 19
4.1 Module Structure of the C^D-torsion 22
4.2 Galois Action on the C^D-torsion 25
4.3 Galois Representation of H-B-C Module 28
Reference 31

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David Goss, Basic structures of function field arithmetic, Springer, Berlin, Hei- delberg, 1998.

D.R. Hayes, Explicit class field theory for rational function fields, Transactions of the American Mathematical Society 189 (1974), 77–91.

Kenneth Ireland and Michael Rosen, A classical introduction to modern num- ber theory, vol. 84, Springer Science, 1990.

Serge Lang, Algebraic number theory, vol. 110, Springer Science, 1986. Mihran Papikian, Drinfeld module.

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