伽羅瓦表現與模形式

Abstract

本論文主要在介紹伽羅瓦表現與模形式之間的關係。主要的來源是Deligne和Serre在1974年所發表的論文“Formes modulaires de poid 1”。 在第二節我們先敘述了關於伽羅瓦表現與模形式的必要知識。在第三、第四節我們簡略地說明了Deligne和Serre的證明。補充的第一部分則給了一個Wiles的應用結果，剩下的部分則證明了許多我們在第二節所提到的敘述，讓這份論文更加完整。

In this thesis, I briefly give an introduction to the relation between Galois representations and modular forms, especially the modular forms of weight one. A main source is the paper “Formes modulaires de poid 1” written by Deligne and Serre in 1974. In section 2, we first state some required knowledge of modular forms and Galois representations. In section 3 and 4, we give an outline of the original proof of Deligne and Serre and make some remarks. The first part of the appendix part contains a short introduction of Wiles’ work, which concerns the weight two case. The rest part of the appendix contains some computations and proof of several results mentioned in section 2, so that the thesis is more self-contained.