伽羅瓦表現與模型式

Abstract

我們想要探討的問題，是如何去尋找一個簡單的方法來討論一個特別的函數x4-2在模掉一個質數p之後解的個數。這是一個跟Hecke L-函數、伽羅瓦群、群的表現有關的應用問題。 更進一步來說，我們可利用這個多項式解空間的伽羅瓦表現來找出一個weight為12，level為256的cusp form。

The problem we want to discuss in this thesis is trying to find a simple description “How the polynomial splits modulo a prime p for a special polynomial x4-2.” This is an application of Hecke L-function, the Galois theorem and the group representation. We will try to connect them by some well-known knowledge, and use them to solve the problem in our discussion. Moreover, we will use the Galois representation of the splitting filed of the polynomial x4-2 to construct a cusp form of weight 1 with level 256.