在這篇論文裡，我要探討的是某一種特定型式的Galois表現的像。
因為這種表現的像是2維方陣的型式，根據Serre-Deligne定理(Theorem 4.1)，可以知道它的trace和determinant。更進一步，可以得到一個同餘等式和交換群的等價關係(Proposition 5.5)。應用此關係，我們只要再做簡單的計算(Lemma 5.6)，就可以證明這些表現的像是不可交換群(Theorem 4.2 , Theorem 4.3)。

In this thesis , we study the images of certain Galois representations attached to the weight 12 modular form △ .
We show that these images are not abelian groups .

1 Notations............................................2
2 Introduction.........................................5
3 Preliminary .........................................7
4 Modular forms and Galois representations............14
5 The image of certain Galois representations..........17
A Appendix 22
B Appendix 23

H.P.F.Swinnerton-Dyer : On -adic representations and congruences for modular forms , International Summer School
on Modular Functions , Antwerp 1972 , p3 - p42 .
R.C.Gunning : Lectures on modular forms , Princton , New Jersey , Princton University Press , 1962 .
Erich Hecke : Lectures on the theory of algebraic numbers , Springer-Verlag , New York Heidelberg Berlin , 1981 .
Richard A.Mollin : Algebraic Number Theory , Chapman & Hall , CRC , 1999 .
Serge.Lang : Algebra Number Theory , Springer-Verlag , 1986 .
Jean-Pierre Serre : A Course in Arithmetic , Springer-Verlag , New York Heidelberg Berlin , 1973 .
J.W.S.Cassels : Local Fields , London Mathematical Society student texts 3 , 1986 .
Z.I.Borevich and I.R.Shafarevich : Number Theory , New York , New York , 1966 .