Some studies on eichler-shimura theory

Abstract

In this thesis, I studied the correspondence between the L-function L(s, E) of an elliptic curve over Q and that of a cusp form of weight 2. Eichler and Shimura give a clue to the nature of the automorphic object to use. Their theory takes certain cusp form f in $S_2(Γ_0(N))$ and gives a geometric construction of elliptic curves over Q such that L(s,~E)=L(s,~f). In Chapter 1,2. I review the compact Riemann sufface $X_0(N)$, it``s canonical model over Q, abelian variety and Jacobian variety. In Chapter 3, I examine the abstract elliptic curves and in chapter 4, I present the Eichler-Shimura thoery and it``s proof. And chapter 5, I explain the Fermat``s Last Theorem as it``s application.