Modularity of Semi-stable Elliptic Curves, M.Sc. Thesis Sharif University of Technology ; Jafari, Amir (Supervisor)
Abstract
Let E be a semi-stable elliptic curve over Q with conductor N and L(E,s) as its L-function. We say that E is modular if there exists an eigenform of weight 2 and level N over X_0(N) such that L(E,s) equals the L-function of f or L(E,s)=L(f,s). Wilse's method to prove this was by proving that the Galois representations induced by E and f are equivalent. First we will discuss that equivalency of those representations is equivalent to equality of L-functions and after that we will use deformation theory for Galois representations to overlook the main ideas of the proof of the modularity theorem. Wiles using the idea of deformation of Galois representaions proved that under proper conditions...
Cataloging briefModularity of Semi-stable Elliptic Curves, M.Sc. Thesis Sharif University of Technology ; Jafari, Amir (Supervisor)
Abstract
Let E be a semi-stable elliptic curve over Q with conductor N and L(E,s) as its L-function. We say that E is modular if there exists an eigenform of weight 2 and level N over X_0(N) such that L(E,s) equals the L-function of f or L(E,s)=L(f,s). Wilse's method to prove this was by proving that the Galois representations induced by E and f are equivalent. First we will discuss that equivalency of those representations is equivalent to equality of L-functions and after that we will use deformation theory for Galois representations to overlook the main ideas of the proof of the modularity theorem. Wiles using the idea of deformation of Galois representaions proved that under proper conditions...
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