Modularity of Semi-stable Elliptic Curves

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Mahkam, Roohallah | 2018

1924 Viewed

Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 51109 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Jafari, Amir
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 modularity of a Galois representation is equivalent to modularity of all its deformations and knowing that the representation induced by an elliptic curve is a deformation of its Galois representation modulo 3 or 5 and that every irreducible Galois representation modulo 3 is modular. He proved modularity theorem by finding an irreducible Galois representaion modulo 3
Keywords:
Modularity ; Galois Representations ; Modular Curve ; Elliptic Curve ; Modular Forms ; Hecke Operators ; Galois Representations Deformation