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The Weil group WF is assigne to every local field F. Langlands Correspondence is a bijetion between semisimple two-dimensional Deligne representations of Weil Group WF and irreducible representations of GL2(F) which fixes assigned L-functions and local constants. In this thesis we have classified an important class of irreducible representations of GL(2) over local fields called Principal Series. We also tried to give a taste of Langlands Correspondence to the reader. First we explain preliminaries and introduce locally profinite groups and their smooth representations. Then we classify representations of GL2(F) when F is finite. Then we study GL2(F) when F is a local field. Finally we... 

Let K, k be two fields and GK be the Galois group of separable closure of K. By a Galois representation of dimension n of GK we mean a homomorphism ρ : GK → GLn(k) of topological groups. In this thesis we want to introduce the theory of Galois representations according to the Taylor’s paper [20] which has written based on his speech in international mathematical conference (ICM 2004). In general we want to build a suitable background for the reader who wants to read the Taylor’s paper with presenting the concepts related to Galois representations such as p-adic and l-adic representations, geometric representations, modular forms, L-function of a Galois representation, Weil-Deligne... 

Using interpolation and starting with Bernoulli numbers, posed by Leopold and Kubota, the aspect of adic function was constructed as adic analogues of the Dirichlet functions.Studing Galois module theory of ideal class group and his favorite structure extensions and modules related to them,Iwasawa found a new method for constructing adic functions by using Stickelberger’s elements.These results wich established by Iwasawa are known as Iwasawa Theory and they have many application in Algebriac Number Theory. Iwasawa’s most remarkable disconvry is the facet that at least in some important cases, there is a similar deep algebraic and analytic dichotomy in arithmetic of extensions. A...