The goal of this thesis project is to study the Cp-semilinear representation given by the p-Tate module of an abelian variety X over a local field K, following Fontaine’s paper “Formes différentielles et modules de Tate des variétés abéliennes sur les corps locaux”. 
The main aim is to prove the Hodge-Tate decomposition for Tp(X)⊗Cp in terms of the Lie algebra of X and Xv. Thus, in the first chapter we compute the continuous cohomology groups of Cp(n) with respect to the absolute Galois group of K. 
In the second chapter, Fontaine’s work analyse the module of Kähler differentials Ω of the extension  using an integration of invariant differentials along elements of the Tate module of a Zp-module Γ.
Finally, in the third chapter we prove Tate-Raynaud theorem using the identification Vp(Ω) = Cp(1) and then we derive the Hodge-Tate decomposition.