Local Indecomposability of Hilbert Modular representations and Mumford-Tate conjecture

In this thesis, we use the Serre-Tate deformation theory for ordinary abelian varieties to study its associated p-adic Galois representations. As applications, we study two types of questions. The first is to determine the indecomposability of the Galois representations restricted to the p-decomposition group attached to a non CM nearly ordinary weight two Hilbert modular form over a totally real field. Then second is to study the Mumford-Tate conjecture for absolutely simple abelian fourfolds with trivial endomorphism algebras.