Modular approach to diophantine equations

In this thesis, we study the modular approach to the Fermat's equation x^p+y^p=z^p, where x,y and z are coprime integers, and p is a prime, and some generalizations. After reviewing and explaining how the modular approach can be used to deal with the Fermat-type equations, following the paper of Emmanuel Halberstadt and Alain Karus, we prove that there exists a dense subset of the set of prime numbers such that the equation ax^p+by^p=cz^p has no non-trivial primitive solution. Here a,b,c are fixed pairwise co-prime odd integers and p>3 is a prime. Then we show that the equation x^4+y^2=z^p has no solutions in co-prime integers when p>211 due to Jordan Ellenberg's article. The main idea to deal with this equation is based on the modularity of Q-curves and the images of Galois representations attached such curves.