Abstract : This thesis has two independant parts. In the first one, we are interested in solving some diophantine equations using the modular method. We especially focus on Fermat equations of type (5,5,p) and equations of the shape F(x,y)=z^p where p is prime number and F a rational cubic. The second part deals with arithmetic of elliptic curves. We are interested in calculating the defect of semi-stability of an elliptic curve defined over a finite extension of Q_2 and having additive bad reduction with potentially good reduction. We state a partial result valid for every finite extension of Q_2. In the case of quadratic extensions, we get a complete result. Besides, let E be an elliptic curve defined over a number field K. In the last chapter, we look at prime numbers p such that the Galois action of K on the group of p-torsion points of E is reducible. In case this set is finite, we state a result which allows us in practice to determine it explicitely.