Abstract：In this lecture, I will explain the philosophy of the theory of Galois representations and of the Fontaine-Mazur Conjecture (FM), specially when the image is potentially everywhere unramified. By the way I will explain why unramified extensions of number fields are important. Then I will give some basic facts concerning p-adic Lie groups in relation with Galois representations. To conclude, I will give an improvement of a result of Nigel Boson (in 90’s) in the unramified context of the FM Conjecture. This is a joint work with Hajir (UMASS). The lecture will be accessible to non-specialists.
