  In this note we prove the non existence of certain irreducible two
dimensional representations of the the absolute Galois group. Such results are
predicted by Serre's conjecture and we use Fontaine's methods to verify these
predictions in a small number of cases. We also use a methods of Faltings to
study certain finiteness conjectures for Galois representations.
  Key words: Serre's conjecture, Fontaine-Mazur conjectures, Galois
representations.
