arXiv:0907.3427 Even Galois Representations and the Fontaine-Mazur Conjecture from arXiv Front: math.NT by Frank Calegari We prove some cases of the Fontaine-Mazur conjecture for even Galois representations. In particular, we prove, under mild hypotheses, that there are no irreducible two-dimensional ordinary even Galois representations of with distinct Hodge-Tate weights. If is an imaginary quadratic field, we also prove (again, under certain hypotheses) that does not admit irreducible two-dimensional ordinary Galois representations of non-parallel weight. Finally, we prove that any weakly compatible family of two dimensional irreducible Galois representations of is, up to twist, either modular or finite.
nLab page on Fontaine-Mazur conjecture