Even Galois representations and the Fontaine–Mazur conjecture. II

Source abstract

We prove, under mild hypotheses, that there are no irreducible two-dimensional potentially semi-stable even p p -adic Galois representations of G a l ( Q ¯ ) \mathrm {Gal}(\overline {\mathbf {Q}}) with distinct Hodge–Tate weights. This removes the ordinary hypotheses required in the author’s previous work. We construct examples of irreducible two-dimensional residual representations that have no characteristic zero geometric deformations.