Even Galois Representations and the Fontaine--Mazur conjecture II

TL;DR

This paper proves the non-existence of certain irreducible even Galois representations with specific properties, removing previous hypotheses and constructing examples of residual representations lacking geometric deformations.

Contribution

It removes the 'ordinary' hypothesis from prior results and constructs examples of residual representations with no geometric deformations.

Findings

No irreducible even Galois representations are de Rham with distinct Hodge--Tate weights.

Constructed residual representations with no characteristic zero geometric deformations.

Extended understanding of the Fontaine--Mazur conjecture in the context of Galois representations.

Abstract

We prove, under mild hypotheses, that there are no irreducible two-dimensional_even_ Galois representations of $\Gal(\Qbar/\Q)$ which are de Rham with distinct Hodge--Tate weights. This removes the "ordinary" hypothesis required in previous work of the author. We construct examples of irreducible two-dimensional residual representations that have no characteristic zero geometric (= de Rham) deformations.