Even Galois Representations and the Fontaine-Mazur Conjecture

TL;DR

This paper advances the understanding of the Fontaine-Mazur conjecture by proving non-existence results for certain even Galois representations and classifying compatible families as modular or finite.

Contribution

It proves new cases of the Fontaine-Mazur conjecture for even Galois representations, including non-existence results and classification of compatible families.

Findings

No irreducible two-dimensional ordinary even Galois representations with distinct Hodge-Tate weights exist under mild hypotheses.

Gal(Q/K) does not admit irreducible two-dimensional ordinary Galois representations of non-parallel weight over imaginary quadratic fields.

Weakly compatible families of two-dimensional irreducible Galois representations are, up to twist, either modular or finite.

Abstract

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 $\Gal(\Qbar/\Q)$ with distinct Hodge-Tate weights. If $K/\Q$ is an imaginary quadratic field, we also prove (again, under certain hypotheses) that $\Gal(\Qbar/K)$ 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 $\Gal(\Qbar/\Q)$ is, up to twist, either modular or finite.